| Title: | Design and Analysis for Clinical Trials |
|---|---|
| Description: | The applications and evaluation of the operating characteristics of many statistical methodologies require the use of sophisticated software or extensive simulations. 'DACT' is designed to serve a wide range of innovative statistical designs and analyses. The primary objective of the 'DACT' software is to promote the understanding and application of cutting-edge statistical solutions in clinical trials. For this reason, the software is free for non-commercial scientific research, including but not limited to academic researchers and research/teaching institutions. Computing codes are available upon request. For more details see P. Gao (2024) <doi:10.1080/10543406.2024.2341673>. Gao, P., Zhang, W. (2024) <doi:10.1080/10543406.2024.2358796>. P. Gao & Y. Li (2024) <doi:10.1080/10543406.2023.2233590>. P. Gao, Y. Li (2024) <doi:10.1080/10543406.2024.2342518>. Gao, P., L. Liu, and C. Mehta. (2013) <doi:10.1002/sim.5847>. |
| Authors: | Ping Gao [aut, cre] |
| Maintainer: | Ping Gao <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 1.0.4 |
| Built: | 2026-11-09 13:26:08 UTC |
| Source: | https://github.com/innovatiostat/rcode |
Group sequential designs / Adaptive sequential designs: Final analysis, With one sample size change
asd_ci_est_back( s, c_bry, snew, c_bry_new, theta_hat_inter, theta_hat_inter_sd, inter_vt, theta_hat_last_ad, theta_hat_last_ad_sd, last_vt_ad, alpha_2 )asd_ci_est_back( s, c_bry, snew, c_bry_new, theta_hat_inter, theta_hat_inter_sd, inter_vt, theta_hat_last_ad, theta_hat_last_ad_sd, last_vt_ad, alpha_2 )
s |
Vector of information fractions of the original design, e.g. c(0.3, 0.7, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries of the original design, one per analysis |
snew |
Vector of information fractions after the sample size is changed; the last element must be 1 |
c_bry_new |
Vector of critical boundaries after the sample size is changed, one per remaining analysis |
theta_hat_inter |
Treatment effect estimate observed at the interim analysis at which the sample size was re-estimated |
theta_hat_inter_sd |
Standard error of theta_hat_inter |
inter_vt |
Index (1-based) of the interim analysis at which the sample size was re-estimated |
theta_hat_last_ad |
Treatment effect estimate observed at the last analysis after the sample size change |
theta_hat_last_ad_sd |
Standard error of theta_hat_last_ad |
last_vt_ad |
Index (1-based) of the last analysis after the sample size change |
alpha_2 |
Type I error for the two-sided test; the confidence interval has coverage 1 - alpha_2 |
A named list with the following components:
Two-sided p-value for the null hypothesis of no treatment effect, obtained by inverting the adaptive sequential boundary under the Brownian motion approximation
Numeric vector of length 3 giving the lower confidence limit, the point estimate and the upper confidence limit of the two-sided (1 - alpha_2) confidence interval
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
asd_ci_est_back(s = c(0.3,0.7,1), c_bry = c(4.347011, 2.845787, 2.380956), snew = c(0.5,1), c_bry_new = c(3.310743, 2.781890), theta_hat_inter = 1.2, theta_hat_inter_sd = 1, inter_vt = 1, theta_hat_last_ad = 2.2, theta_hat_last_ad_sd = 1, last_vt_ad = 2, alpha_2 = 0.05)asd_ci_est_back(s = c(0.3,0.7,1), c_bry = c(4.347011, 2.845787, 2.380956), snew = c(0.5,1), c_bry_new = c(3.310743, 2.781890), theta_hat_inter = 1.2, theta_hat_inter_sd = 1, inter_vt = 1, theta_hat_last_ad = 2.2, theta_hat_last_ad_sd = 1, last_vt_ad = 2, alpha_2 = 0.05)
Single arm phase 2 oncology trial: Final analysis: two stage design, with sample size change
asd_ci_est_one_arm_2_stage(p_0, n_1, n_2, r_1_e, r_2, n_new, x_1, x_new, alpha)asd_ci_est_one_arm_2_stage(p_0, n_1, n_2, r_1_e, r_2, n_new, x_1, x_new, alpha)
p_0 |
Response rate indicating low activity with insufficient clinical benefit |
n_1 |
Number of patients at first look |
n_2 |
Number of patients at second look in the original design |
r_1_e |
Superiority-stopping threshold at the first look: the trial would be stopped for superiority if at least r_1_e responses are observed at the first look. Enter NA when the design uses no efficacy boundary at the first look; the function then bases the final boundary on r_2 alone |
r_2 |
If no more than r_2 responses are observed at the second look, then the null hypothesis will not be rejected |
n_new |
New sample size after the interim analysis, used as the final sample size |
x_1 |
Number of responses at the first look |
x_new |
Number of responses at the final visit |
alpha |
Type I error for the one-sided test |
The results with continuity correction are not necessarily different from those without continuity correction. The need for continuity correction can be determined through simulations on the type I error (see Gao and Zhang, 2004).
A named list whose components depend on whether a first-look efficacy boundary was supplied. When r_1_e is NA the list contains p_value and Confidence interval; otherwise it contains four components, reported without and with continuity correction:
P-value of the test of the null hypothesis that the response rate equals p_0 at the final analysis; returned when r_1_e is NA
Numeric vector of length 3 holding the lower limit, the point estimate and the upper limit of the two-sided (1 - 2*alpha) confidence interval for the response rate; returned when r_1_e is NA
P-value at the final analysis computed without continuity correction
The same p-value computed with the 0.5 continuity correction
Numeric vector of length 3 holding the lower limit, the point estimate and the upper limit of the two-sided (1 - 2*alpha) confidence interval for the response rate, computed without continuity correction
The same confidence interval computed with the 0.5 continuity correction
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
asd_ci_est_one_arm_2_stage( p_0=0.2,n_1=100,n_2=105,r_1_e=33,r_2=29,n_new=115, x_1=28,x_new=34,alpha=0.025)asd_ci_est_one_arm_2_stage( p_0=0.2,n_1=100,n_2=105,r_1_e=33,r_2=29,n_new=115, x_1=28,x_new=34,alpha=0.025)
Single arm phase 2 oncology trial: Final analysis: three stage design, with sample size change
asd_ci_est_one_arm_3_stage( p_0, n_2, n_3, r_2_e, r_2, r_3, n_new, x_2, x_new, alpha )asd_ci_est_one_arm_3_stage( p_0, n_2, n_3, r_2_e, r_2, r_3, n_new, x_2, x_new, alpha )
p_0 |
Response rate indicating low activity with insufficient clinical benefit |
n_2 |
Number of patients at second look |
n_3 |
Number of patients at third look |
r_2_e |
Superiority-stopping threshold at the second look: the trial would be stopped for superiority if at least r_2_e responses are observed at the second look. If r_2_e was not used in the design, enter NA; the function then bases the final boundary on r_3 alone |
r_2 |
If no more than r_2 responses are observed at the second look, then the trial would be stopped for futility; kept for consistency with the design of the trial and not used by this function, which conditions on the observed responses at the second look |
r_3 |
If no more than r_3 responses are observed at the third look, then the null hypothesis will not be rejected |
n_new |
New sample size after the interim analysis, taken at the second look and used as the final sample size |
x_2 |
Number of responses at the second look |
x_new |
Number of responses at the final visit |
alpha |
Type I error for the one-sided test |
The results with continuity correction are not necessarily different from those without continuity correction. The need for continuity correction can be determined through simulations on the type I error (see Gao and Zhang, 2004).
A named list whose components depend on whether a second-look efficacy boundary was supplied. When r_2_e is NA the list contains p_value and Confidence interval; otherwise it contains four components, reported without and with continuity correction:
P-value of the test of the null hypothesis that the response rate equals p_0 at the final analysis; returned when r_2_e is NA
Numeric vector of length 3 holding the lower limit, the point estimate and the upper limit of the two-sided (1 - 2*alpha) confidence interval for the response rate; returned when r_2_e is NA
P-value at the final analysis computed without continuity correction
The same p-value computed with the 0.5 continuity correction
Numeric vector of length 3 holding the lower limit, the point estimate and the upper limit of the two-sided (1 - 2*alpha) confidence interval for the response rate, computed without continuity correction
The same confidence interval computed with the 0.5 continuity correction
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
asd_ci_est_one_arm_3_stage( p_0=0.2,n_2=105,n_3=110,r_2_e=33,r_3=29,n_new=115, x_2=28,x_new=34,alpha=0.025)asd_ci_est_one_arm_3_stage( p_0=0.2,n_2=105,n_3=110,r_2_e=33,r_3=29,n_new=115, x_2=28,x_new=34,alpha=0.025)
Group sequential designs / Adaptive sequential designs: Interim analysis, Non-survival analysis, Use theta_interim
asd_new_design( s, c_bry, inter_vt, snew, theta_hat_inter, theta_hat_inter_sd, n_control_inter, beta )asd_new_design( s, c_bry, inter_vt, snew, theta_hat_inter, theta_hat_inter_sd, n_control_inter, beta )
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
inter_vt |
Index of the interim analysis at which the sample size is re-estimated (1-based); must be less than length(s) |
snew |
Vector of new information fractions after the sample size is changed, e.g. c(0.5, 1); the fractions are relative to the new total information and the last element must be 1 |
theta_hat_inter |
Treatment effect estimate observed at the interim analysis, e.g. the difference of the observed means or event rates between the test arm and the control arm |
theta_hat_inter_sd |
Standard error of 'theta_hat_inter' |
n_control_inter |
Sample size of the control arm at the interim analysis |
beta |
Type II error rate, i.e. 1 - power |
A named list with the following components:
Character string giving the outcome of the interim hypothesis test: whether the null hypothesis was rejected at the interim analysis
Wald statistic of the treatment effect observed at the interim analysis
Planned critical boundary at the interim analysis at which the sample size is re-estimated
The effect size used for the conditional power calculation, taken to be the treatment effect estimate observed at the interim analysis
Conditional type I error at the interim analysis, calculated by setting theta = 0
Conditional power at the interim analysis using the estimated effect size theta
New sample size for the control arm after the sample size re-estimation, or 'NA' if the null hypothesis was rejected at the interim analysis or if the estimated effect size is < 0.001
The new information fractions after the sample size change
New critical boundaries for the new interim analyses (O'Brien-Fleming type), or 'NA' if they were not calculated
asd_new_design(s = c(1/3, 2/3, 1), c_bry = c(3.984298, 3.961146, 2.349682), inter_vt = 1, snew = c(0.5, 1), theta_hat_inter = 0.3, theta_hat_inter_sd = 0.5, n_control_inter = 100, beta = 0.1)asd_new_design(s = c(1/3, 2/3, 1), c_bry = c(3.984298, 3.961146, 2.349682), inter_vt = 1, snew = c(0.5, 1), theta_hat_inter = 0.3, theta_hat_inter_sd = 0.5, n_control_inter = 100, beta = 0.1)
Group sequential designs / Adaptive sequential designs: Interim analysis, Survival analysis, Use theta_interim
asd_new_design_survival( s, c_bry, inter_vt, snew, theta_hat_inter, theta_hat_inter_sd, events_inter, beta )asd_new_design_survival( s, c_bry, inter_vt, snew, theta_hat_inter, theta_hat_inter_sd, events_inter, beta )
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
inter_vt |
Index of the interim analysis at which the sample size is re-estimated (1-based); must be less than length(s) |
snew |
Vector of new information fractions after the sample size is changed, e.g. c(0.5, 1); the fractions are relative to the new total information and the last element must be 1 |
theta_hat_inter |
Treatment effect estimate observed at the interim analysis, i.e. minus the log of the observed hazard ratio |
theta_hat_inter_sd |
Standard error of 'theta_hat_inter' |
events_inter |
Number of events observed at the interim analysis |
beta |
Type II error rate, i.e. 1 - power |
A named list with the following components:
Character string giving the outcome of the interim hypothesis test: whether the null hypothesis was rejected at the interim analysis
Wald statistic of the treatment effect observed at the interim analysis
Planned critical boundary at the interim analysis at which the sample size is re-estimated
The effect size used for the conditional power calculation, taken to be the treatment effect estimate observed at the interim analysis
Conditional type I error at the interim analysis, calculated by setting theta = 0
Conditional power at the interim analysis using the estimated effect size theta
New total number of events required after the sample size re-estimation, or 'NA' if the null hypothesis was rejected at the interim analysis or if the estimated effect size is < 0.001
The new information fractions after the sample size change
New critical boundaries for the new interim analyses (O'Brien-Fleming type), or 'NA' if they were not calculated
asd_new_design_survival(s = c(1/3, 2/3, 1), c_bry = c(3.984298, 3.961146, 2.349682), inter_vt = 1, snew = c(0.5, 1), theta_hat_inter = 0.3, theta_hat_inter_sd = 0.5, events_inter = 100, beta = 0.1)asd_new_design_survival(s = c(1/3, 2/3, 1), c_bry = c(3.984298, 3.961146, 2.349682), inter_vt = 1, snew = c(0.5, 1), theta_hat_inter = 0.3, theta_hat_inter_sd = 0.5, events_inter = 100, beta = 0.1)
Group sequential designs / Adaptive sequential designs: Interim analysis, Non-survival analysis, Input theta
asd_new_design_theta( theta, s, c_bry, inter_vt, snew, theta_hat_inter, theta_hat_inter_sd, n_control_inter, beta )asd_new_design_theta( theta, s, c_bry, inter_vt, snew, theta_hat_inter, theta_hat_inter_sd, n_control_inter, beta )
theta |
Assumed effect size used for the conditional power calculation; different values of theta can be entered to check the sensitivity of the conditional power to the effect size; if theta < 0.001 the new sample size and the new critical boundary are returned as 'NA' |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
inter_vt |
Index of the interim analysis at which the sample size is re-estimated (1-based); must be less than length(s) |
snew |
Vector of new information fractions after the sample size is changed, e.g. c(0.5, 1); the fractions are relative to the new total information and the last element must be 1 |
theta_hat_inter |
Treatment effect estimate observed at the interim analysis, e.g. the difference of the observed means or event rates between the test arm and the control arm |
theta_hat_inter_sd |
Standard error of 'theta_hat_inter' |
n_control_inter |
Sample size of the control arm at the interim analysis |
beta |
Type II error rate, i.e. 1 - power |
A named list with the following components:
Character string giving the outcome of the interim hypothesis test: whether the null hypothesis was rejected at the interim analysis
Wald statistic of the treatment effect observed at the interim analysis
Planned critical boundary at the interim analysis at which the sample size is re-estimated
The assumed effect size theta used for the conditional power calculation
Conditional type I error at the interim analysis, calculated by setting theta = 0
Conditional power at the interim analysis using the assumed effect size theta
New sample size for the control arm after the sample size re-estimation, or 'NA' if the null hypothesis was rejected at the interim analysis or if theta < 0.001
The new information fractions after the sample size change
New critical boundaries for the new interim analyses (O'Brien-Fleming type), or 'NA' if they were not calculated
asd_new_design_theta(theta = 0.2, s = c(1/3, 2/3, 1), c_bry = c(3.984298, 3.961146, 2.349682), inter_vt = 1, snew = c(0.5, 1), theta_hat_inter = 0.3, theta_hat_inter_sd = 0.5, n_control_inter = 100, beta = 0.1)asd_new_design_theta(theta = 0.2, s = c(1/3, 2/3, 1), c_bry = c(3.984298, 3.961146, 2.349682), inter_vt = 1, snew = c(0.5, 1), theta_hat_inter = 0.3, theta_hat_inter_sd = 0.5, n_control_inter = 100, beta = 0.1)
Group sequential designs / Adaptive sequential designs: Interim analysis, Survival analysis, Input theta
asd_new_design_theta_survival( theta, s, c_bry, inter_vt, snew, theta_hat_inter, theta_hat_inter_sd, events_inter, beta )asd_new_design_theta_survival( theta, s, c_bry, inter_vt, snew, theta_hat_inter, theta_hat_inter_sd, events_inter, beta )
theta |
Assumed effect size used for the conditional power calculation, i.e. minus the log of the hazard ratio; different values of theta can be entered to check the sensitivity of the conditional power to the effect size; if theta < 0.001 the new number of events and the new critical boundary are returned as 'NA' |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
inter_vt |
Index of the interim analysis at which the sample size is re-estimated (1-based); must be less than length(s) |
snew |
Vector of new information fractions after the sample size is changed, e.g. c(0.5, 1); the fractions are relative to the new total information and the last element must be 1 |
theta_hat_inter |
Treatment effect estimate observed at the interim analysis, i.e. minus the log of the observed hazard ratio |
theta_hat_inter_sd |
Standard error of 'theta_hat_inter' |
events_inter |
Number of events observed at the interim analysis |
beta |
Type II error rate, i.e. 1 - power |
A named list with the following components:
Character string giving the outcome of the interim hypothesis test: whether the null hypothesis was rejected at the interim analysis
Wald statistic of the treatment effect observed at the interim analysis
Planned critical boundary at the interim analysis at which the sample size is re-estimated
The assumed effect size theta used for the conditional power calculation
Conditional type I error at the interim analysis, calculated by setting theta = 0
Conditional power at the interim analysis using the assumed effect size theta
New total number of events required after the sample size re-estimation, or 'NA' if the null hypothesis was rejected at the interim analysis or if theta < 0.001
The new information fractions after the sample size change
New critical boundaries for the new interim analyses (O'Brien-Fleming type), or 'NA' if they were not calculated
asd_new_design_theta_survival(theta = 0.2, s = c(1/3, 2/3, 1), c_bry = c(3.984298, 3.961146, 2.349682), inter_vt = 1, snew = c(0.5, 1), theta_hat_inter = 0.3, theta_hat_inter_sd = 0.5, events_inter = 100, beta = 0.1)asd_new_design_theta_survival(theta = 0.2, s = c(1/3, 2/3, 1), c_bry = c(3.984298, 3.961146, 2.349682), inter_vt = 1, snew = c(0.5, 1), theta_hat_inter = 0.3, theta_hat_inter_sd = 0.5, events_inter = 100, beta = 0.1)
Group sequential designs / Adaptive sequential designs: Simulations, Binary distribution, Operating characteristics
asd_power_simu_binary( p_t, p_c, direction, s, bry_type, alpha_2, samsz, N_max, theta_cut, beta, sim_num )asd_power_simu_binary( p_t, p_c, direction, s, bry_type, alpha_2, samsz, N_max, theta_cut, beta, sim_num )
p_t |
Event rate of the test arm; must be between 0 and 1 |
p_c |
Event rate of the control arm; must be between 0 and 1 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock, 3 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
samsz |
Planned sample size of the control arm; the trial starts from this sample size and it is increased only if the conditional power at the interim analysis is below the target |
N_max |
Maximum sample size of the control arm allowed for the sample size change; the re-estimated sample size is truncated at N_max |
theta_cut |
Lack of efficacy threshold on the effect-size scale: if the observed interim effect size is below theta_cut, the trial continues to the planned sample size with no sample size increase and no early termination for futility |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; must be between 0 and 1 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with the following components:
Data frame with one row per analysis and the columns Looks, Information fraction, Critical boundaries, Sample size increase (the rate of sample size increase attributed to each look) and Rejection by look (the rejection rate at each interim analysis)
Named vector with the components Rejection rate without SSR, Rejection rate with SSR, Rate of sample size increase, Rejection rate without SSR/(meeting sample size incerase criteria), Rejection rate with SSR/(sample size incerase), Futility rate, Planned control sample size, Maximum control sample size, Mean control sample size and mean control sample size conditional on increase
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
asd_power_simu_binary(p_t = 0.4, p_c = 0.2, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 107, N_max = 500, theta_cut = 0.05, beta = 0.1, sim_num = 20)asd_power_simu_binary(p_t = 0.4, p_c = 0.2, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 107, N_max = 500, theta_cut = 0.05, beta = 0.1, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Negative binomial distribution, Operating characteristics
asd_power_simu_neg_binomial( r_t, r_c, kappa, direction, s, bry_type, alpha_2, samsz, N_max, theta_cut, beta, sim_num )asd_power_simu_neg_binomial( r_t, r_c, kappa, direction, s, bry_type, alpha_2, samsz, N_max, theta_cut, beta, sim_num )
r_t |
Mean of the test arm; must be > 0 |
r_c |
Mean of the control arm; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock, 3 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
samsz |
Planned sample size of the control arm; the trial starts from this sample size and it is increased only if the conditional power at the interim analysis is below the target |
N_max |
Maximum sample size of the control arm allowed for the sample size change; the re-estimated sample size is truncated at N_max |
theta_cut |
Lack of efficacy threshold on the effect-size scale: if the observed interim effect size is below theta_cut, the trial continues to the planned sample size with no sample size increase and no early termination for futility |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; must be between 0 and 1 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with the following components:
Data frame with one row per analysis and the columns Looks, Information fraction, Critical boundaries, Sample size increase (the rate of sample size increase attributed to each look) and Rejection by look (the rejection rate at each interim analysis)
Named vector with the components Rejection rate without SSR, Rejection rate with SSR, Rate of sample size increase, Rejection rate without SSR/(meeting sample size incerase criteria), Rejection rate with SSR/(sample size incerase), Futility rate, Planned control sample size, Maximum control sample size, Mean control sample size and mean control sample size conditional on increase
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
asd_power_simu_neg_binomial(r_t = 0.3, r_c = 0.2, kappa = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 112, N_max = 200, theta_cut = 0.01, beta = 0.1, sim_num = 20)asd_power_simu_neg_binomial(r_t = 0.3, r_c = 0.2, kappa = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 112, N_max = 200, theta_cut = 0.01, beta = 0.1, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Normal distribution, Operating characteristics
asd_power_simu_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, samsz, N_max, theta_cut, beta, sim_num )asd_power_simu_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, samsz, N_max, theta_cut, beta, sim_num )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
sigma_t |
Common standard deviation of the test arm and the control arm; must be > 0 |
sigma_c |
Common standard deviation of the test arm and the control arm; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock, 3 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
samsz |
Planned sample size of the control arm; the trial starts from this sample size and it is increased only if the conditional power at the interim analysis is below the target |
N_max |
Maximum sample size of the control arm allowed for the sample size change; the re-estimated sample size is truncated at N_max |
theta_cut |
Lack of efficacy threshold on the effect-size scale: if the observed interim effect size is below theta_cut, the trial continues to the planned sample size with no sample size increase and no early termination for futility |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; must be between 0 and 1 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with the following components:
Data frame with one row per analysis and the columns Looks, Information fraction, Critical boundaries, Sample size increase (the rate of sample size increase attributed to each look) and Rejection by look (the rejection rate at each interim analysis)
Named vector with the components Rejection rate without SSR, Rejection rate with SSR, Rate of sample size increase, Rejection rate without SSR/(meeting sample size incerase criteria), Rejection rate with SSR/(sample size incerase), Futility rate, Planned control sample size, Maximum control sample size, Mean control sample size and mean control sample size conditional on increase
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
asd_power_simu_normal(mu_t = 0, mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, N_max = 500, theta_cut = 0.05, beta = 0.1, sim_num = 20)asd_power_simu_normal(mu_t = 0, mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, N_max = 500, theta_cut = 0.05, beta = 0.1, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Poisson distribution, Operating characteristics
asd_power_simu_poisson( lambda_t, lambda_c, direction, s, bry_type, alpha_2, samsz, N_max, theta_cut, beta, sim_num )asd_power_simu_poisson( lambda_t, lambda_c, direction, s, bry_type, alpha_2, samsz, N_max, theta_cut, beta, sim_num )
lambda_t |
Mean event rate of the test arm; must be > 0 |
lambda_c |
Mean event rate of the control arm; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock, 3 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
samsz |
Planned sample size of the control arm; the trial starts from this sample size and it is increased only if the conditional power at the interim analysis is below the target |
N_max |
Maximum sample size of the control arm allowed for the sample size change; the re-estimated sample size is truncated at N_max |
theta_cut |
Lack of efficacy threshold on the effect-size scale: if the observed interim effect size is below theta_cut, the trial continues to the planned sample size with no sample size increase and no early termination for futility |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; must be between 0 and 1 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with the following components:
Data frame with one row per analysis and the columns Looks, Information fraction, Critical boundaries, Sample size increase (the rate of sample size increase attributed to each look) and Rejection by look (the rejection rate at each interim analysis)
Named vector with the components Rejection rate without SSR, Rejection rate with SSR, Rate of sample size increase, Rejection rate without SSR/(meeting sample size incerase criteria), Rejection rate with SSR/(sample size incerase), Futility rate, Planned control sample size, Maximum control sample size, Mean control sample size and mean control sample size conditional on increase
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
asd_power_simu_poisson(lambda_t = 1.1, lambda_c = 0.8, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, N_max = 200, theta_cut = 0.01, beta = 0.1, sim_num = 20)asd_power_simu_poisson(lambda_t = 1.1, lambda_c = 0.8, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, N_max = 200, theta_cut = 0.01, beta = 0.1, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Survival analysis, Operating characteristics
asd_power_simu_survival( HR, s, bry_type, alpha_2, evt_num, evt_max, HR_cut, surv_rate, beta, sim_num )asd_power_simu_survival( HR, s, bry_type, alpha_2, evt_num, evt_max, HR_cut, surv_rate, beta, sim_num )
HR |
Hazards ratio of the test arm vs. the control arm; HR < 1 favours the test arm |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock, 3 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
evt_num |
Planned total number of events at the final analysis; the trial starts from this number of events and it is increased only if the conditional power at the interim analysis is below the target |
evt_max |
Maximum number of events allowed for the sample size change; the re-estimated number of events is truncated at evt_max |
HR_cut |
Lack of efficacy threshold on the hazards-ratio scale; the corresponding threshold on the log-hazards-ratio scale is theta_cut = -log(HR_cut). If the observed interim log-hazards-ratio is below theta_cut (i.e. the observed HR is greater than HR_cut), the trial continues to the planned number of events with no sample size increase and no early termination for futility |
surv_rate |
Survival rate at the end of the trial, used together with the number of events to derive the sample size; must be between 0 and 1 |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; must be between 0 and 1 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with the following components:
Data frame with one row per analysis and the columns Looks, Information fraction, Critical boundaries, Sample size increase (the rate of sample size increase attributed to each look) and Rejection by look (the rejection rate at each interim analysis)
Named vector with the components Rejection rate without SSR, Rejection rate with SSR, Rate of sample size increase, Rejection rate without SSR/(meeting sample size incerase criteria), Rejection rate with SSR/(sample size incerase), Futility rate, Planned control sample size (the planned number of events), Maximum control sample size (the maximum number of events allowed), Mean control sample size (the mean simulated number of events) and mean control sample size conditional on increase
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
asd_power_simu_survival(HR = 0.8, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, evt_num = 845, evt_max = 1500, HR_cut = 0.95, surv_rate = 0.5, beta = 0.1, sim_num = 20)asd_power_simu_survival(HR = 0.8, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, evt_num = 845, evt_max = 1500, HR_cut = 0.95, surv_rate = 0.5, beta = 0.1, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Binary distribution, Optimizing adaptive sequential design
asd_simu_OC_binary( p_t, p_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )asd_simu_OC_binary( p_t, p_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
p_t |
Event rate of the test arm; must be between 0 and 1. The simulation grid varies p_t over the values supplied |
p_c |
Event rate of the control arm; must be between 0 and 1 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s1 |
Vector of information fractions of the first interim analysis schedule, e.g. c(1/3, 2/3, 1); the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule, e.g. c(0.5, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
alpha_2_alps_s1 |
Two-sided type I error spent at the interim analyses of schedule s1 with the alpha-spending boundary; each element must be between 0 and 1 and the elements must sum to alpha_2 |
alpha_2_alps_s2 |
Two-sided type I error spent at the interim analyses of schedule s2 with the alpha-spending boundary; each element must be between 0 and 1 and the elements must sum to alpha_2 |
theta_cut |
Lack of efficacy threshold on the effect-size scale: if the observed interim effect size is below theta_cut, the trial continues to the planned sample size with no sample size increase and no early termination for futility |
samsz_1 |
Planned sample size of the control arm for the first scenario of the simulation grid |
samsz_2 |
Planned sample size of the control arm for the second scenario of the simulation grid |
N_max_1 |
Maximum sample size of the control arm allowed for the sample size change in the first scenario of the simulation grid |
N_max_2 |
Maximum sample size of the control arm allowed for the sample size change in the second scenario of the simulation grid |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; must be between 0 and 1 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with six components, one for each combination of critical boundary type (O'Brien-Fleming, Pocock, alpha-spending) and interim analysis schedule (s1, s2):
Named list with the group sequential design (Looks, Information fractions, O'Brien-Fleming boundaries) and the simulation results for the four combinations of planned sample size (samsz_1, samsz_2) and maximum sample size (N_max_1, N_max_2), each a data frame with one row per test-arm event rate and the columns p<sub>t1</sub>,..., p<sub>c</sub>, sample size for fixed design, Rejection rate without SSR, Rejection rate with SSR, Rate of sample size increase, Rejection rate without SSR/(meeting sample size incerase criteria), Rejection rate with SSR/(sample size incerase), Futility rate, Planned control sample size, Maximum control sample size, Mean control sample size and mean control sample size conditional on increase
The same group sequential design and simulation results computed with the information fractions s2
The same group sequential design and simulation results computed with Pocock boundaries and the information fractions s1
The same group sequential design and simulation results computed with Pocock boundaries and the information fractions s2
The same group sequential design and simulation results computed with alpha-spending boundaries and the information fractions s1
The same group sequential design and simulation results computed with alpha-spending boundaries and the information fractions s2
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
set.seed(1) asd_simu_OC_binary(p_t = c(0.3,0.35,0.4), p_c = 0.2, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 20)set.seed(1) asd_simu_OC_binary(p_t = c(0.3,0.35,0.4), p_c = 0.2, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Negative binomial distribution, Optimizing adaptive sequential design
asd_simu_OC_neg_binomial( r_t, r_c, kappa, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )asd_simu_OC_neg_binomial( r_t, r_c, kappa, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
r_t |
Mean of the test arm; must be > 0. The simulation grid varies r_t over the values supplied |
r_c |
Mean of the control arm; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s1 |
Vector of information fractions of the first interim analysis schedule, e.g. c(1/3, 2/3, 1); the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule, e.g. c(0.5, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
alpha_2_alps_s1 |
Two-sided type I error spent at the interim analyses of schedule s1 with the alpha-spending boundary; each element must be between 0 and 1 and the elements must sum to alpha_2 |
alpha_2_alps_s2 |
Two-sided type I error spent at the interim analyses of schedule s2 with the alpha-spending boundary; each element must be between 0 and 1 and the elements must sum to alpha_2 |
theta_cut |
Lack of efficacy threshold on the effect-size scale: if the observed interim effect size is below theta_cut, the trial continues to the planned sample size with no sample size increase and no early termination for futility |
samsz_1 |
Planned sample size of the control arm for the first scenario of the simulation grid |
samsz_2 |
Planned sample size of the control arm for the second scenario of the simulation grid |
N_max_1 |
Maximum sample size of the control arm allowed for the sample size change in the first scenario of the simulation grid |
N_max_2 |
Maximum sample size of the control arm allowed for the sample size change in the second scenario of the simulation grid |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; must be between 0 and 1 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with six components, one for each combination of critical boundary type (O'Brien-Fleming, Pocock, alpha-spending) and interim analysis schedule (s1, s2):
Named list with the group sequential design (Looks, Information fractions, O'Brien-Fleming boundaries) and the simulation results for the four combinations of planned sample size (samsz_1, samsz_2) and maximum sample size (N_max_1, N_max_2), each a data frame with one row per test-arm mean and the columns r<sub>t1</sub>,..., r<sub>c</sub>, sample size for fixed design, Rejection rate without SSR, Rejection rate with SSR, Rate of sample size increase, Rejection rate without SSR/(meeting sample size incerase criteria), Rejection rate with SSR/(sample size incerase), Futility rate, Planned control sample size, Maximum control sample size, Mean control sample size and mean control sample size conditional on increase
The same group sequential design and simulation results computed with the information fractions s2
The same group sequential design and simulation results computed with Pocock boundaries and the information fractions s1
The same group sequential design and simulation results computed with Pocock boundaries and the information fractions s2
The same group sequential design and simulation results computed with alpha-spending boundaries and the information fractions s1
The same group sequential design and simulation results computed with alpha-spending boundaries and the information fractions s2
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
set.seed(1) asd_simu_OC_neg_binomial(r_t = c(0.3,0.4,0.5), r_c = 0.2, kappa = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 20)set.seed(1) asd_simu_OC_neg_binomial(r_t = c(0.3,0.4,0.5), r_c = 0.2, kappa = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Normal distribution, Optimizing adaptive sequential design
asd_simu_OC_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )asd_simu_OC_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
sigma_t |
Common standard deviation of the test arm and the control arm; must be > 0 |
sigma_c |
Common standard deviation of the test arm and the control arm; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s1 |
Vector of information fractions of the first interim analysis schedule, e.g. c(1/3, 2/3, 1); the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule, e.g. c(0.5, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
alpha_2_alps_s1 |
Two-sided type I error spent at the interim analyses of schedule s1 with the alpha-spending boundary; each element must be between 0 and 1 and the elements must sum to alpha_2 |
alpha_2_alps_s2 |
Two-sided type I error spent at the interim analyses of schedule s2 with the alpha-spending boundary; each element must be between 0 and 1 and the elements must sum to alpha_2 |
theta_cut |
Lack of efficacy threshold on the effect-size scale: if the observed interim effect size is below theta_cut, the trial continues to the planned sample size with no sample size increase and no early termination for futility |
samsz_1 |
Planned sample size of the control arm for the first stage of the simulation grid |
samsz_2 |
Planned sample size of the control arm for the second stage of the simulation grid |
N_max_1 |
Maximum sample size of the control arm allowed for the sample size change in the first scenario of the simulation grid |
N_max_2 |
Maximum sample size of the control arm allowed for the sample size change in the second scenario of the simulation grid |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; must be between 0 and 1 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with six components, one for each combination of critical boundary type (O'Brien-Fleming, Pocock, alpha-spending) and interim analysis schedule (s1, s2):
Named list with the group sequential design (Looks, Information fractions, O'Brien-Fleming boundaries) and the simulation results for the four combinations of planned sample size (samsz_1, samsz_2) and maximum sample size (N_max_1, N_max_2), each a data frame with one row per test-arm mean and the columns mu_t1,..., mu_c, sample size for fixed design, Rejection rate without SSR, Rejection rate with SSR, Rate of sample size increase, Rejection rate without SSR/(meeting sample size incerase criteria), Rejection rate with SSR/(sample size incerase), Futility rate, Planned control sample size, Maximum control sample size, Mean control sample size, mean control sample size conditional on increase and Ratio of adaptive sample size over fixed sample size
The same group sequential design and simulation results computed with the information fractions s2
The same group sequential design and simulation results computed with Pocock boundaries and the information fractions s1
The same group sequential design and simulation results computed with Pocock boundaries and the information fractions s2
The same group sequential design and simulation results computed with alpha-spending boundaries and the information fractions s1
The same group sequential design and simulation results computed with alpha-spending boundaries and the information fractions s2
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
set.seed(1) asd_simu_OC_normal(mu_t = c(0.3,0.4,0.5), mu_c = 0.2, sigma_t = 1, sigma_c = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 20)set.seed(1) asd_simu_OC_normal(mu_t = c(0.3,0.4,0.5), mu_c = 0.2, sigma_t = 1, sigma_c = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Poisson distribution, Optimizing adaptive sequential design
asd_simu_OC_poisson( lambda_t, lambda_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )asd_simu_OC_poisson( lambda_t, lambda_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
lambda_t |
Mean event rate of the test arm; must be > 0. The simulation grid varies lambda_t over the values supplied |
lambda_c |
Mean event rate of the control arm; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s1 |
Vector of information fractions of the first interim analysis schedule, e.g. c(1/3, 2/3, 1); the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule, e.g. c(0.5, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
alpha_2_alps_s1 |
Two-sided type I error spent at the interim analyses of schedule s1 with the alpha-spending boundary; each element must be between 0 and 1 and the elements must sum to alpha_2 |
alpha_2_alps_s2 |
Two-sided type I error spent at the interim analyses of schedule s2 with the alpha-spending boundary; each element must be between 0 and 1 and the elements must sum to alpha_2 |
theta_cut |
Lack of efficacy threshold on the effect-size scale: if the observed interim effect size is below theta_cut, the trial continues to the planned sample size with no sample size increase and no early termination for futility |
samsz_1 |
Planned sample size of the control arm for the first scenario of the simulation grid |
samsz_2 |
Planned sample size of the control arm for the second scenario of the simulation grid |
N_max_1 |
Maximum sample size of the control arm allowed for the sample size change in the first scenario of the simulation grid |
N_max_2 |
Maximum sample size of the control arm allowed for the sample size change in the second scenario of the simulation grid |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; must be between 0 and 1 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with six components, one for each combination of critical boundary type (O'Brien-Fleming, Pocock, alpha-spending) and interim analysis schedule (s1, s2):
Named list with the group sequential design (Looks, Information fractions, O'Brien-Fleming boundaries) and the simulation results for the four combinations of planned sample size (samsz_1, samsz_2) and maximum sample size (N_max_1, N_max_2), each a data frame with one row per test-arm event rate and the columns lambda_t1,..., lambda_c, sample size for fixed design, Rejection rate without SSR, Rejection rate with SSR, Rate of sample size increase, Rejection rate without SSR/(meeting sample size incerase criteria), Rejection rate with SSR/(sample size incerase), Futility rate, Planned control sample size, Maximum control sample size, Mean control sample size and mean control sample size conditional on increase
The same group sequential design and simulation results computed with the information fractions s2
The same group sequential design and simulation results computed with Pocock boundaries and the information fractions s1
The same group sequential design and simulation results computed with Pocock boundaries and the information fractions s2
The same group sequential design and simulation results computed with alpha-spending boundaries and the information fractions s1
The same group sequential design and simulation results computed with alpha-spending boundaries and the information fractions s2
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
asd_simu_OC_poisson(lambda_t = c(0.3,0.4,0.5), lambda_c = 0.2, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 50, samsz_2 = 80, N_max_1 = 120, N_max_2 = 150, beta = 0.1, sim_num = 20)asd_simu_OC_poisson(lambda_t = c(0.3,0.4,0.5), lambda_c = 0.2, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 50, samsz_2 = 80, N_max_1 = 120, N_max_2 = 150, beta = 0.1, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Survival analysis, Optimizing adaptive sequential design
asd_simu_OC_survival( HR, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, surv_rate, evt_num_1, evt_num_2, evt_max_1, evt_max_2, HR_cut, beta, sim_num )asd_simu_OC_survival( HR, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, surv_rate, evt_num_1, evt_num_2, evt_max_1, evt_max_2, HR_cut, beta, sim_num )
HR |
Hazards ratio of the test arm vs. the control arm; HR < 1 favours the test arm. The simulation grid varies HR over the values supplied |
s1 |
Vector of information fractions of the first interim analysis schedule, e.g. c(1/3, 2/3, 1); the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule, e.g. c(0.5, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
alpha_2_alps_s1 |
Two-sided type I error spent at the interim analyses of schedule s1 with the alpha-spending boundary; each element must be between 0 and 1 and the elements must sum to alpha_2 |
alpha_2_alps_s2 |
Two-sided type I error spent at the interim analyses of schedule s2 with the alpha-spending boundary; each element must be between 0 and 1 and the elements must sum to alpha_2 |
surv_rate |
Survival rate at the end of the trial, used together with the number of events to derive the sample size; must be between 0 and 1 |
evt_num_1 |
Planned number of events for the first scenario of the simulation grid |
evt_num_2 |
Planned number of events for the second scenario of the simulation grid |
evt_max_1 |
Maximum number of events allowed for the sample size change in the first scenario of the simulation grid |
evt_max_2 |
Maximum number of events allowed for the sample size change in the second scenario of the simulation grid |
HR_cut |
Lack of efficacy threshold on the hazards-ratio scale; the corresponding threshold on the log-hazards-ratio scale is theta_cut = -log(HR_cut). If the observed interim log-hazards-ratio is below theta_cut (i.e. the observed HR is greater than HR_cut), the trial continues to the planned number of events with no sample size increase and no early termination for futility |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; must be between 0 and 1 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with six components, one for each combination of critical boundary type (O'Brien-Fleming, Pocock, alpha-spending) and interim analysis schedule (s1, s2):
Named list with the group sequential design (Looks, Information fractions, O'Brien-Fleming boundaries) and the simulation results for the four combinations of planned number of events (evt_num_1, evt_num_2) and maximum number of events (evt_max_1, evt_max_2), each a data frame with one row per hazards ratio and the columns HR<sub>1</sub>,..., sample size for fixed design, Rejection rate without SSR, Rejection rate with SSR, Rate of sample size increase, Rejection rate without SSR/(meeting sample size incerase criteria), Rejection rate with SSR/(sample size incerase), Futility rate, Planned sample size (the planned number of events), Maximum sample size (the maximum number of events allowed), Mean sample size (the mean simulated number of events), mean sample size conditional on increase and Ratio of adaptive sample size over fixed sample size
The same group sequential design and simulation results computed with the information fractions s2
The same group sequential design and simulation results computed with Pocock boundaries and the information fractions s1
The same group sequential design and simulation results computed with Pocock boundaries and the information fractions s2
The same group sequential design and simulation results computed with alpha-spending boundaries and the information fractions s1
The same group sequential design and simulation results computed with alpha-spending boundaries and the information fractions s2
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
set.seed(1) asd_simu_OC_survival(HR = c(0.75,0.8,0.85), s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), surv_rate = 0.5, evt_num_1 = 200, evt_num_2 = 300, evt_max_1 = 2000, evt_max_2 = 3000, HR_cut = 0.95, beta = 0.1, sim_num = 20)set.seed(1) asd_simu_OC_survival(HR = c(0.75,0.8,0.85), s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), surv_rate = 0.5, evt_num_1 = 200, evt_num_2 = 300, evt_max_1 = 2000, evt_max_2 = 3000, HR_cut = 0.95, beta = 0.1, sim_num = 20)
Single arm phase 2 oncology trial: Three-stage design
EXP_3_stg_sz(alpha, beta, p_0, p_low, p_1, d_23_cut)EXP_3_stg_sz(alpha, beta, p_0, p_low, p_1, d_23_cut)
alpha |
Type I error for the one-sided test |
beta |
Type II error rate, i.e. 1 - power |
p_0 |
Response rate indicating low activity with insufficient clinical benefit |
p_low |
Minimally clinically beneficial response rate |
p_1 |
Assumed response rate |
d_23_cut |
Sets the minimal difference between n_3 and n_2 for the design; min(n_3 - n_2) |
A named list with the following components:
A matrix with one row per design (minimax, optimal and average) for Simon's design using p_1 as the target response rate, with columns r1, n1, r_2, n_2, EN(p0) (expected sample size under p_0), PET(p0) (probability of early termination under p_0), t1_err, PET_p (PET under p_1), PET_p_low (PET under p_low), power_p (power under p_1) and power_p_low (power under p_low)
The corresponding matrix for Simon's design using p_low as the target response rate
A matrix with one row per design (minimax, optimal and average) and columns r_1, r_2, r_3, n_1, n_2, n_3 (n_3 is also the total sample size), r_2_e (the trial is stopped for superiority if at least r_2_e responses are observed at the second look), t1_err, PET_p1, PET_plow, power (power under p_1), EN_p0 (expected sample size under p_0) and PW_p_low (power under p_low)
R. Simon. Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10:1-10 (1989).
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon's design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
EXP_3_stg_sz( alpha = c(0.001,0.024), beta = 0.1, p_0 = 0.2, p_low = 0.4, p_1 = 0.55, d_23_cut = 0)EXP_3_stg_sz( alpha = c(0.001,0.024), beta = 0.1, p_0 = 0.2, p_low = 0.4, p_1 = 0.55, d_23_cut = 0)
Fixed sample designs: Negative binomial distribution, Simulations
fixed_rej_nbglm_simu(alpha_2, r_c, r_t, rand_ratio, kappa, samsz, sim_num)fixed_rej_nbglm_simu(alpha_2, r_c, r_t, rand_ratio, kappa, samsz, sim_num)
alpha_2 |
Type I error for the two-sided test |
r_c |
Event rate of the control arm; must be > 0 |
r_t |
Event rate of the test arm; the null hypothesis is r_t >= r_c and the alternative hypothesis is r_t < r_c; must be > 0 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
kappa |
Dispersion parameter of the negative binomial distribution; the negative binomial size parameter used in the simulation is 1/kappa, so kappa must be > 0 and a larger kappa means more overdispersion |
samsz |
Sample size of the control arm used in the simulation |
sim_num |
Number of simulation replicates |
A named list with the following components:
Simulated rejection rate of the two-sided Wald test from the negative binomial log-link GLM; this is the simulated power, and the type I error rate when r_t = r_c
Mean over the simulation replicates of the estimated negative binomial size (dispersion) parameter theta of the fitted GLM
fixed_rej_nbglm_simu(alpha_2 = 0.05, r_c = 1, r_t = 1*0.6, rand_ratio = 1, kappa = 1, samsz = 188, sim_num = 20)fixed_rej_nbglm_simu(alpha_2 = 0.05, r_c = 1, r_t = 1*0.6, rand_ratio = 1, kappa = 1, samsz = 188, sim_num = 20)
Fixed sample designs: Binary distribution, Simulations
fixed_rej_rate_binary(p_t, p_c, rand_ratio, alpha_2, samsz, sim_num)fixed_rej_rate_binary(p_t, p_c, rand_ratio, alpha_2, samsz, sim_num)
p_t |
Event rate (probability of an event) of the test arm; the null hypothesis is p_t <= p_c and the alternative hypothesis is p_t > p_c; 0 < p_t < 1 |
p_c |
Event rate (probability of an event) of the control arm; 0 < p_c < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
alpha_2 |
Type I error for the two-sided test |
samsz |
Sample size of the control arm used in the simulation |
sim_num |
Number of simulation replicates |
A named numeric value with the following component:
Simulated rejection rate of the two-sided fixed-sample Wald test of p_t = p_c; this is the simulated power, and the type I error rate when p_t = p_c
fixed_rej_rate_binary(p_t = 0.4, p_c = 0.2, rand_ratio = 1, alpha_2 = 0.05, samsz = 106, sim_num = 20)fixed_rej_rate_binary(p_t = 0.4, p_c = 0.2, rand_ratio = 1, alpha_2 = 0.05, samsz = 106, sim_num = 20)
Fixed sample designs: Normal distribution, Simulations
fixed_rej_rate_normal( mu_t, mu_c, sigma_t, sigma_c, alpha_2, rand_ratio, samsz, sim_num )fixed_rej_rate_normal( mu_t, mu_c, sigma_t, sigma_c, alpha_2, rand_ratio, samsz, sim_num )
mu_t |
Mean of the test arm; the null hypothesis is mu_t <= mu_c and the alternative hypothesis is mu_t > mu_c |
mu_c |
Mean of the control arm |
sigma_t |
Standard deviation of the test arm |
sigma_c |
Standard deviation of the control arm |
alpha_2 |
Type I error for the two-sided test |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
samsz |
Sample size of the control arm used in the simulation |
sim_num |
Number of simulation replicates |
A named numeric value with the following component:
Simulated rejection rate of the two-sided fixed-sample Wald test of mu_t = mu_c; this is the simulated power, and the type I error rate when mu_t = mu_c
fixed_rej_rate_normal(mu_t = 0, mu_c = 0, sigma_t = 1, sigma_c = 1, alpha_2 = 0.05, rand_ratio = 1, samsz = 82, sim_num = 20)fixed_rej_rate_normal(mu_t = 0, mu_c = 0, sigma_t = 1, sigma_c = 1, alpha_2 = 0.05, rand_ratio = 1, samsz = 82, sim_num = 20)
Fixed sample designs: Poisson distribution, Simulations
fixed_rej_rate_poisson(lambda_t, lambda_c, rand_ratio, alpha_2, samsz, sim_num)fixed_rej_rate_poisson(lambda_t, lambda_c, rand_ratio, alpha_2, samsz, sim_num)
lambda_t |
Event rate of the test arm; the null hypothesis is lambda_t <= lambda_c and the alternative hypothesis is lambda_t > lambda_c; must be > 0 |
lambda_c |
Event rate of the control arm; must be > 0 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
alpha_2 |
Type I error for the two-sided test |
samsz |
Sample size of the control arm used in the simulation |
sim_num |
Number of simulation replicates |
A named numeric value with the following component:
Simulated rejection rate of the two-sided fixed-sample Wald test of lambda_t = lambda_c; this is the simulated power, and the type I error rate when lambda_t = lambda_c
fixed_rej_rate_poisson(lambda_t = 11, lambda_c = 10, rand_ratio = 2, alpha_2 = 0.05, samsz = 163, sim_num = 20)fixed_rej_rate_poisson(lambda_t = 11, lambda_c = 10, rand_ratio = 2, alpha_2 = 0.05, samsz = 163, sim_num = 20)
Fixed sample designs: Survival analysis, Simulations
fixed_rej_rate_surv_cox(HR, evt_num, alpha_2, sim_num)fixed_rej_rate_surv_cox(HR, evt_num, alpha_2, sim_num)
HR |
Hazard ratio of the test arm vs. the control arm; the null hypothesis is HR >= 1 and the alternative hypothesis is HR < 1; must be > 0 |
evt_num |
Total number of events at the final analysis; the simulated data are truncated at the first observation at which this number of events is reached |
alpha_2 |
Type I error for the two-sided test |
sim_num |
Number of simulation replicates |
A named numeric value with the following component:
Simulated rejection rate of the two-sided Wald test from the Cox proportional hazards model; this is the simulated power, and the type I error rate when HR = 1
fixed_rej_rate_surv_cox(HR = 0.8, evt_num = 845, alpha_2 = 0.05, sim_num = 20)fixed_rej_rate_surv_cox(HR = 0.8, evt_num = 845, alpha_2 = 0.05, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Final analysis, Without sample size change
gsd_ci_est(s, c_bry, theta_hat_last, theta_hat_last_sd, last_vt, alpha_2)gsd_ci_est(s, c_bry, theta_hat_last, theta_hat_last_sd, last_vt, alpha_2)
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
theta_hat_last |
Treatment effect estimate observed at the final analysis, i.e. the naive point estimate of the parameter of interest at the analysis at which the trial terminated |
theta_hat_last_sd |
Standard error of 'theta_hat_last' |
last_vt |
Index of the analysis at which the trial stopped (1-based); this can be the final analysis, at which s = 1, or an interim analysis, and it must not exceed length(s) |
alpha_2 |
Type I error for the two-sided test |
A named list with the following components:
The p-value of the test of the null hypothesis that the treatment effect is 0
Numeric vector of length 3 giving the lower confidence limit, the median-unbiased point estimate and the upper confidence limit of the two-sided (1 - alpha_2) confidence interval
gsd_ci_est(s = c(1/3,2/3,1), c_bry = c(3.471095, 2.454434,2.004037), theta_hat_last = 2.004, theta_hat_last_sd = 1, last_vt = 3, alpha_2 = 0.05)gsd_ci_est(s = c(1/3,2/3,1), c_bry = c(3.471095, 2.454434,2.004037), theta_hat_last = 2.004, theta_hat_last_sd = 1, last_vt = 3, alpha_2 = 0.05)
Single arm phase 2 oncology trial: Final analysis: two stage design, no sample size change
gsd_ci_est_2_stage(p_0, n_1, n_2, r_1_e, r_2, x_last, last_vt, alpha)gsd_ci_est_2_stage(p_0, n_1, n_2, r_1_e, r_2, x_last, last_vt, alpha)
p_0 |
Response rate indicating low activity with insufficient clinical benefit |
n_1 |
Number of patients at first look |
n_2 |
Number of patients at second look |
r_1_e |
Superiority-stopping threshold at the first look: the trial would be stopped for superiority if at least r_1_e responses are observed at the first look. Enter NA when the design uses no efficacy boundary at the first look; the function then substitutes a boundary that cannot be crossed |
r_2 |
If no more than r_2 responses are observed at the second look, then the null hypothesis will not be rejected |
x_last |
Number of responses at the final visit |
last_vt |
Index of the analysis at which the trial stopped: 1 = first look, 2 = second (final) look |
alpha |
Type I error for the one-sided test |
The results with continuity correction are not necessarily different from those without continuity correction. The need for continuity correction can be determined through simulations on the type I error (see Gao and Zhang, 2004).
A named list with the following components:
P-value of the test of the null hypothesis that the response rate equals p_0, calculated at the analysis at which the trial stopped without continuity correction
The same p-value calculated with the 0.5 continuity correction
Numeric vector of length 3 holding the lower limit, the point estimate and the upper limit of the two-sided (1 - 2*alpha) confidence interval for the response rate, computed without continuity correction
The same confidence interval computed with the 0.5 continuity correction
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
gsd_ci_est_2_stage(p_0=0.2,n_1=100,n_2=105,r_1_e=NA,r_2=29,x_last=31,last_vt=1,alpha=0.025)gsd_ci_est_2_stage(p_0=0.2,n_1=100,n_2=105,r_1_e=NA,r_2=29,x_last=31,last_vt=1,alpha=0.025)
Single arm phase 2 oncology trial: Final analysis: three stage design, no sample size change
gsd_ci_est_3_stage(p_0, n_2, n_3, r_2_e, r_3, x_last, last_vt, alpha)gsd_ci_est_3_stage(p_0, n_2, n_3, r_2_e, r_3, x_last, last_vt, alpha)
p_0 |
Response rate indicating low activity with insufficient clinical benefit |
n_2 |
Number of patients at second look; because the trial has proceeded to the third look, n_1, r_1 and r_2 of the original design are not needed for the final analysis |
n_3 |
Number of patients at third look |
r_2_e |
Superiority-stopping threshold at the second look: the trial would be stopped for superiority if at least r_2_e responses are observed at the second look. If r_2_e was not used in the design, enter NA; the function then substitutes a boundary that cannot be crossed |
r_3 |
If no more than r_3 responses are observed at the third look, then the null hypothesis will not be rejected |
x_last |
Number of responses at the final visit |
last_vt |
Index of the analysis at which the trial stopped: 2 = second look, 3 = third (final) look |
alpha |
Type I error for the one-sided test |
The results with continuity correction are not necessarily different from those without continuity correction. The need for continuity correction can be determined through simulations on the type I error (see Gao and Zhang, 2004).
A named list with the following components:
P-value of the test of the null hypothesis that the response rate equals p_0, calculated at the analysis at which the trial stopped without continuity correction
The same p-value calculated with the 0.5 continuity correction
Numeric vector of length 3 holding the lower limit, the point estimate and the upper limit of the two-sided (1 - 2*alpha) confidence interval for the response rate, computed without continuity correction
The same confidence interval computed with the 0.5 continuity correction
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
gsd_ci_est_3_stage(p_0=0.2,n_2=100,n_3=105,r_2_e=31,r_3=29,x_last=31,last_vt=3,alpha=0.025)gsd_ci_est_3_stage(p_0=0.2,n_2=100,n_3=105,r_2_e=31,r_3=29,x_last=31,last_vt=3,alpha=0.025)
Group sequential designs / Adaptive sequential designs: Simulations, Binary distribution, Type I error and power
gsd_power_simu_binary( p_t, p_c, direction, s, bry_type, alpha_2, samsz, sim_num )gsd_power_simu_binary( p_t, p_c, direction, s, bry_type, alpha_2, samsz, sim_num )
p_t |
Event rate of the test arm; must be between 0 and 1 |
p_c |
Event rate of the control arm; must be between 0 and 1 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock, 3 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
samsz |
Planned sample size of the control arm used in the simulation; must be > 0 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with the following components:
Data frame with one row per analysis and the columns Looks, Information fraction, Critical boundaries and Rejection by look (the rejection rate at each interim analysis)
Named vector with the components Planned control sample size (the sample size used in the simulation), Rejection rate (the simulated power, i.e. the type I error rate when p_t = p_c) and Mean control sample size (the mean simulated sample size)
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
gsd_power_simu_binary(p_t = 0.4, p_c = 0.4, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)gsd_power_simu_binary(p_t = 0.4, p_c = 0.4, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Negative binomial distribution, Type I error and power
gsd_power_simu_negbinom( r_t, r_c, kappa, direction, s, bry_type, alpha_2, samsz, sim_num )gsd_power_simu_negbinom( r_t, r_c, kappa, direction, s, bry_type, alpha_2, samsz, sim_num )
r_t |
Mean of the test arm; must be > 0 |
r_c |
Mean of the control arm; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock, 3 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
samsz |
Planned sample size of the control arm used in the simulation; must be > 0 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with the following components:
Data frame with one row per analysis and the columns Looks, Information fraction, Critical boundaries and Rejection by look (the rejection rate at each interim analysis)
Named vector with the components Planned control sample size (the sample size used in the simulation), Rejection rate (the simulated power, i.e. the type I error rate when r_t = r_c) and Mean control sample size (the mean simulated sample size)
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
gsd_power_simu_negbinom(r_t = 1.1, r_c = 1.1, kappa = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)gsd_power_simu_negbinom(r_t = 1.1, r_c = 1.1, kappa = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Normal distribution, Type I error and power
gsd_power_simu_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, samsz, sim_num )gsd_power_simu_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, samsz, sim_num )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
sigma_t |
Common standard deviation of the test arm and the control arm; must be > 0 |
sigma_c |
Common standard deviation of the test arm and the control arm; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock, 3 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
samsz |
Planned sample size of the control arm used in the simulation; must be > 0 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with the following components:
Data frame with one row per analysis and the columns Looks, Information fraction, Critical boundaries and Rejection by look (the rejection rate at each interim analysis)
Named vector with the components Rejection rate (the simulated power, i.e. the type I error rate when mu_t = mu_c), Planned control sample size (the sample size used in the simulation) and Mean control sample size (the mean simulated sample size)
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
gsd_power_simu_normal(mu_t = 0.4, mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/4,1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)gsd_power_simu_normal(mu_t = 0.4, mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/4,1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Poisson distribution, Type I error and power
gsd_power_simu_poisson( lambda_c, lambda_t, direction, s, bry_type, alpha_2, samsz, sim_num )gsd_power_simu_poisson( lambda_c, lambda_t, direction, s, bry_type, alpha_2, samsz, sim_num )
lambda_c |
Mean event rate of the control arm; must be > 0 |
lambda_t |
Mean event rate of the test arm; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock, 3 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
samsz |
Planned sample size of the control arm used in the simulation; must be > 0 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with the following components:
Data frame with one row per analysis and the columns Looks, Information fraction, Critical boundaries and Rejection by look (the rejection rate at each interim analysis)
Named vector with the components Planned control sample size (the sample size used in the simulation), Rejection rate (the simulated power, i.e. the type I error rate when lambda_t = lambda_c) and Mean control sample size (the mean simulated sample size)
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
gsd_power_simu_poisson(lambda_c = 0.5, lambda_t = 0.8, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)gsd_power_simu_poisson(lambda_c = 0.5, lambda_t = 0.8, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Group sequential designs / Adaptive sequential designs: Simulations, Survival analysis, Type I error and power
gsd_power_simu_survival(HR, s, bry_type, alpha_2, surv_rate, evt_num, sim_num)gsd_power_simu_survival(HR, s, bry_type, alpha_2, surv_rate, evt_num, sim_num)
HR |
Hazards ratio of the test arm vs. the control arm; HR < 1 favours the test arm |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock, 3 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; must be between 0 and 1, and alpha_2/2 is the one-sided type I error used for the critical boundaries |
surv_rate |
Survival rate at the end of the trial, used together with the number of events to derive the sample size; must be between 0 and 1 |
evt_num |
Planned total number of events at the final analysis; must be > 0 |
sim_num |
Number of simulation replicates; must be > 0 |
A named list with the following components:
Data frame with one row per analysis and the columns Looks, Information fraction, Critical boundaries and Rejection by look (the rejection rate at each interim analysis)
Named vector with the components Rejection rate (the simulated power, i.e. the type I error rate when HR = 1), Planned sample size (the planned number of events) and Mean sample size (the mean simulated number of events)
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
Gao, P., L. Liu, and C. Mehta. 2013 Oct 15. Exact inference for adaptive group sequential designs. Statistics in Medicine 32(23):3991–4005. doi:10.1002/sim.5847.
gsd_power_simu_survival(HR = 0.8, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, surv_rate = 0.5, evt_num = 845, sim_num = 20)gsd_power_simu_survival(HR = 0.8, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, surv_rate = 0.5, evt_num = 845, sim_num = 20)
Hybrid: Simulations, Binary distribution, Using given estimates of informative prior
hybrid_fixed_prior_rej_binary_simu( t_prior, theta_prior, p_t, p_c, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )hybrid_fixed_prior_rej_binary_simu( t_prior, theta_prior, p_t, p_c, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )
t_prior |
Prior information time: the prior distribution of theta = p_t - p_c is assumed to have a normal distribution, and the Fisher's information time is the reciprocal of the variance of the prior distribution; set t_prior = 0 for a non-informative prior |
theta_prior |
Mean effect size of prior: the mean of the normal prior distribution of theta = p_t - p_c; set theta_prior = 0 for a non-informative prior |
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1. The effect size is theta = p_t - p_c. Enter p_t = p_c to verify type I error control and p_t > p_c for the power simulation |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1. The effect size is theta = p_t - p_c |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz |
Planned sample size: the planned sample size at the final analysis |
N_max |
Maximum sample size: a user chosen, maximum allowable sample size for sample size modification |
theta_cut |
Futility threshold: a cut-off for theta_hat. In the simulations, if theta_hat at the interim analysis is at or below theta_cut the trial is considered futile and proceeds to the planned sample size without any change |
cp_min |
Promising zone cp_min: CP_min in the promising zone sample size re-estimation; a conditional power below cp_min is the unfavorable zone. The user should select this value from Mehta, Pocock (2011) |
gamma |
Parameter chosen for the credible interval modified predictive power (CI_MPP); it defines the (1 - gamma) credible interval of the posterior for theta, e.g. gamma = 0.5 gives a 50 percent interval |
alpha_2 |
Type I error for the two-sided test |
beta |
Conditional type II error; 1 - beta is the target conditional power used for the sample size re-estimation |
sim_num |
Number of simulation replicates |
A named list with the following components:
A data frame with the columns "Prior type", "SSR method", "Rejection rate" and "Mean sample size", with one row per sample size re-estimation (SSR) method. The methods compared are the planned sample size with no SSR, the promising zone SSR, the SSR using the true effect size with conditional power, and, for both the non-informative prior and the informative prior, the cp SSR, pp SSR, mpp SSR and ci_mpp SSR methods
The rate at which the observed theta_hat at the interim analysis is at or below theta_cut, so that the trial proceeds without any sample size change
A data frame of the references for the methods compared
Gao, P. Improving the effectiveness of sample size re-estimation: an operating characteristic focused, hybrid frequentist-Bayesian approach, Statistics in Medicine, 2025; 44:e10310. To link to this article: https://onlinelibrary.wiley.com/doi/10.1002/sim.10310.
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
C. R. Mehta, S. J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine. Volume30, Issue28, 10 December 2011, Pages 3267-3284
hybrid_fixed_prior_rej_binary_simu(t_prior = 5, theta_prior = 0.2, p_t = 0.3, p_c = 0.3, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)hybrid_fixed_prior_rej_binary_simu(t_prior = 5, theta_prior = 0.2, p_t = 0.3, p_c = 0.3, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)
Hybrid: Simulations, Negative binomial distribution, Using given estimates of informative prior
hybrid_fixed_prior_rej_negbinom_simu( t_prior, theta_prior, r_t, r_c, kappa, nu_t, rand_ratio, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )hybrid_fixed_prior_rej_negbinom_simu( t_prior, theta_prior, r_t, r_c, kappa, nu_t, rand_ratio, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )
t_prior |
Prior information time: the prior distribution of theta = -log(r_t/r_c) is assumed to have a normal distribution, and the Fisher's information time is the reciprocal of the variance of the prior distribution; set t_prior = 0 for a non-informative prior |
theta_prior |
Mean effect size of prior: the mean of the normal prior distribution of theta = -log(r_t/r_c); set theta_prior = 0 for a non-informative prior |
r_t |
Event rate in the test arm; the effect size is theta = -log(r_t/r_c) and a smaller event rate is assumed to be associated with better efficacy. Enter r_t = r_c to verify type I error control and r_t < r_c for the power simulation |
r_c |
Event rate in the control arm; the effect size is theta = -log(r_t/r_c) |
kappa |
Dispersion parameter of the negative binomial distribution; must be > 0 |
nu_t |
Exposure time for the negative binomial distribution; must be > 0 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz |
Planned sample size: the planned sample size at the final analysis |
N_max |
Maximum sample size: a user chosen, maximum allowable sample size for sample size modification |
theta_cut |
Futility threshold: a cut-off for theta_hat. In the simulations, if theta_hat at the interim analysis is at or below theta_cut the trial is considered futile and proceeds to the planned sample size without any change |
cp_min |
Promising zone cp_min: CP_min in the promising zone sample size re-estimation; a conditional power below cp_min is the unfavorable zone. The user should select this value from Mehta, Pocock (2011) |
gamma |
Parameter chosen for the credible interval modified predictive power (CI_MPP); it defines the (1 - gamma) credible interval of the posterior for theta, e.g. gamma = 0.5 gives a 50 percent interval |
alpha_2 |
Type I error for the two-sided test |
beta |
Conditional type II error; 1 - beta is the target conditional power used for the sample size re-estimation |
sim_num |
Number of simulation replicates |
A named list with the following components:
A data frame with the columns "Prior type", "SSR method", "Rejection rate" and "Mean sample size", with one row per sample size re-estimation (SSR) method. The methods compared are the planned sample size with no SSR, the promising zone SSR, the SSR using the true effect size with conditional power, and, for both the non-informative prior and the informative prior, the cp SSR, pp SSR, mpp SSR and ci_mpp SSR methods
The rate at which the observed theta_hat at the interim analysis is at or below theta_cut, so that the trial proceeds without any sample size change
A data frame of the references for the methods compared
Gao, P. Improving the effectiveness of sample size re-estimation: an operating characteristic focused, hybrid frequentist-Bayesian approach, Statistics in Medicine, 2025; 44:e10310. To link to this article: https://onlinelibrary.wiley.com/doi/10.1002/sim.10310.
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
C. R. Mehta, S. J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine. Volume30, Issue28, 10 December 2011, Pages 3267-3284
hybrid_fixed_prior_rej_negbinom_simu(t_prior = 5, theta_prior = 0.2, r_t = 1, r_c = 1, kappa = 1, nu_t = 1, rand_ratio = 1, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)hybrid_fixed_prior_rej_negbinom_simu(t_prior = 5, theta_prior = 0.2, r_t = 1, r_c = 1, kappa = 1, nu_t = 1, rand_ratio = 1, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)
Hybrid: Simulations, Normal distribution, Using given estimates of informative prior
hybrid_fixed_prior_rej_normal_simu( t_prior, theta_prior, mu, sigma, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )hybrid_fixed_prior_rej_normal_simu( t_prior, theta_prior, mu, sigma, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )
t_prior |
Prior information time: the prior distribution of theta = mu_t - mu_c is assumed to have a normal distribution, and the Fisher's information time is the reciprocal of the variance of the prior distribution; set t_prior = 0 for a non-informative prior |
theta_prior |
Mean effect size of prior: the mean of the normal prior distribution of theta = mu_t - mu_c; set theta_prior = 0 for a non-informative prior |
mu |
Effect size (difference in means) of the current trial: theta = mu_t - mu_c, where mu_t is the mean of the treatment arm and mu_c is the mean of the control arm. In the simulations the test arm is generated from a normal distribution with mean mu and the control arm from a normal distribution with mean 0, both with standard deviation sigma. Enter mu = 0 to verify type I error control and mu > 0 for the power simulation |
sigma |
Common standard deviation shared by the test and control arms; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz |
Planned sample size: the planned sample size at the final analysis |
N_max |
Maximum sample size: a user chosen, maximum allowable sample size for sample size modification |
theta_cut |
Futility threshold: a cut-off for theta_hat. In the simulations, if theta_hat at the interim analysis is at or below theta_cut the trial is considered futile and proceeds to the planned sample size without any change |
cp_min |
Promising zone cp_min: CP_min in the promising zone sample size re-estimation; a conditional power below cp_min is the unfavorable zone. The user should select this value from Mehta, Pocock (2011) |
gamma |
Parameter chosen for the credible interval modified predictive power (CI_MPP); it defines the (1 - gamma) credible interval of the posterior for theta, e.g. gamma = 0.5 gives a 50 percent interval |
alpha_2 |
Type I error for the two-sided test |
beta |
Conditional type II error; 1 - beta is the target conditional power used for the sample size re-estimation |
sim_num |
Number of simulation replicates |
A named list with the following components:
A data frame with the columns "Prior type", "SSR method", "Rejection rate" and "Mean sample size", with one row per sample size re-estimation (SSR) method. The methods compared are the planned sample size with no SSR, the promising zone SSR, the SSR using the true effect size with conditional power, and, for both the non-informative prior and the informative prior, the cp SSR, pp SSR, mpp SSR and ci_mpp SSR methods
The rate at which the observed theta_hat at the interim analysis is at or below theta_cut, so that the trial proceeds without any sample size change
A data frame of the references for the methods compared
Gao, P. Improving the effectiveness of sample size re-estimation: an operating characteristic focused, hybrid frequentist-Bayesian approach, Statistics in Medicine, 2025; 44:e10310. To link to this article: https://onlinelibrary.wiley.com/doi/10.1002/sim.10310.
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
C. R. Mehta, S. J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine. Volume30, Issue28, 10 December 2011, Pages 3267-3284
hybrid_fixed_prior_rej_normal_simu(t_prior = 5, theta_prior = 0.2, mu = 0, sigma = 1, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)hybrid_fixed_prior_rej_normal_simu(t_prior = 5, theta_prior = 0.2, mu = 0, sigma = 1, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)
Hybrid: Simulations, Poisson distribution, Using given estimates of informative prior
hybrid_fixed_prior_rej_poisson_simu( t_prior, theta_prior, lambda_t, lambda_c, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )hybrid_fixed_prior_rej_poisson_simu( t_prior, theta_prior, lambda_t, lambda_c, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )
t_prior |
Prior information time: the prior distribution of theta = lambda_t - lambda_c is assumed to have a normal distribution, and the Fisher's information time is the reciprocal of the variance of the prior distribution; set t_prior = 0 for a non-informative prior |
theta_prior |
Mean effect size of prior: the mean of the normal prior distribution of theta = lambda_t - lambda_c; set theta_prior = 0 for a non-informative prior |
lambda_t |
Event rate in the test arm of the current trial: the number of events per subject, generated as rpois(lambda_t). The effect size is theta = lambda_t - lambda_c. Enter lambda_t = lambda_c to verify type I error control and lambda_t > lambda_c for the power simulation |
lambda_c |
Event rate in the control arm of the current trial, generated as rpois(lambda_c); the effect size is theta = lambda_t - lambda_c |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz |
Planned sample size: the planned sample size at the final analysis |
N_max |
Maximum sample size: a user chosen, maximum allowable sample size for sample size modification |
theta_cut |
Futility threshold: a cut-off for theta_hat. In the simulations, if theta_hat at the interim analysis is at or below theta_cut the trial is considered futile and proceeds to the planned sample size without any change |
cp_min |
Promising zone cp_min: CP_min in the promising zone sample size re-estimation; a conditional power below cp_min is the unfavorable zone. The user should select this value from Mehta, Pocock (2011) |
gamma |
Parameter chosen for the credible interval modified predictive power (CI_MPP); it defines the (1 - gamma) credible interval of the posterior for theta, e.g. gamma = 0.5 gives a 50 percent interval |
alpha_2 |
Type I error for the two-sided test |
beta |
Conditional type II error; 1 - beta is the target conditional power used for the sample size re-estimation |
sim_num |
Number of simulation replicates |
A named list with the following components:
A data frame with the columns "Prior type", "SSR method", "Rejection rate" and "Mean sample size", with one row per sample size re-estimation (SSR) method. The methods compared are the planned sample size with no SSR, the promising zone SSR, the SSR using the true effect size with conditional power, and, for both the non-informative prior and the informative prior, the cp SSR, pp SSR, mpp SSR and ci_mpp SSR methods
The rate at which the observed theta_hat at the interim analysis is at or below theta_cut, so that the trial proceeds without any sample size change
A data frame of the references for the methods compared
Gao, P. Improving the effectiveness of sample size re-estimation: an operating characteristic focused, hybrid frequentist-Bayesian approach, Statistics in Medicine, 2025; 44:e10310. To link to this article: https://onlinelibrary.wiley.com/doi/10.1002/sim.10310.
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
C. R. Mehta, S. J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine. Volume30, Issue28, 10 December 2011, Pages 3267-3284
hybrid_fixed_prior_rej_poisson_simu(t_prior = 5, theta_prior = 0.2, lambda_t = 0.5, lambda_c = 0.5, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)hybrid_fixed_prior_rej_poisson_simu(t_prior = 5, theta_prior = 0.2, lambda_t = 0.5, lambda_c = 0.5, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)
Hybrid: Simulations, Survival analysis, Using given estimates of informative prior
hybrid_fixed_prior_rej_survival_simu( t_prior, theta_prior, HR, lambda, s, alpha_2, surv_rate, evt_num, evt_max, HR_cut, cp_min, gamma, samp_multip, sim_num, beta )hybrid_fixed_prior_rej_survival_simu( t_prior, theta_prior, HR, lambda, s, alpha_2, surv_rate, evt_num, evt_max, HR_cut, cp_min, gamma, samp_multip, sim_num, beta )
t_prior |
Prior information time: the prior distribution of theta = -log(HR) is assumed to have a normal distribution, and the Fisher's information time is the reciprocal of the variance of the prior distribution; set t_prior = 0 for a non-informative prior |
theta_prior |
Mean effect size of prior: the mean of the normal prior distribution of theta = -log(HR); set theta_prior = 0 for a non-informative prior |
HR |
Hazards ratio of test arm vs. the control arm. The effect size is theta = -log(HR) and a smaller HR indicates better efficacy. Enter HR = 1 to verify type I error control and HR < 1 for the power simulation |
lambda |
Baseline hazard rate of the control arm for the exponential survival times: the control arm is generated as rexp(lambda) and the test arm as rexp(HR * lambda), so lambda is a rate (not a log-hazard) and must be > 0. It also sets the censoring time T = -log(surv_rate)/lambda |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test |
surv_rate |
Survival rate used to translate the number of events into a sample size; it is the survival probability at the (administrative) censoring time T = -log(surv_rate)/lambda, 0 < surv_rate < 1 |
evt_num |
Planned number of events: the planned number of events at the final analysis |
evt_max |
Maximum number of events: a user chosen, maximum allowable number of events for sample size modification |
HR_cut |
Futility hazards ratio threshold: a cut-off for HR_hat. In the simulations, if the estimated hazards ratio at the interim analysis is at or above HR_cut the trial is considered futile and proceeds to the planned number of events without any change; it is converted internally to theta_cut = -log(HR_cut) |
cp_min |
Promising zone cp_min: CP_min in the promising zone sample size re-estimation; a conditional power below cp_min is the unfavorable zone. The user should select this value from Mehta, Pocock (2011) |
gamma |
Parameter chosen for the credible interval modified predictive power (CI_MPP); it defines the (1 - gamma) credible interval of the posterior for theta, e.g. gamma = 0.5 gives a 50 percent interval |
samp_multip |
Multiplication factor applied when translating the maximum number of events into the sample size used in the simulations, sampsz = ceiling(samp_multip * evt_max / surv_rate); it must be large enough for the required number of events to be observed |
sim_num |
Number of simulation replicates |
beta |
Conditional type II error; 1 - beta is the target conditional power used for the sample size re-estimation |
A named list with the following components:
A data frame with the columns "Prior type", "SSR method", "Rejection rate" and "Mean sample size", with one row per sample size re-estimation (SSR) method. The methods compared are the planned sample size with no SSR, the promising zone SSR, the SSR using the true effect size with conditional power, and, for both the non-informative prior and the informative prior, the cp SSR, pp SSR, mpp SSR and ci_mpp SSR methods. For this survival design the mean sample size is the mean number of events
The rate at which the estimated hazards ratio at the interim analysis is at or above HR_cut, so that the trial proceeds without any change to the number of events
A data frame of the references for the methods compared
Gao, P. Improving the effectiveness of sample size re-estimation: an operating characteristic focused, hybrid frequentist-Bayesian approach, Statistics in Medicine, 2025; 44:e10310. To link to this article: https://onlinelibrary.wiley.com/doi/10.1002/sim.10310.
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
C. R. Mehta, S. J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine. Volume30, Issue28, 10 December 2011, Pages 3267-3284
hybrid_fixed_prior_rej_survival_simu(t_prior = 10, theta_prior = 0.8, HR = 1, lambda = 1, s = c(0.5,1), alpha_2 = 0.05, surv_rate = 0.5, evt_num = 88, evt_max = 331, HR_cut = 0.95, cp_min = 0.27, gamma = 0.5, samp_multip = 5, beta = 0.1, sim_num = 20)hybrid_fixed_prior_rej_survival_simu(t_prior = 10, theta_prior = 0.8, HR = 1, lambda = 1, s = c(0.5,1), alpha_2 = 0.05, surv_rate = 0.5, evt_num = 88, evt_max = 331, HR_cut = 0.95, cp_min = 0.27, gamma = 0.5, samp_multip = 5, beta = 0.1, sim_num = 20)
Hybrid: Simulations, Binary distribution, Using random informative prior from previous trial
hybrid_random_prior_rej_binary_simu( p_t_0, p_c_0, sampsz_0, p_t, p_c, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )hybrid_random_prior_rej_binary_simu( p_t_0, p_c_0, sampsz_0, p_t, p_c, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )
p_t_0 |
Response rate (probability of response) in the test arm of the previous trial, p_t0; the previous-trial effect size is theta_0 = p_t_0 - p_c_0 |
p_c_0 |
Response rate (probability of response) in the control arm of the previous trial |
sampsz_0 |
Sample size per arm from the previous trial, N_0; random samples of this size are drawn from the previous trial to form the random prior |
p_t |
Response rate (probability of response) in the test arm of the current trial; 0 < p_t < 1. The effect size is theta = p_t - p_c. Enter p_t = p_c to verify type I error control and p_t > p_c for the power simulation |
p_c |
Response rate (probability of response) in the control arm of the current trial; 0 < p_c < 1. The effect size is theta = p_t - p_c |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz |
Planned sample size: the planned sample size at the final analysis |
N_max |
Maximum sample size: a user chosen, maximum allowable sample size for sample size modification |
theta_cut |
Futility threshold: a cut-off for theta_hat. In the simulations, if theta_hat at the interim analysis is at or below theta_cut the trial is considered futile and proceeds to the planned sample size without any change |
cp_min |
Promising zone cp_min: CP_min in the promising zone sample size re-estimation; a conditional power below cp_min is the unfavorable zone. The user should select this value from Mehta, Pocock (2011) |
gamma |
Parameter chosen for the credible interval modified predictive power (CI_MPP); it defines the (1 - gamma) credible interval of the posterior for theta, e.g. gamma = 0.5 gives a 50 percent interval |
alpha_2 |
Type I error for the two-sided test |
beta |
Conditional type II error; 1 - beta is the target conditional power used for the sample size re-estimation |
sim_num |
Number of simulation replicates |
A named list with the following components:
A data frame with the columns "Prior type", "SSR method", "Rejection rate" and "Mean sample size", with one row per sample size re-estimation (SSR) method. The methods compared are the planned sample size with no SSR, the promising zone SSR, the SSR using the true effect size with conditional power, and, for both the non-informative prior and the informative prior, the cp SSR, pp SSR, mpp SSR and ci_mpp SSR methods
The rate at which the observed theta_hat at the interim analysis is at or below theta_cut, so that the trial proceeds without any sample size change
A data frame of the references for the methods compared
Gao, P. Improving the effectiveness of sample size re-estimation: an operating characteristic focused, hybrid frequentist-Bayesian approach, Statistics in Medicine, 2025; 44:e10310. To link to this article: https://onlinelibrary.wiley.com/doi/10.1002/sim.10310.
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
C. R. Mehta, S. J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine. Volume30, Issue28, 10 December 2011, Pages 3267-3284
hybrid_random_prior_rej_binary_simu(p_t_0 = 0.4, p_c_0 = 0.3, sampsz_0 = 60, p_t = 0.3, p_c = 0.3, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)hybrid_random_prior_rej_binary_simu(p_t_0 = 0.4, p_c_0 = 0.3, sampsz_0 = 60, p_t = 0.3, p_c = 0.3, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)
Hybrid: Simulations, Negative binomial distribution, Using random informative prior from previous trial
hybrid_random_prior_rej_negbinom_simu( r_t_0, r_c_0, sampsz_0, r_t, r_c, kappa, nu_t, rand_ratio, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )hybrid_random_prior_rej_negbinom_simu( r_t_0, r_c_0, sampsz_0, r_t, r_c, kappa, nu_t, rand_ratio, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )
r_t_0 |
Event rate in the test arm of the previous trial, r_t0; the previous-trial effect size is theta_0 = -log(r_t_0/r_c_0) |
r_c_0 |
Event rate in the control arm of the previous trial, r_c0 |
sampsz_0 |
Sample size per arm from the previous trial, N_0; random samples of this size are drawn from the previous trial to form the random prior |
r_t |
Event rate in the test arm of the current trial; the effect size is theta = -log(r_t/r_c) and a smaller event rate is assumed to be associated with better efficacy. Enter r_t = r_c to verify type I error control and r_t < r_c for the power simulation |
r_c |
Event rate in the control arm of the current trial; the effect size is theta = -log(r_t/r_c) |
kappa |
Dispersion parameter of the negative binomial distribution; must be > 0 |
nu_t |
Exposure time for the negative binomial distribution; must be > 0 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz |
Planned sample size: the planned sample size at the final analysis |
N_max |
Maximum sample size: a user chosen, maximum allowable sample size for sample size modification |
theta_cut |
Futility threshold: a cut-off for theta_hat. In the simulations, if theta_hat at the interim analysis is at or below theta_cut the trial is considered futile and proceeds to the planned sample size without any change |
cp_min |
Promising zone cp_min: CP_min in the promising zone sample size re-estimation; a conditional power below cp_min is the unfavorable zone. The user should select this value from Mehta, Pocock (2011) |
gamma |
Parameter chosen for the credible interval modified predictive power (CI_MPP); it defines the (1 - gamma) credible interval of the posterior for theta, e.g. gamma = 0.5 gives a 50 percent interval |
alpha_2 |
Type I error for the two-sided test |
beta |
Conditional type II error; 1 - beta is the target conditional power used for the sample size re-estimation |
sim_num |
Number of simulation replicates |
A named list with the following components:
A data frame with the columns "Prior type", "SSR method", "Rejection rate" and "Mean sample size", with one row per sample size re-estimation (SSR) method. The methods compared are the planned sample size with no SSR, the promising zone SSR, the SSR using the true effect size with conditional power, and, for both the non-informative prior and the informative prior, the cp SSR, pp SSR, mpp SSR and ci_mpp SSR methods
The rate at which the observed theta_hat at the interim analysis is at or below theta_cut, so that the trial proceeds without any sample size change
A data frame of the references for the methods compared
Gao, P. Improving the effectiveness of sample size re-estimation: an operating characteristic focused, hybrid frequentist-Bayesian approach, Statistics in Medicine, 2025; 44:e10310. To link to this article: https://onlinelibrary.wiley.com/doi/10.1002/sim.10310.
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
C. R. Mehta, S. J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine. Volume30, Issue28, 10 December 2011, Pages 3267-3284
hybrid_random_prior_rej_negbinom_simu(r_t_0 = 1.2, r_c_0 = 1, sampsz_0 = 50, r_t = 1, r_c = 1, kappa = 1, nu_t = 1, rand_ratio = 1, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)hybrid_random_prior_rej_negbinom_simu(r_t_0 = 1.2, r_c_0 = 1, sampsz_0 = 50, r_t = 1, r_c = 1, kappa = 1, nu_t = 1, rand_ratio = 1, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)
Hybrid: Simulations, Normal distribution, Using random informative prior from previous trial
hybrid_random_prior_rej_normal_simu( mu_0, sigma_0, sampsz_0, mu, sigma, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )hybrid_random_prior_rej_normal_simu( mu_0, sigma_0, sampsz_0, mu, sigma, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )
mu_0 |
Effect size (difference in means) of the previous trial: theta_0 = mu_t0 - mu_c0; in the simulations the previous-trial test arm is generated from a normal distribution with mean mu_0 and the previous-trial control arm from a normal distribution with mean 0, both with standard deviation sigma_0 |
sigma_0 |
Common standard deviation of the test and control arms in the previous trial; must be > 0 |
sampsz_0 |
Sample size per arm from the previous trial, N_0; random samples of this size are drawn from the previous trial to form the random prior |
mu |
Effect size (difference in means) of the current trial: theta = mu_t - mu_c, where mu_t is the mean of the treatment arm and mu_c is the mean of the control arm. In the simulations the test arm is generated from a normal distribution with mean mu and the control arm from a normal distribution with mean 0, both with standard deviation sigma. Enter mu = 0 to verify type I error control and mu > 0 for the power simulation |
sigma |
Common standard deviation shared by the test and control arms; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz |
Planned sample size: the planned sample size at the final analysis |
N_max |
Maximum sample size: a user chosen, maximum allowable sample size for sample size modification |
theta_cut |
Futility threshold: a cut-off for theta_hat. In the simulations, if theta_hat at the interim analysis is at or below theta_cut the trial is considered futile and proceeds to the planned sample size without any change |
cp_min |
Promising zone cp_min: CP_min in the promising zone sample size re-estimation; a conditional power below cp_min is the unfavorable zone. The user should select this value from Mehta, Pocock (2011) |
gamma |
Parameter chosen for the credible interval modified predictive power (CI_MPP); it defines the (1 - gamma) credible interval of the posterior for theta, e.g. gamma = 0.5 gives a 50 percent interval |
alpha_2 |
Type I error for the two-sided test |
beta |
Conditional type II error; 1 - beta is the target conditional power used for the sample size re-estimation |
sim_num |
Number of simulation replicates |
A named list with the following components:
A data frame with the columns "Prior type", "SSR method", "Rejection rate" and "Mean sample size", with one row per sample size re-estimation (SSR) method. The methods compared are the planned sample size with no SSR, the promising zone SSR, the SSR using the true effect size with conditional power, and, for both the non-informative prior and the informative prior, the cp SSR, pp SSR, mpp SSR and ci_mpp SSR methods
The rate at which the observed theta_hat at the interim analysis is at or below theta_cut, so that the trial proceeds without any sample size change
A data frame of the references for the methods compared
Gao, P. Improving the effectiveness of sample size re-estimation: an operating characteristic focused, hybrid frequentist-Bayesian approach, Statistics in Medicine, 2025; 44:e10310. To link to this article: https://onlinelibrary.wiley.com/doi/10.1002/sim.10310.
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
C. R. Mehta, S. J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine. Volume30, Issue28, 10 December 2011, Pages 3267-3284
hybrid_random_prior_rej_normal_simu(mu_0 = 0.3, sigma_0 = 1, sampsz_0 = 60, mu = 0, sigma = 1, s = c(0.5,1), sampsz = 100, N_max = 250, theta_cut = 0.01, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)hybrid_random_prior_rej_normal_simu(mu_0 = 0.3, sigma_0 = 1, sampsz_0 = 60, mu = 0, sigma = 1, s = c(0.5,1), sampsz = 100, N_max = 250, theta_cut = 0.01, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)
Hybrid: Simulations, Poisson distribution, Using random informative prior from previous trial
hybrid_random_prior_rej_poisson_simu( lambda_t_0, lambda_c_0, sampsz_0, lambda_t, lambda_c, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )hybrid_random_prior_rej_poisson_simu( lambda_t_0, lambda_c_0, sampsz_0, lambda_t, lambda_c, s, sampsz, N_max, theta_cut, cp_min, gamma, alpha_2, beta, sim_num )
lambda_t_0 |
Event rate in the test arm of the previous trial, generated as rpois(lambda_t_0); the previous-trial effect size is theta_0 = lambda_t_0 - lambda_c_0 |
lambda_c_0 |
Event rate in the control arm of the previous trial, generated as rpois(lambda_c_0) |
sampsz_0 |
Sample size per arm from the previous trial, N_0; random samples of this size are drawn from the previous trial to form the random prior |
lambda_t |
Event rate in the test arm of the current trial: the number of events per subject, generated as rpois(lambda_t). The effect size is theta = lambda_t - lambda_c. Enter lambda_t = lambda_c to verify type I error control and lambda_t > lambda_c for the power simulation |
lambda_c |
Event rate in the control arm of the current trial, generated as rpois(lambda_c); the effect size is theta = lambda_t - lambda_c |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz |
Planned sample size: the planned sample size at the final analysis |
N_max |
Maximum sample size: a user chosen, maximum allowable sample size for sample size modification |
theta_cut |
Futility threshold: a cut-off for theta_hat. In the simulations, if theta_hat at the interim analysis is at or below theta_cut the trial is considered futile and proceeds to the planned sample size without any change |
cp_min |
Promising zone cp_min: CP_min in the promising zone sample size re-estimation; a conditional power below cp_min is the unfavorable zone. The user should select this value from Mehta, Pocock (2011) |
gamma |
Parameter chosen for the credible interval modified predictive power (CI_MPP); it defines the (1 - gamma) credible interval of the posterior for theta, e.g. gamma = 0.5 gives a 50 percent interval |
alpha_2 |
Type I error for the two-sided test |
beta |
Conditional type II error; 1 - beta is the target conditional power used for the sample size re-estimation |
sim_num |
Number of simulation replicates |
A named list with the following components:
A data frame with the columns "Prior type", "SSR method", "Rejection rate" and "Mean sample size", with one row per sample size re-estimation (SSR) method. The methods compared are the planned sample size with no SSR, the promising zone SSR, the SSR using the true effect size with conditional power, and, for both the non-informative prior and the informative prior, the cp SSR, pp SSR, mpp SSR and ci_mpp SSR methods
The rate at which the observed theta_hat at the interim analysis is at or below theta_cut, so that the trial proceeds without any sample size change
A data frame of the references for the methods compared
Gao, P. Improving the effectiveness of sample size re-estimation: an operating characteristic focused, hybrid frequentist-Bayesian approach, Statistics in Medicine, 2025; 44:e10310. To link to this article: https://onlinelibrary.wiley.com/doi/10.1002/sim.10310.
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
C. R. Mehta, S. J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine. Volume30, Issue28, 10 December 2011, Pages 3267-3284
hybrid_random_prior_rej_poisson_simu(lambda_t_0 = 0.5, lambda_c_0 = 0.3, sampsz_0 = 60, lambda_t = 0.5, lambda_c = 0.5, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)hybrid_random_prior_rej_poisson_simu(lambda_t_0 = 0.5, lambda_c_0 = 0.3, sampsz_0 = 60, lambda_t = 0.5, lambda_c = 0.5, s = c(0.5,1), sampsz = 100, N_max = 150, theta_cut = 0.001, cp_min = 0.27, gamma = 0.5, alpha_2 = 0.05, beta = 0.1, sim_num = 20)
Hybrid: Simulations, Survival analysis, Using random informative prior from previous trial
hybrid_random_prior_rej_survival_simu( HR_0, evt_num_0, HR, lambda, s, alpha_2, surv_rate, evt_num, evt_max, HR_cut, cp_min, gamma, samp_multip, sim_num, beta )hybrid_random_prior_rej_survival_simu( HR_0, evt_num_0, HR, lambda, s, alpha_2, surv_rate, evt_num, evt_max, HR_cut, cp_min, gamma, samp_multip, sim_num, beta )
HR_0 |
Hazards ratio of test vs. control in the previous trial, HR_0; the previous-trial effect size is theta_0 = -log(HR_0) |
evt_num_0 |
Number of events from the previous trial, N_0; random samples are drawn from the previous trial up to this number of events to form the random prior |
HR |
Hazards ratio of test arm vs. the control arm for the current trial. The effect size is theta = -log(HR) and a smaller HR indicates better efficacy. Enter HR = 1 to verify type I error control and HR < 1 for the power simulation |
lambda |
Baseline hazard rate of the control arm for the exponential survival times: the control arm is generated as rexp(lambda) and the test arm as rexp(HR * lambda), so lambda is a rate (not a log-hazard) and must be > 0. It also sets the censoring time T = -log(surv_rate)/lambda for both the previous and the current trial |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test |
surv_rate |
Survival rate used to translate the number of events into a sample size; it is the survival probability at the (administrative) censoring time T = -log(surv_rate)/lambda, 0 < surv_rate < 1 |
evt_num |
Planned number of events: the planned number of events at the final analysis of the current trial |
evt_max |
Maximum number of events: a user chosen, maximum allowable number of events for sample size modification |
HR_cut |
Futility hazards ratio threshold: a cut-off for HR_hat. In the simulations, if the estimated hazards ratio at the interim analysis is at or above HR_cut the trial is considered futile and proceeds to the planned number of events without any change; it is converted internally to theta_cut = -log(HR_cut) |
cp_min |
Promising zone cp_min: CP_min in the promising zone sample size re-estimation; a conditional power below cp_min is the unfavorable zone. The user should select this value from Mehta, Pocock (2011) |
gamma |
Parameter chosen for the credible interval modified predictive power (CI_MPP); it defines the (1 - gamma) credible interval of the posterior for theta, e.g. gamma = 0.5 gives a 50 percent interval |
samp_multip |
Multiplication factor applied when translating the number of events of the previous and the current trial into sample sizes, for example sampsz = ceiling(samp_multip * evt_max / surv_rate); it must be large enough for the required number of events to be observed |
sim_num |
Number of simulation replicates |
beta |
Conditional type II error; 1 - beta is the target conditional power used for the sample size re-estimation |
A named list with the following components:
A data frame with the columns "Prior type", "SSR method", "Rejection rate" and "Mean sample size", with one row per sample size re-estimation (SSR) method. The methods compared are the planned sample size with no SSR, the promising zone SSR, the SSR using the true effect size with conditional power, and, for both the non-informative prior and the informative prior, the cp SSR, pp SSR, mpp SSR and ci_mpp SSR methods. For this survival design the mean sample size is the mean number of events
The rate at which the estimated hazards ratio at the interim analysis is at or above HR_cut, so that the trial proceeds without any change to the number of events
A data frame of the references for the methods compared
Gao, P. Improving the effectiveness of sample size re-estimation: an operating characteristic focused, hybrid frequentist-Bayesian approach, Statistics in Medicine, 2025; 44:e10310. To link to this article: https://onlinelibrary.wiley.com/doi/10.1002/sim.10310.
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
C. R. Mehta, S. J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine. Volume30, Issue28, 10 December 2011, Pages 3267-3284
hybrid_random_prior_rej_survival_simu(HR_0 = 0.63, evt_num_0 = 40, HR = 1, lambda = 1, s = c(0.5,1), alpha_2 = 0.05, surv_rate = 0.5, evt_num = 88, evt_max = 331, HR_cut = 0.95, cp_min = 0.27, gamma = 0.5, samp_multip = 5, beta = 0.1, sim_num = 20)hybrid_random_prior_rej_survival_simu(HR_0 = 0.63, evt_num_0 = 40, HR = 1, lambda = 1, s = c(0.5,1), alpha_2 = 0.05, surv_rate = 0.5, evt_num = 88, evt_max = 331, HR_cut = 0.95, cp_min = 0.27, gamma = 0.5, samp_multip = 5, beta = 0.1, sim_num = 20)
Single arm phase 2 oncology trial: Two-stage design, the hybrid design
hybrid_sz(alpha, beta, p_0, p_low, p_1, d_12_cut)hybrid_sz(alpha, beta, p_0, p_low, p_1, d_12_cut)
alpha |
Type I error for the one-sided test |
beta |
Type II error rate, i.e. 1 - power |
p_0 |
Response rate indicating low activity with insufficient clinical benefit |
p_low |
Minimally clinically beneficial response rate |
p_1 |
Assumed response rate |
d_12_cut |
Sets the minimal difference between n_2 and n_1 for the design; min(n_2 - n_1) |
A named list with the following components:
A matrix with one row per design (minimax, optimal and average) for Simon's design using p_1 as the target response rate, with columns r1, n1, r_2, n_2, EN(p0) (expected sample size under p_0), PET(p0) (probability of early termination under p_0), t1_err, PET_p (PET under p_1), PET_p_low (PET under p_low), power_p (power under p_1) and power_p_low (power under p_low)
The corresponding matrix for Simon's design using p_low as the target response rate
A matrix with one row per design (minimax, optimal and average) and columns n_1, r_1, r_1_e (the trial is stopped for superiority if at least r_1_e responses are observed at the first look), r_2, n_2, t1_err, power (power under p_1), PET_p1, PET_p_low, PET_p0, En_p0 (expected sample size under p_0) and PW_p_low (power under p_low)
R. Simon. Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10:1-10 (1989).
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon's design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
hybrid_sz(alpha=c(0.001,0.024),beta=0.1,p_0=0.2,p_low=0.4,p_1=0.55,d_12_cut=0)hybrid_sz(alpha=c(0.001,0.024),beta=0.1,p_0=0.2,p_low=0.4,p_1=0.55,d_12_cut=0)
Single arm phase 2 oncology trial: The interim analysis can be conducted for both two-stage and three-stage designs
interim_analysis(p_0, r_final, n_inter, n_final, x_inter, beta, N_max)interim_analysis(p_0, r_final, n_inter, n_final, x_inter, beta, N_max)
p_0 |
Response rate indicating low activity with insufficient clinical benefit |
r_final |
Rejection threshold of the original design in terms of the number of responses: r_final = r_2 for a two-stage design and r_final = r_3 for a three-stage design |
n_inter |
Number of patients at the interim look: n_inter = n_1 for a two-stage design and n_inter = n_2 for a three-stage design |
n_final |
Number of patients at the final look: n_final = n_2 for a two-stage design and n_final = n_3 for a three-stage design; must be greater than n_inter |
x_inter |
Number of responses observed at the interim look |
beta |
Conditional type II error rate: 1 - beta is the desired conditional power, which drives the re-estimated sample size |
N_max |
Pre-selected maximum sample size for the trial; the new sample size is truncated at N_max |
A named list with the following components:
Re-estimated total sample size N_new, obtained by converting the required information time into a number of patients and truncating at N_max
Threshold r_new such that the null hypothesis is rejected at the new sample size if at least r_new responses are observed; no continuity correction
The same rejection threshold computed with the 0.5 continuity correction applied to the response count
R. Simon. Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10:1-10 (1989).
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
P. Gao, J.H. Ware, C. Mehta, (2008), Sample size re-estimation for adaptive sequential designs. Journal of Biopharmaceutical Statistics, 18: 1184–1196, 2008.
interim_analysis(p_0=0.2,r_final=17,n_inter=22,n_final=56,x_inter=7,beta=0.1,N_max=120)interim_analysis(p_0=0.2,r_final=17,n_inter=22,n_final=56,x_inter=7,beta=0.1,N_max=120)
Multiple comparisons Group sequential design: Final analysis, With one sample size change, orgin_comp==comp2
masd_est_back_no_drop( s, c_bry, snew, c_bry_new, theta_hat_inter_m1, theta_hat_inter_m1_sd, inter_vt, theta_hat_last_ad_m1, theta_hat_last_ad_m1_sd, last_vt_ad, alpha_2, dist )masd_est_back_no_drop( s, c_bry, snew, c_bry_new, theta_hat_inter_m1, theta_hat_inter_m1_sd, inter_vt, theta_hat_last_ad_m1, theta_hat_last_ad_m1_sd, last_vt_ad, alpha_2, dist )
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
snew |
Vector of new information fractions after the sample size is changed, e.g. c(0.5, 1); the fractions are relative to the new total information and the last element must be 1 |
c_bry_new |
Vector of new critical boundaries after the sample size is changed, one per element of 'snew' |
theta_hat_inter_m1 |
Numeric vector of treatment effect estimates observed at the interim analysis at which the sample size was changed, one per original comparison (dose), e.g. the difference of the observed means or event rates between a test arm and the common control arm |
theta_hat_inter_m1_sd |
Standard error of 'theta_hat_inter_m1'; either a single value shared by all comparisons or a vector with one value per comparison; must be > 0 |
inter_vt |
Index of the interim analysis at which the sample size was changed (1-based); must be less than length(s) |
theta_hat_last_ad_m1 |
Numeric vector of treatment effect estimates observed at the analysis after the sample size change at which the trial was terminated, one per original comparison (dose), so that no doses were dropped; the comparison with the largest estimate is the one for which the point estimate and confidence interval are returned |
theta_hat_last_ad_m1_sd |
Standard error of 'theta_hat_last_ad_m1'; either a single value shared by all comparisons or a vector with one value per comparison; must be > 0 |
last_vt_ad |
Index of the interim or final analysis after the sample size change at which the trial was terminated (1-based); it indexes 'snew' |
alpha_2 |
Type I error for the two-sided test |
dist |
Distribution indicator: 1 = the comparisons (doses) share one common control arm and the endpoint is normally distributed, so that the treatment effect estimates are correlated with correlation 1/2; otherwise the comparisons are treated as independent |
A named list with the following components:
The p-value of the test of the null hypothesis that the treatment effect of the comparison (dose) with the best efficacy is 0
Numeric vector of length 3 giving the lower confidence limit, the conservatively median unbiased point estimate and the upper confidence limit for the comparison (dose) with the best efficacy; this is the two-sided (1 - alpha_2) * 100 percent confidence interval, and its lower limit is consistent with the p-value, in the sense that the p-value < alpha_2/2 if and only if the lower limit is greater than 0
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
masd_est_back_no_drop(s = c(0.3,0.7,1), c_bry = c(4.347011, 2.845787, 2.380956), snew = c(0.5,1), c_bry_new = c(3.310743,2.781890), theta_hat_inter_m1 = c(1.2,1.3,1.5), theta_hat_inter_m1_sd = 1, inter_vt = 1, theta_hat_last_ad_m1 = c(2.2,1.4,1.8), theta_hat_last_ad_m1_sd = 1, last_vt_ad = 2, alpha_2 = 0.05, dist = 1)masd_est_back_no_drop(s = c(0.3,0.7,1), c_bry = c(4.347011, 2.845787, 2.380956), snew = c(0.5,1), c_bry_new = c(3.310743,2.781890), theta_hat_inter_m1 = c(1.2,1.3,1.5), theta_hat_inter_m1_sd = 1, inter_vt = 1, theta_hat_last_ad_m1 = c(2.2,1.4,1.8), theta_hat_last_ad_m1_sd = 1, last_vt_ad = 2, alpha_2 = 0.05, dist = 1)
Multiple comparisons Group sequential design: Final analysis, With one sample size change, orgin_comp>comp2
masd_est_back_with_drop( s, c_bry, snew, c_bry_new, theta_hat_inter_m1, theta_hat_inter_m1_sd, theta_hat_inter_m2, theta_hat_inter_m2_sd, inter_vt, theta_hat_last_ad_m2, theta_hat_last_ad_m2_sd, last_vt_ad, alpha_2, dist )masd_est_back_with_drop( s, c_bry, snew, c_bry_new, theta_hat_inter_m1, theta_hat_inter_m1_sd, theta_hat_inter_m2, theta_hat_inter_m2_sd, inter_vt, theta_hat_last_ad_m2, theta_hat_last_ad_m2_sd, last_vt_ad, alpha_2, dist )
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
snew |
Vector of new information fractions after the sample size is changed, e.g. c(0.5, 1); the fractions are relative to the new total information and the last element must be 1 |
c_bry_new |
Vector of new critical boundaries after the sample size is changed, one per element of 'snew' |
theta_hat_inter_m1 |
Numeric vector of treatment effect estimates observed at the interim analysis at which the sample size was changed, one per original comparison (dose), e.g. the difference of the observed means or event rates between a test arm and the common control arm |
theta_hat_inter_m1_sd |
Standard error of 'theta_hat_inter_m1'; either a single value shared by all comparisons or a vector with one value per comparison; must be > 0 |
theta_hat_inter_m2 |
Numeric vector of treatment effect estimates at the interim analysis for the M^(2) comparisons that remain after the doses (comparisons) dropped for lack of efficacy are removed; each element must appear in 'theta_hat_inter_m1', which is used to map the remaining comparisons back onto the original ones |
theta_hat_inter_m2_sd |
Standard error of 'theta_hat_inter_m2'; either a single value shared by all comparisons or a vector with one value per comparison; must be > 0 |
inter_vt |
Index of the interim analysis at which the sample size was changed (1-based); must be less than length(s) |
theta_hat_last_ad_m2 |
Numeric vector of treatment effect estimates observed at the analysis after the sample size change at which the trial was terminated, one per remaining comparison (dose); the remaining comparison with the largest estimate is the one for which the point estimate and confidence interval are returned |
theta_hat_last_ad_m2_sd |
Standard error of 'theta_hat_last_ad_m2'; either a single value shared by all comparisons or a vector with one value per comparison; must be > 0 |
last_vt_ad |
Index of the interim or final analysis after the sample size change at which the trial was terminated (1-based); it indexes 'snew' |
alpha_2 |
Type I error for the two-sided test |
dist |
Distribution indicator: 1 = the comparisons (doses) share one common control arm and the endpoint is normally distributed, so that the treatment effect estimates are correlated with correlation 1/2; otherwise the comparisons are treated as independent |
A named list with the following components:
The p-value of the test of the null hypothesis that the treatment effect of the remaining comparison (dose) with the best efficacy is 0
Numeric vector of length 3 giving the lower confidence limit, the conservatively median unbiased point estimate and the upper confidence limit for the remaining comparison (dose) with the best efficacy; this is the two-sided (1 - alpha_2) * 100 percent confidence interval, and its lower limit is consistent with the p-value, in the sense that the p-value < alpha_2/2 if and only if the lower limit is greater than 0
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
masd_est_back_with_drop(s = c(0.3,0.7,1), c_bry = c(4.347011,2.845787,2.380956), snew = c(0.5,1), c_bry_new = c(3.310743,2.781890), theta_hat_inter_m1 = c(1.2,1.3,1.5), theta_hat_inter_m1_sd = 1, theta_hat_inter_m2 = c(1.3,1.5), theta_hat_inter_m2_sd = 1, inter_vt = 1, theta_hat_last_ad_m2 = c(2.2,1.4), theta_hat_last_ad_m2_sd = 1, last_vt_ad = 2, alpha_2 = 0.05, dist = 1)masd_est_back_with_drop(s = c(0.3,0.7,1), c_bry = c(4.347011,2.845787,2.380956), snew = c(0.5,1), c_bry_new = c(3.310743,2.781890), theta_hat_inter_m1 = c(1.2,1.3,1.5), theta_hat_inter_m1_sd = 1, theta_hat_inter_m2 = c(1.3,1.5), theta_hat_inter_m2_sd = 1, inter_vt = 1, theta_hat_last_ad_m2 = c(2.2,1.4), theta_hat_last_ad_m2_sd = 1, last_vt_ad = 2, alpha_2 = 0.05, dist = 1)
Multiple comparisons Group sequential design: Final analysis, Without sample size change
masd_est_ci( s, c_bry, theta_hat_last_m1, theta_hat_last_m1_sd, last_vt, alpha_2, dist )masd_est_ci( s, c_bry, theta_hat_last_m1, theta_hat_last_m1_sd, last_vt, alpha_2, dist )
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
theta_hat_last_m1 |
Numeric vector of treatment effect estimates observed at the analysis at which the trial stopped, one per comparison (dose), e.g. the difference of the observed means or event rates between a test arm and the common control arm, or minus the log of the observed hazard ratio; its length gives the number of comparisons |
theta_hat_last_m1_sd |
Standard error of 'theta_hat_last_m1'; either a single value shared by all comparisons or a vector with one value per comparison; must be > 0 |
last_vt |
Index of the analysis at which the trial stopped (1-based); this can be the final analysis, at which s = 1, or an interim analysis, and it must not exceed length(s) |
alpha_2 |
Type I error for the two-sided test |
dist |
Distribution indicator: 1 = the comparisons (doses) share one common control arm and the endpoint is normally distributed, so that the treatment effect estimates are correlated with correlation 1/2; otherwise the comparisons are treated as independent |
A named list with the following components:
The p-value of the test of the null hypothesis that the treatment effect of the dose (comparison) with the best efficacy is 0
Numeric vector of length 3 giving the lower confidence limit, the conservatively median unbiased point estimate and the upper confidence limit for the dose (comparison) with the best efficacy; the lower limit is consistent with the p-value, in the sense that the p-value < alpha_2/2 if and only if the lower limit is greater than 0
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
masd_est_ci(s = c(1/3, 2/3, 1), c_bry = c(3.984298, 3.961146, 2.349682), theta_hat_last_m1 = 0.35, theta_hat_last_m1_sd = 0.6, last_vt = 1, alpha_2 = 0.05, dist = 1)masd_est_ci(s = c(1/3, 2/3, 1), c_bry = c(3.984298, 3.961146, 2.349682), theta_hat_last_m1 = 0.35, theta_hat_last_m1_sd = 0.6, last_vt = 1, alpha_2 = 0.05, dist = 1)
Multiple comparisons Group sequential design: Interim analysis, Non-survival analysis, Use theta_interim
masd_new_samsz( s, c_bry, snew, inter_vt, theta_hat_inter_m1, theta_hat_inter_m1_sd, theta_hat_inter_m2, theta_hat_inter_m2_sd, beta, n_control_inter, dist )masd_new_samsz( s, c_bry, snew, inter_vt, theta_hat_inter_m1, theta_hat_inter_m1_sd, theta_hat_inter_m2, theta_hat_inter_m2_sd, beta, n_control_inter, dist )
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
snew |
Vector of new information fractions after the sample size is changed, e.g. c(0.5, 1); the fractions are relative to the new total information and the last element must be 1 |
inter_vt |
Index of the interim analysis at which the sample size is re-estimated (1-based); must be less than length(s) |
theta_hat_inter_m1 |
Numeric vector of treatment effect estimates observed at the interim analysis for all M^(1) comparisons, e.g. the difference of the observed means or event rates between a test arm and the common control arm, or minus the log of the observed hazard ratio; it is also used as the estimated effect size for the conditional power calculation |
theta_hat_inter_m1_sd |
Standard error of 'theta_hat_inter_m1'; must be > 0 |
theta_hat_inter_m2 |
Numeric vector of treatment effect estimates at the interim analysis for the M^(2) comparisons that remain after the comparisons (doses) dropped for lack of efficacy are removed; each element must appear in 'theta_hat_inter_m1', which is used to map the remaining comparisons back onto the original ones |
theta_hat_inter_m2_sd |
Standard error of 'theta_hat_inter_m2'; must be > 0 |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation |
n_control_inter |
Sample size of the control arm at the interim analysis |
dist |
Distribution indicator: 1 = the comparisons (doses) share one common control arm and the endpoint is normally distributed, so that the treatment effect estimates are correlated with correlation 1/2; otherwise the comparisons are treated as independent |
A named list with the following components:
Character string giving the outcome of the interim hypothesis test: whether the null hypothesis was rejected at the interim analysis, which is the case if the largest Wald statistic reaches the planned critical boundary
Numeric vector of the Wald statistics of the treatment effects observed at the interim analysis, one per original (M^(1)) comparison
Planned critical boundary at the interim analysis at which the sample size is re-estimated
The effect size used for the conditional power calculation, taken to be the treatment effect estimates observed at the interim analysis, one value per original (M^(1)) comparison
The elements of the estimated effect size belonging to the M^(2) comparisons that remain after the doses (comparisons) are dropped
Conditional type I error at the interim analysis, calculated by setting theta = 0 for all comparisons
Conditional power at the interim analysis using all M^(1) comparisons, the estimated effect size and the original critical boundaries
Conditional power for the remaining M^(2) comparisons using the new critical boundary evaluated at the planned sample size (ratio of new to original sample size 1), or 'NA' if the null hypothesis was rejected at the interim analysis or if the largest observed effect estimate is < 0.001
New sample size for the control arm after the sample size re-estimation, or 'NA' if the null hypothesis was rejected at the interim analysis or if the largest observed effect estimate is < 0.001
The new information fractions after the sample size change, i.e. 'snew'
Numeric vector of the new critical boundaries for the new interim analyses (O'Brien-Fleming type), one per element of 'snew', or 'NA' if they were not calculated
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
masd_new_samsz( s = c(1/3,2/3,1), c_bry = c(3.984298, 3.961146, 2.349682), snew = 1, inter_vt = 2, theta_hat_inter_m1 = c(0.2,0.5,0.8), theta_hat_inter_m1_sd = 0.5, theta_hat_inter_m2 = c(0.5,0.8), theta_hat_inter_m2_sd = 0.2, beta = 0.1, n_control_inter = 65, dist = 1)masd_new_samsz( s = c(1/3,2/3,1), c_bry = c(3.984298, 3.961146, 2.349682), snew = 1, inter_vt = 2, theta_hat_inter_m1 = c(0.2,0.5,0.8), theta_hat_inter_m1_sd = 0.5, theta_hat_inter_m2 = c(0.5,0.8), theta_hat_inter_m2_sd = 0.2, beta = 0.1, n_control_inter = 65, dist = 1)
Multiple comparisons Group sequential design: Interim analysis, Survival analysis, Use theta_interim
masd_new_samsz_survival( s, c_bry, snew, inter_vt, theta_hat_inter_m1, theta_hat_inter_m1_sd, theta_hat_inter_m2, theta_hat_inter_m2_sd, beta, tot_events_inter_m1, dist )masd_new_samsz_survival( s, c_bry, snew, inter_vt, theta_hat_inter_m1, theta_hat_inter_m1_sd, theta_hat_inter_m2, theta_hat_inter_m2_sd, beta, tot_events_inter_m1, dist )
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
snew |
Vector of new information fractions after the sample size is changed, e.g. c(0.5, 1); the fractions are relative to the new total information and the last element must be 1 |
inter_vt |
Index of the interim analysis at which the sample size is re-estimated (1-based); must be less than length(s) |
theta_hat_inter_m1 |
Numeric vector of treatment effect estimates observed at the interim analysis for all M^(1) comparisons, i.e. minus the log of the observed hazard ratio for each comparison; it is also used as the estimated effect size for the conditional power calculation |
theta_hat_inter_m1_sd |
Standard error of 'theta_hat_inter_m1'; must be > 0 |
theta_hat_inter_m2 |
Numeric vector of treatment effect estimates at the interim analysis for the M^(2) comparisons that remain after the comparisons (doses) dropped for lack of efficacy are removed; each element must appear in 'theta_hat_inter_m1', which is used to map the remaining comparisons back onto the original ones |
theta_hat_inter_m2_sd |
Standard error of 'theta_hat_inter_m2'; must be > 0 |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation |
tot_events_inter_m1 |
Total number of events observed at the interim analysis across all M^(1) test arms and the common control arm |
dist |
Distribution indicator: 1 = the comparisons (doses) share one common control arm and the endpoint is normally distributed, so that the treatment effect estimates are correlated with correlation 1/2; otherwise the comparisons are treated as independent |
A named list with the following components:
Character string giving the outcome of the interim hypothesis test: whether the null hypothesis was rejected at the interim analysis, which is the case if the largest Wald statistic reaches the planned critical boundary
Numeric vector of the Wald statistics of the treatment effects observed at the interim analysis, one per original (M^(1)) comparison
Planned critical boundary at the interim analysis at which the sample size is re-estimated
The effect size used for the conditional power calculation, taken to be the treatment effect estimates observed at the interim analysis (minus the log of the observed hazard ratio), one value per original (M^(1)) comparison
The elements of the effect size belonging to the M^(2) comparisons that remain after the doses (comparisons) are dropped
Conditional type I error at the interim analysis, calculated by setting theta = 0 for all comparisons
Conditional power at the interim analysis using all M^(1) comparisons, the observed effect size and the original critical boundaries
Conditional power for the remaining M^(2) comparisons using the new critical boundary evaluated at the planned sample size (ratio of new to original sample size 1), or 'NA' if the null hypothesis was rejected at the interim analysis or if max(theta) < 0.001
Number of events between one test arm and the common control arm required after the sample size re-estimation, derived from the interim total events and the ratio of the new to the planned information, or 'NA' if the null hypothesis was rejected at the interim analysis or if max(theta) < 0.001
New total number of events after the sample size re-estimation, across the M^(2) remaining test arms and the common control arm, or 'NA' if the null hypothesis was rejected at the interim analysis or if max(theta) < 0.001
The new information fractions after the sample size change, i.e. 'snew'
Numeric vector of the new critical boundaries for the new interim analyses (O'Brien-Fleming type), one per element of 'snew', or 'NA' if they were not calculated
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
masd_new_samsz_survival(s = c(1/3,2/3,1), c_bry = c(3.984298, 3.961146, 2.349682), snew = 1, inter_vt = 2, theta_hat_inter_m1 = c(1.2,1.5,1.1), theta_hat_inter_m1_sd = 1.2, theta_hat_inter_m2 = c(1.2,1.5), theta_hat_inter_m2_sd = 1.2, beta = 0.1, tot_events_inter_m1 = 65, dist = 1)masd_new_samsz_survival(s = c(1/3,2/3,1), c_bry = c(3.984298, 3.961146, 2.349682), snew = 1, inter_vt = 2, theta_hat_inter_m1 = c(1.2,1.5,1.1), theta_hat_inter_m1_sd = 1.2, theta_hat_inter_m2 = c(1.2,1.5), theta_hat_inter_m2_sd = 1.2, beta = 0.1, tot_events_inter_m1 = 65, dist = 1)
Multiple comparisons Group sequential design: Interim analysis, Non-survival analysis, Input theta
masd_new_samsz_theta( theta, s, c_bry, snew, inter_vt, theta_hat_inter_m1, theta_hat_inter_m1_sd, theta_hat_inter_m2, theta_hat_inter_m2_sd, beta, n_control_inter, dist )masd_new_samsz_theta( theta, s, c_bry, snew, inter_vt, theta_hat_inter_m1, theta_hat_inter_m1_sd, theta_hat_inter_m2, theta_hat_inter_m2_sd, beta, n_control_inter, dist )
theta |
Assumed effect size for each of the M^(1) comparisons, i.e. a vector with length(theta) = length(theta_hat_inter_m1); different values can be entered to check the sensitivity of the conditional power to the effect size |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
snew |
Vector of new information fractions after the sample size is changed, e.g. c(0.5, 1); the fractions are relative to the new total information and the last element must be 1 |
inter_vt |
Index of the interim analysis at which the sample size is re-estimated (1-based); must be less than length(s) |
theta_hat_inter_m1 |
Numeric vector of treatment effect estimates observed at the interim analysis for all M^(1) comparisons, e.g. the difference of the observed means or event rates between a test arm and the common control arm, or minus the log of the observed hazard ratio |
theta_hat_inter_m1_sd |
Standard error of 'theta_hat_inter_m1'; must be > 0 |
theta_hat_inter_m2 |
Numeric vector of treatment effect estimates at the interim analysis for the M^(2) comparisons that remain after the comparisons (doses) dropped for lack of efficacy are removed; each element must appear in 'theta_hat_inter_m1', which is used to map the remaining comparisons back onto the original ones |
theta_hat_inter_m2_sd |
Standard error of 'theta_hat_inter_m2'; must be > 0 |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation |
n_control_inter |
Sample size of the control arm at the interim analysis |
dist |
Distribution indicator: 1 = the comparisons (doses) share one common control arm and the endpoint is normally distributed, so that the treatment effect estimates are correlated with correlation 1/2; otherwise the comparisons are treated as independent |
A named list with the following components:
Character string giving the outcome of the interim hypothesis test: whether the null hypothesis was rejected at the interim analysis, which is the case if the largest Wald statistic reaches the planned critical boundary
Numeric vector of the Wald statistics of the treatment effects observed at the interim analysis, one per original (M^(1)) comparison
Planned critical boundary at the interim analysis at which the sample size is re-estimated
The assumed effect size theta entered by the user, one value per original (M^(1)) comparison
The elements of the assumed effect size belonging to the M^(2) comparisons that remain after the doses (comparisons) are dropped
Conditional type I error at the interim analysis, calculated by setting theta = 0 for all comparisons
Conditional power at the interim analysis using all M^(1) comparisons, the assumed effect size theta and the original critical boundaries
Conditional power for the remaining M^(2) comparisons using the new critical boundary evaluated at the planned sample size (ratio of new to original sample size 1), or 'NA' if the null hypothesis was rejected at the interim analysis or if max(theta) < 0.001
New sample size for the control arm after the sample size re-estimation, or 'NA' if the null hypothesis was rejected at the interim analysis or if max(theta) < 0.001
The new information fractions after the sample size change, i.e. 'snew'
Numeric vector of the new critical boundaries for the new interim analyses (O'Brien-Fleming type), one per element of 'snew', or 'NA' if they were not calculated
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
masd_new_samsz_theta( theta = c(0,0.2,0), s = c(1/3,2/3,1), c_bry = c(3.984298, 3.961146, 2.349682), snew = 1, inter_vt = 2, theta_hat_inter_m1 = c(0.2,0.5,0.8), theta_hat_inter_m1_sd = 0.5, theta_hat_inter_m2 = c(0.5,0.8), theta_hat_inter_m2_sd = 0.2, beta = 0.1, n_control_inter = 65, dist = 1)masd_new_samsz_theta( theta = c(0,0.2,0), s = c(1/3,2/3,1), c_bry = c(3.984298, 3.961146, 2.349682), snew = 1, inter_vt = 2, theta_hat_inter_m1 = c(0.2,0.5,0.8), theta_hat_inter_m1_sd = 0.5, theta_hat_inter_m2 = c(0.5,0.8), theta_hat_inter_m2_sd = 0.2, beta = 0.1, n_control_inter = 65, dist = 1)
Multiple comparisons Group sequential design: Interim analysis, Survival analysis, Input theta
masd_new_samsz_theta_survival( theta, s, c_bry, snew, inter_vt, theta_hat_inter_m1, theta_hat_inter_m1_sd, theta_hat_inter_m2, theta_hat_inter_m2_sd, beta, tot_events_inter_m1, dist )masd_new_samsz_theta_survival( theta, s, c_bry, snew, inter_vt, theta_hat_inter_m1, theta_hat_inter_m1_sd, theta_hat_inter_m2, theta_hat_inter_m2_sd, beta, tot_events_inter_m1, dist )
theta |
Assumed effect size (minus the log of the hazard ratio) for each of the M^(1) comparisons, i.e. a vector with length(theta) = length(theta_hat_inter_m1); different values can be entered to check the sensitivity of the conditional power to the effect size |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry |
Vector of critical boundaries for each analysis |
snew |
Vector of new information fractions after the sample size is changed, e.g. c(0.5, 1); the fractions are relative to the new total information and the last element must be 1 |
inter_vt |
Index of the interim analysis at which the sample size is re-estimated (1-based); must be less than length(s) |
theta_hat_inter_m1 |
Numeric vector of treatment effect estimates observed at the interim analysis for all M^(1) comparisons, i.e. minus the log of the observed hazard ratio for each comparison |
theta_hat_inter_m1_sd |
Standard error of 'theta_hat_inter_m1'; must be > 0 |
theta_hat_inter_m2 |
Numeric vector of treatment effect estimates at the interim analysis for the M^(2) comparisons that remain after the comparisons (doses) dropped for lack of efficacy are removed; each element must appear in 'theta_hat_inter_m1', which is used to map the remaining comparisons back onto the original ones |
theta_hat_inter_m2_sd |
Standard error of 'theta_hat_inter_m2'; must be > 0 |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation |
tot_events_inter_m1 |
Total number of events observed at the interim analysis across all M^(1) test arms and the common control arm |
dist |
Distribution indicator: 1 = the comparisons (doses) share one common control arm and the endpoint is normally distributed, so that the treatment effect estimates are correlated with correlation 1/2; otherwise the comparisons are treated as independent |
A named list with the following components:
Character string giving the outcome of the interim hypothesis test: whether the null hypothesis was rejected at the interim analysis, which is the case if the largest Wald statistic reaches the planned critical boundary
Numeric vector of the Wald statistics of the treatment effects observed at the interim analysis, one per original (M^(1)) comparison
Planned critical boundary at the interim analysis at which the sample size is re-estimated
The assumed effect size theta entered by the user, one value per original (M^(1)) comparison
The elements of the assumed effect size belonging to the M^(2) comparisons that remain after the doses (comparisons) are dropped
Conditional type I error at the interim analysis, calculated by setting theta = 0 for all comparisons
Conditional power at the interim analysis using all M^(1) comparisons, the assumed effect size theta and the original critical boundaries
Conditional power for the remaining M^(2) comparisons using the new critical boundary evaluated at the planned sample size (ratio of new to original sample size 1), or 'NA' if the null hypothesis was rejected at the interim analysis or if max(theta) < 0.001
Number of events between one test arm and the common control arm available for the re-estimation, calculated from the interim total events as 2 * tot_events_inter_m1 / (M^(1) + 1), or 'NA' if the null hypothesis was rejected at the interim analysis or if max(theta) < 0.001
New total number of events after the sample size re-estimation, across the M^(2) remaining test arms and the common control arm, or 'NA' if the null hypothesis was rejected at the interim analysis or if max(theta) < 0.001
The new information fractions after the sample size change, i.e. 'snew'
Numeric vector of the new critical boundaries for the new interim analyses (O'Brien-Fleming type), one per element of 'snew', or 'NA' if they were not calculated
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
masd_new_samsz_theta_survival( theta = c(0,0.2,0), s = c(1/3,2/3,1), c_bry = c(3.984298, 3.961146, 2.349682), snew = 1, inter_vt = 2, theta_hat_inter_m1 = c(1.2,1.5,1.1), theta_hat_inter_m1_sd = 1.2, theta_hat_inter_m2 = c(1.2,1.5), theta_hat_inter_m2_sd = 1.2, beta = 0.1, tot_events_inter_m1 = 65, dist = 1)masd_new_samsz_theta_survival( theta = c(0,0.2,0), s = c(1/3,2/3,1), c_bry = c(3.984298, 3.961146, 2.349682), snew = 1, inter_vt = 2, theta_hat_inter_m1 = c(1.2,1.5,1.1), theta_hat_inter_m1_sd = 1.2, theta_hat_inter_m2 = c(1.2,1.5), theta_hat_inter_m2_sd = 1.2, beta = 0.1, tot_events_inter_m1 = 65, dist = 1)
Multiple comparisons Group sequential design: Simulations, Binary distribution, Operating characteristics
masd_power_simu_adapt_binary( p_t, p_c, direction, s, bry_type, alpha_2, theta_cut, beta, samsz, N_max, sim_num )masd_power_simu_adapt_binary( p_t, p_c, direction, s, bry_type, alpha_2, theta_cut, beta, samsz, N_max, sim_num )
p_t |
Vector of true event rates of the test arms, one per dose; 0 < p_t < 1 |
p_c |
Event rate of the control arm; 0 < p_c < 1 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; when bry_type = 2 this is instead the vector of two-sided type I error levels spent at each look, which must sum to the total two-sided type I error |
theta_cut |
Lack-of-efficacy threshold on the effect-size scale: a test arm is dropped at the interim analysis when its observed effect size is below this value |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
samsz |
Sample size of the control arm used in the simulation; must be > 0 |
N_max |
Maximum sample size for the control arm allowed by the sample size re-estimation; must be >= samsz |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix with one row per look giving the look number, the information fraction, the critical boundary, the rejection rate by look and the rate of sample size increase
Matrix with one row per test arm giving the test-arm event rate, the control-arm event rate, the drop rate by arm and the rejection rate by arm
Named numeric vector holding the rejection rate without and with sample size re-estimation, the rate of sample size increase, the rejection rates conditional on meeting the sample size increase criteria, the simulated type I error rate, the non-type I error rejection rate, the futility rate, the planned and maximum control sample sizes, the mean simulated control sample size and the mean simulated control sample size conditional on an increase
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
masd_power_simu_adapt_binary(p_t = c(0.4,0.4,0.4), p_c = 0.4, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, theta_cut = 0.01, beta = 0.1, samsz = 100, N_max = 200, sim_num = 20)masd_power_simu_adapt_binary(p_t = c(0.4,0.4,0.4), p_c = 0.4, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, theta_cut = 0.01, beta = 0.1, samsz = 100, N_max = 200, sim_num = 20)
Multiple comparisons Group sequential design: Simulations, Negative binomial distribution, Operating characteristics
masd_power_simu_adapt_neg_binomial( r_t, r_c, kappa, direction, s, bry_type, alpha_2, beta, theta_cut, samsz, N_max, sim_num )masd_power_simu_adapt_neg_binomial( r_t, r_c, kappa, direction, s, bry_type, alpha_2, beta, theta_cut, samsz, N_max, sim_num )
r_t |
Vector of true mean rates of the test arms, one per dose; must be > 0 |
r_c |
Mean rate of the control arm; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; when bry_type = 2 this is instead the vector of two-sided type I error levels spent at each look, which must sum to the total two-sided type I error |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
theta_cut |
Lack-of-efficacy threshold on the effect-size scale: a test arm is dropped at the interim analysis when its observed effect size is below this value |
samsz |
Sample size of the control arm used in the simulation; must be > 0 |
N_max |
Maximum sample size for the control arm allowed by the sample size re-estimation; must be >= samsz |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix with one row per look giving the look number, the information fraction, the critical boundary, the rejection rate by look and the rate of sample size increase
Matrix with one row per test arm giving the test-arm mean rate, the control-arm mean rate, the drop rate by arm and the rejection rate by arm
Named numeric vector holding the rejection rate without and with sample size re-estimation, the rate of sample size increase, the rejection rates conditional on meeting the sample size increase criteria, the simulated type I error rate, the non-type I error rejection rate, the futility rate, the planned and maximum control sample sizes, the mean simulated control sample size and the mean simulated control sample size conditional on an increase
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
masd_power_simu_adapt_neg_binomial(r_t = c(1.1,1.1,1.1), r_c = 1.1, kappa = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, beta = 0.1, theta_cut = 0.01, samsz = 100, N_max = 500, sim_num = 20)masd_power_simu_adapt_neg_binomial(r_t = c(1.1,1.1,1.1), r_c = 1.1, kappa = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, beta = 0.1, theta_cut = 0.01, samsz = 100, N_max = 500, sim_num = 20)
Multiple comparisons Group sequential design: Simulations, Normal distribution, Operating characteristics
masd_power_simu_adapt_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, beta, theta_cut, samsz, N_max, sim_num )masd_power_simu_adapt_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, beta, theta_cut, samsz, N_max, sim_num )
mu_t |
Vector of true means of the test arms, one per dose; setting mu_t = mu_c gives the simulated type I error rate |
mu_c |
Mean of the control arm |
sigma_t |
Standard deviation of the test arm; must be > 0 |
sigma_c |
Standard deviation of the control arm; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; when bry_type = 2 this is instead the vector of two-sided type I error levels spent at each look, which must sum to the total two-sided type I error |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
theta_cut |
Lack-of-efficacy threshold on the effect-size scale: a test arm is dropped at the interim analysis when its observed effect size is below this value |
samsz |
Sample size of the control arm used in the simulation; must be > 0 |
N_max |
Maximum sample size for the control arm allowed by the sample size re-estimation; must be >= samsz |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix with one row per look giving the look number, the information fraction, the critical boundary, the rejection rate by look and the rate of sample size increase
Matrix with one row per test arm giving the test-arm mean, the control-arm mean, the drop rate by arm and the rejection rate by arm
Named numeric vector holding the rejection rate without and with sample size re-estimation, the rate of sample size increase, the rejection rates conditional on meeting the sample size increase criteria, the simulated type I error rate, the non-type I error rejection rate, the futility rate, the planned and maximum control sample sizes, the mean simulated control sample size and the mean simulated control sample size conditional on an increase
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
masd_power_simu_adapt_normal(mu_t = c(0,0,0), mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, beta = 0.1, theta_cut = 0.01, samsz = 100, N_max = 500, sim_num = 20)masd_power_simu_adapt_normal(mu_t = c(0,0,0), mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, beta = 0.1, theta_cut = 0.01, samsz = 100, N_max = 500, sim_num = 20)
Multiple comparisons Group sequential design: Simulations, Poisson distribution, Operating characteristics
masd_power_simu_adapt_poisson( lambda_t, lambda_c, direction, s, bry_type, alpha_2, beta, theta_cut, samsz, N_max, sim_num )masd_power_simu_adapt_poisson( lambda_t, lambda_c, direction, s, bry_type, alpha_2, beta, theta_cut, samsz, N_max, sim_num )
lambda_t |
Vector of true mean event counts of the test arms, one per dose; must be > 0 |
lambda_c |
Mean event count of the control arm; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; when bry_type = 2 this is instead the vector of two-sided type I error levels spent at each look, which must sum to the total two-sided type I error |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
theta_cut |
Lack-of-efficacy threshold on the effect-size scale: a test arm is dropped at the interim analysis when its observed effect size is below this value |
samsz |
Sample size of the control arm used in the simulation; must be > 0 |
N_max |
Maximum sample size for the control arm allowed by the sample size re-estimation; must be >= samsz |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix with one row per look giving the look number, the information fraction, the critical boundary, the rejection rate by look and the rate of sample size increase
Matrix with one row per test arm giving the test-arm mean, the control-arm mean, the drop rate by arm and the rejection rate by arm
Named numeric vector holding the rejection rate without and with sample size re-estimation, the rate of sample size increase, the rejection rates conditional on meeting the sample size increase criteria, the simulated type I error rate, the non-type I error rejection rate, the futility rate, the planned and maximum control sample sizes, the mean simulated control sample size and the mean simulated control sample size conditional on an increase
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
masd_power_simu_adapt_poisson(lambda_t = c(1.1,1.1,1.1), lambda_c = 1.1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, beta = 0.1, theta_cut = 0.01, samsz = 100, N_max = 500, sim_num = 20)masd_power_simu_adapt_poisson(lambda_t = c(1.1,1.1,1.1), lambda_c = 1.1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, beta = 0.1, theta_cut = 0.01, samsz = 100, N_max = 500, sim_num = 20)
Multiple comparisons Group sequential design: Simulations, Survival analysis, Operating characteristics
masd_power_simu_adapt_survival( HR, s, bry_type, alpha_2, surv_rate, HR_cut, evt_num, evt_max, beta, sim_num )masd_power_simu_adapt_survival( HR, s, bry_type, alpha_2, surv_rate, HR_cut, evt_num, evt_max, beta, sim_num )
HR |
Vector of hazard ratios of each test arm vs. the control arm; HR < 1 favours the test arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; when bry_type = 2 this is instead the vector of two-sided type I error levels spent at each look, which must sum to the total two-sided type I error |
surv_rate |
Survival rate at the end of follow-up; 0 < surv_rate < 1 |
HR_cut |
Hazard-ratio cut-off for the lack-of-efficacy rule at the interim analysis: a test arm is dropped when its observed hazard ratio is greater than HR_cut |
evt_num |
Planned number of events between one test arm and the control arm by the final analysis; must be > 0 |
evt_max |
Maximum number of events between one test arm and the control arm allowed by the sample size re-estimation; must be >= evt_num |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix with one row per look giving the look number, the information fraction, the critical boundary, the rejection rate by look and the rate of sample size increase
Matrix with one row per test arm giving the hazard ratio, the drop rate by arm and the rejection rate by arm
Named numeric vector holding the rejection rate without and with sample size re-estimation, the rate of sample size increase, the rejection rates conditional on meeting the sample size increase criteria, the simulated type I error rate, the non-type I error rejection rate, the futility rate, the planned and maximum two-arm event numbers and the mean simulated two-arm event numbers with and without conditioning on an increase
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
set.seed(1) masd_power_simu_adapt_survival(HR = c(0.63, 0.65, 0.7), s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, surv_rate = 0.5, HR_cut = 0.95, evt_num = 200, evt_max = 500, beta = 0.1, sim_num = 2)set.seed(1) masd_power_simu_adapt_survival(HR = c(0.63, 0.65, 0.7), s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, surv_rate = 0.5, HR_cut = 0.95, evt_num = 200, evt_max = 500, beta = 0.1, sim_num = 2)
Multiple comparisons Group sequential design: Simulations, Binary distribution, Optimizing adaptive sequential design
masd_simu_OC_binary( groups, p_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )masd_simu_OC_binary( groups, p_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
groups |
Matrix of true event rates of the test arms: each row is one simulation scenario and each column one test arm/dose; 0 < groups < 1 |
p_c |
Event rate of the control arm; 0 < p_c < 1 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s1 |
Vector of information fractions of the first interim analysis schedule, e.g. c(1/3, 2/3, 1); the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule, e.g. c(0.5, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
alpha_2_alps_s1 |
Two-sided type I error levels spent at each look of the schedule s1; must sum to alpha_2 |
alpha_2_alps_s2 |
Two-sided type I error levels spent at each look of the schedule s2; must sum to alpha_2 |
theta_cut |
Lack-of-efficacy threshold on the effect-size scale: a test arm is dropped at the interim analysis when its observed effect size is below this value |
samsz_1 |
Sample size of the control arm used in the simulation for the first sample size scenario; must be > 0 |
samsz_2 |
Sample size of the control arm used in the simulation for the second sample size scenario; must be > 0 |
N_max_1 |
Maximum sample size for the control arm allowed by the sample size re-estimation in the first N_max scenario; must be >= samsz_1 |
N_max_2 |
Maximum sample size for the control arm allowed by the sample size re-estimation in the second N_max scenario; must be >= samsz_2 |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
List holding the O'Brien-Fleming group sequential design for the schedule s1 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the O'Brien-Fleming group sequential design for the schedule s2 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the alpha-spending group sequential design for the schedule s1 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the alpha-spending group sequential design for the schedule s2 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
set.seed(1) masd_simu_OC_binary(groups = rbind(c(0.35,0.4,0.45),c(0.4,0.45,0.5),c(0.35,0.45,0.55), c(0.45,0.5,0.55)), p_c = 0.3, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 50, samsz_2 = 75, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)set.seed(1) masd_simu_OC_binary(groups = rbind(c(0.35,0.4,0.45),c(0.4,0.45,0.5),c(0.35,0.45,0.55), c(0.45,0.5,0.55)), p_c = 0.3, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 50, samsz_2 = 75, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)
Multiple comparisons Group sequential design: Simulations, Negative binomial distribution, Optimizing adaptive sequential design
masd_simu_OC_neg_binomial( groups, r_c, kappa, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )masd_simu_OC_neg_binomial( groups, r_c, kappa, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
groups |
Matrix of true mean rates of the test arms: each row is one simulation scenario and each column one test arm/dose; must be > 0 |
r_c |
Mean rate of the control arm; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s1 |
Vector of information fractions of the first interim analysis schedule, e.g. c(1/3, 2/3, 1); the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule, e.g. c(0.5, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
alpha_2_alps_s1 |
Two-sided type I error levels spent at each look of the schedule s1; must sum to alpha_2 |
alpha_2_alps_s2 |
Two-sided type I error levels spent at each look of the schedule s2; must sum to alpha_2 |
theta_cut |
Lack-of-efficacy threshold on the effect-size scale: a test arm is dropped at the interim analysis when its observed effect size is below this value |
samsz_1 |
Sample size of the control arm used in the simulation for the first sample size scenario; must be > 0 |
samsz_2 |
Sample size of the control arm used in the simulation for the second sample size scenario; must be > 0 |
N_max_1 |
Maximum sample size for the control arm allowed by the sample size re-estimation in the first N_max scenario; must be >= samsz_1 |
N_max_2 |
Maximum sample size for the control arm allowed by the sample size re-estimation in the second N_max scenario; must be >= samsz_2 |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
List holding the O'Brien-Fleming group sequential design for the schedule s1 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the O'Brien-Fleming group sequential design for the schedule s2 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the alpha-spending group sequential design for the schedule s1 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the alpha-spending group sequential design for the schedule s2 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
set.seed(1) masd_simu_OC_neg_binomial( groups = rbind(c(0.2,0.3,0.4),c(0.3,0.4,0.5),c(0.4,0.5,0.6),c(0.5,0.6,0.7)), r_c = 0.35, kappa = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 50, samsz_2 = 75, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)set.seed(1) masd_simu_OC_neg_binomial( groups = rbind(c(0.2,0.3,0.4),c(0.3,0.4,0.5),c(0.4,0.5,0.6),c(0.5,0.6,0.7)), r_c = 0.35, kappa = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 50, samsz_2 = 75, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)
Multiple comparisons Group sequential design: Simulations, Normal distribution, Optimizing adaptive sequential design
masd_simu_OC_normal( groups, mu_c, sigma_t, sigma_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )masd_simu_OC_normal( groups, mu_c, sigma_t, sigma_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
groups |
Matrix of true means of the test arms: each row is one simulation scenario and each column one test arm/dose |
mu_c |
Mean of the control arm |
sigma_t |
Standard deviation of the test arm; must be > 0 |
sigma_c |
Standard deviation of the control arm; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s1 |
Vector of information fractions of the first interim analysis schedule, e.g. c(1/3, 2/3, 1); the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule, e.g. c(0.5, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
alpha_2_alps_s1 |
Two-sided type I error levels spent at each look of the schedule s1; must sum to alpha_2 |
alpha_2_alps_s2 |
Two-sided type I error levels spent at each look of the schedule s2; must sum to alpha_2 |
theta_cut |
Lack-of-efficacy threshold on the effect-size scale: a test arm is dropped at the interim analysis when its observed effect size is below this value |
samsz_1 |
Sample size of the control arm used in the simulation for the first sample size scenario; must be > 0 |
samsz_2 |
Sample size of the control arm used in the simulation for the second sample size scenario; must be > 0 |
N_max_1 |
Maximum sample size for the control arm allowed by the sample size re-estimation in the first N_max scenario; must be >= samsz_1 |
N_max_2 |
Maximum sample size for the control arm allowed by the sample size re-estimation in the second N_max scenario; must be >= samsz_2 |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
List holding the O'Brien-Fleming group sequential design for the schedule s1 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the O'Brien-Fleming group sequential design for the schedule s2 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the alpha-spending group sequential design for the schedule s1 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the alpha-spending group sequential design for the schedule s2 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
set.seed(1) masd_simu_OC_normal(groups = rbind(c(0.3,0.4,0.5),c(0.1,0.2,0.3),c(0.3,0.4,0.5),c(0.1,0.2,0.3)), mu_c = 0.15, sigma_t = 1, sigma_c = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 50, samsz_2 = 75, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)set.seed(1) masd_simu_OC_normal(groups = rbind(c(0.3,0.4,0.5),c(0.1,0.2,0.3),c(0.3,0.4,0.5),c(0.1,0.2,0.3)), mu_c = 0.15, sigma_t = 1, sigma_c = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 50, samsz_2 = 75, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)
Multiple comparisons Group sequential design: Simulations, Poisson distribution, Optimizing adaptive sequential design
masd_simu_OC_poisson( groups, lambda_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )masd_simu_OC_poisson( groups, lambda_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
groups |
Matrix of true mean event counts of the test arms: each row is one simulation scenario and each column one test arm/dose; must be > 0 |
lambda_c |
Mean event count of the control arm; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s1 |
Vector of information fractions of the first interim analysis schedule, e.g. c(1/3, 2/3, 1); the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule, e.g. c(0.5, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
alpha_2_alps_s1 |
Two-sided type I error levels spent at each look of the schedule s1; must sum to alpha_2 |
alpha_2_alps_s2 |
Two-sided type I error levels spent at each look of the schedule s2; must sum to alpha_2 |
theta_cut |
Lack-of-efficacy threshold on the effect-size scale: a test arm is dropped at the interim analysis when its observed effect size is below this value |
samsz_1 |
Sample size of the control arm used in the simulation for the first sample size scenario; must be > 0 |
samsz_2 |
Sample size of the control arm used in the simulation for the second sample size scenario; must be > 0 |
N_max_1 |
Maximum sample size for the control arm allowed by the sample size re-estimation in the first N_max scenario; must be >= samsz_1 |
N_max_2 |
Maximum sample size for the control arm allowed by the sample size re-estimation in the second N_max scenario; must be >= samsz_2 |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
List holding the O'Brien-Fleming group sequential design for the schedule s1 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the O'Brien-Fleming group sequential design for the schedule s2 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the alpha-spending group sequential design for the schedule s1 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
List holding the alpha-spending group sequential design for the schedule s2 and the simulation results for the four combinations of the two planned control sample sizes and the two maximum control sample sizes
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
set.seed(1) masd_simu_OC_poisson( groups = rbind(c(0.2,0.3,0.4),c(0.3,0.4,0.5),c(0.4,0.5,0.6),c(0.5,0.6,0.7)), lambda_c = 0.3, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 50, samsz_2 = 75, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)set.seed(1) masd_simu_OC_poisson( groups = rbind(c(0.2,0.3,0.4),c(0.3,0.4,0.5),c(0.4,0.5,0.6),c(0.5,0.6,0.7)), lambda_c = 0.3, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 50, samsz_2 = 75, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)
Multiple comparisons Group sequential design: Simulations, Survival analysis, Optimizing adaptive sequential design
masd_simu_OC_survival( groups, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, surv_rate, evt_num_1, evt_num_2, evt_max_1, evt_max_2, HR_cut, beta, sim_num )masd_simu_OC_survival( groups, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, surv_rate, evt_num_1, evt_num_2, evt_max_1, evt_max_2, HR_cut, beta, sim_num )
groups |
Matrix of true hazard ratios of each test arm vs. the control arm: each row is one simulation scenario and each column one test arm; must be > 0 |
s1 |
Vector of information fractions of the first interim analysis schedule, e.g. c(1/3, 2/3, 1); the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule, e.g. c(0.5, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
alpha_2_alps_s1 |
Two-sided type I error levels spent at each look of the schedule s1; must sum to alpha_2 |
alpha_2_alps_s2 |
Two-sided type I error levels spent at each look of the schedule s2; must sum to alpha_2 |
surv_rate |
Survival rate at the end of follow-up; 0 < surv_rate < 1 |
evt_num_1 |
Planned number of events between one test arm and the control arm by the final analysis in the first event number scenario; must be > 0 |
evt_num_2 |
Planned number of events between one test arm and the control arm by the final analysis in the second event number scenario; must be > 0 |
evt_max_1 |
Maximum number of events between one test arm and the control arm in the first maximum-event-number scenario; must be >= evt_num_1 |
evt_max_2 |
Maximum number of events between one test arm and the control arm in the second maximum-event-number scenario; must be >= evt_num_2 |
HR_cut |
Hazard-ratio cut-off for the lack-of-efficacy rule at the interim analysis: a test arm is dropped when its observed hazard ratio is greater than HR_cut |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
List holding the O'Brien-Fleming group sequential design for the schedule s1 and the simulation results for the four combinations of the two planned event numbers and the two maximum event numbers
List holding the O'Brien-Fleming group sequential design for the schedule s2 and the simulation results for the four combinations of the two planned event numbers and the two maximum event numbers
List holding the alpha-spending group sequential design for the schedule s1 and the simulation results for the four combinations of the two planned event numbers and the two maximum event numbers
List holding the alpha-spending group sequential design for the schedule s2 and the simulation results for the four combinations of the two planned event numbers and the two maximum event numbers
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
set.seed(1) masd_simu_OC_survival( groups = rbind(c(0.85, 0.87, 0.9), c(0.8, 0.82, 0.85), c(0.75, 0.78, 0.8), c(0.7, 0.75, 0.8)), s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), surv_rate = 0.5, evt_num_1 = 200, evt_num_2 = 300, evt_max_1 = 2000, evt_max_2 = 3000, HR_cut = 0.95, beta = 0.1, sim_num = 2)set.seed(1) masd_simu_OC_survival( groups = rbind(c(0.85, 0.87, 0.9), c(0.8, 0.82, 0.85), c(0.75, 0.78, 0.8), c(0.7, 0.75, 0.8)), s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), surv_rate = 0.5, evt_num_1 = 200, evt_num_2 = 300, evt_max_1 = 2000, evt_max_2 = 3000, HR_cut = 0.95, beta = 0.1, sim_num = 2)
Single arm phase 2 oncology trial: Two-stage design, the midpoint design
midpnt_sz(alpha, beta, p_0, p_low, p_1, q, d_12_cut)midpnt_sz(alpha, beta, p_0, p_low, p_1, q, d_12_cut)
alpha |
Type I error for the one-sided test |
beta |
Type II error rate, i.e. 1 - power |
p_0 |
Response rate indicating low activity with insufficient clinical benefit |
p_low |
Minimally clinically beneficial response rate |
p_1 |
Assumed response rate |
q |
Weight between 0 and 1 used to interpolate the first-look sample size of the mid-point design from the two Simon's designs: n_1 = q * n_1(p_1) + (1 - q) * n_1(p_low), where n_1(p_1) and n_1(p_low) are the first-look sample sizes of Simon's designs with target response rates p_1 and p_low; 0 < q < 1 |
d_12_cut |
Sets the minimal difference between n_2 and n_1 for the design; min(n_2 - n_1) |
A named list with the following components:
A matrix with one row per design (minimax, optimal and average) for Simon's design using p_1 as the target response rate, with columns r1, n1, r_2, n_2, EN(p0) (expected sample size under p_0), PET(p0) (probability of early termination under p_0), t1_err, PET_p (PET under p_1), PET_p_low (PET under p_low), power_p (power under p_1) and power_p_low (power under p_low)
The corresponding matrix for Simon's design using p_low as the target response rate
A matrix with one row per design (minimax, optimal and average) and columns r_1, r_2, n_1, n_2, r_1_e (the trial is stopped for superiority if at least r_1_e responses are observed at the first look), t1_err, PET_p_low, power_p (power under p_1), PET_p0, EN_p0 (expected sample size under p_0), PW_p_low (power under p_low) and q. If q is too large for the probability of early termination under p_low to be below beta, a character string with a warning message is returned instead
R. Simon. Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10:1-10 (1989).
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon's design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
midpnt_sz(alpha = c(0.001,0.024), beta = 0.1, p_0 = 0.2, p_low = 0.33, p_1 = 0.5, q = 0, d_12_cut = 5)midpnt_sz(alpha = c(0.001,0.024), beta = 0.1, p_0 = 0.2, p_low = 0.33, p_1 = 0.5, q = 0, d_12_cut = 5)
Multiple comparisons Group sequential design: Binary distribution, Power calculation, Alpha-spending boundary
MSD_power_alsp_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )MSD_power_alsp_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
p_t |
Vector of event rates of the test arms, one entry per dose or endpoint; 0 < p_t < 1 |
p_c |
Event rate of the common control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Vector of alpha-spending values for the one-sided test, one per analysis and of the same length as s, summing to the overall one-sided type I error; each value must satisfy 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Power of the trial for the given control arm sample size
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_power_alsp_boundary_binary_diff(p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0.1, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.048), sampsz_control = 68, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_power_alsp_boundary_binary_diff(p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0.1, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.048), sampsz_control = 68, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Negative binomial distribution, Power calculation, Alpha-spending boundary
MSD_power_alsp_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa, dist )MSD_power_alsp_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa, dist )
r_t |
Vector of mean rates of the test arms, one entry per dose or endpoint |
r_c |
Mean rate of the common control arm |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Vector of alpha-spending values for the one-sided test, one per analysis and of the same length as s, summing to the overall one-sided type I error; each value must satisfy 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
dist |
Distribution indicator kept for consistency with the sibling functions; it is not used here, because the function always sets dist1 = 2 for the boundary calculation and dist2 = 1 for the information time calculation |
A named list with the following components:
Power of the trial for the given control arm sample size
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
Reference for the negative binomial sample size formula
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_power_alsp_boundary_neg_binomial(r_t=c(0.2,0.3,0.4), r_c=0.6, margin=0.1, direction=1, s=c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 90, rand_ratio = 1, nu_t = 1, kappa = 1, dist = 1)MSD_power_alsp_boundary_neg_binomial(r_t=c(0.2,0.3,0.4), r_c=0.6, margin=0.1, direction=1, s=c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 90, rand_ratio = 1, nu_t = 1, kappa = 1, dist = 1)
Multiple comparisons Group sequential design: Normal distribution, Power calculation, Alpha-spending boundary
MSD_power_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )MSD_power_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
mu_t |
Vector of means of the test arms, one entry per dose or endpoint |
mu_c |
Mean of the common control arm |
std_t |
Common standard deviation of the test arms; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Vector of alpha-spending values for the one-sided test, one per analysis and of the same length as s, summing to the overall one-sided type I error; each value must satisfy 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Power of the trial for the given control arm sample size
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_power_alsp_boundary_normal(mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 148, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_power_alsp_boundary_normal(mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 148, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Poisson distribution, Power calculation, Alpha-spending boundary
MSD_power_alsp_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )MSD_power_alsp_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
lambda_t |
Vector of means (rates) of the test arms, one entry per dose or endpoint |
lambda_c |
Mean (rate) of the common control arm |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Vector of alpha-spending values for the one-sided test, one per analysis and of the same length as s, summing to the overall one-sided type I error; each value must satisfy 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Power of the trial for the given control arm sample size
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_power_alsp_boundary_poisson(lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 67, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_power_alsp_boundary_poisson(lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 67, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Survival analysis, Power calculation, Alpha-spending boundary
MSD_power_alsp_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, events_tot, rand_ratio, dist1, dist2 )MSD_power_alsp_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, events_tot, rand_ratio, dist1, dist2 )
HR |
Vector of hazard ratios of the test arms vs. the control arm, one entry per dose or endpoint |
margin |
Non-inferiority margin on the hazard ratio scale; must be > 0, with margin = 1 for a superiority trial and margin != 1 for a non-inferiority trial (0 < margin < 1 if larger HR is better, margin > 1 if smaller HR is better) |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Vector of alpha-spending values for the one-sided test, one per analysis and of the same length as s, summing to the overall one-sided type I error; each value must satisfy 0 < alpha < 1 |
events_tot |
Total number of events for the trial, used in place of the control arm sample size |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage; dist1 = 2 is the conservative choice |
dist2 |
Distribution indicator for the second stage; dist2 = 1 is the conservative choice |
A named list with the following components:
Power of the trial for the given total number of events
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
Reference for the Schoenfeld event number formula
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_power_alsp_boundary_survival_Schoenfeld(HR = c(0.2,0.3,0.4), margin = 1.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), events_tot = 17, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_power_alsp_boundary_survival_Schoenfeld(HR = c(0.2,0.3,0.4), margin = 1.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), events_tot = 17, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Binary distribution, Power calculation, O'Brien-Fleming boundary
MSD_power_OF_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )MSD_power_OF_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
p_t |
Vector of event rates of the test arms, one entry per dose or endpoint; 0 < p_t < 1 |
p_c |
Event rate of the common control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Power of the trial for the given control arm sample size
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_power_OF_boundary_binary_diff(p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_power_OF_boundary_binary_diff(p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Negative binomial distribution, Power calculation, O'Brien-Fleming boundary
MSD_power_OF_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa, dist1, dist2 )MSD_power_OF_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa, dist1, dist2 )
r_t |
Vector of mean rates of the test arms, one entry per dose or endpoint |
r_c |
Mean rate of the common control arm |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control; 2 = otherwise. It has no effect here: the boundary calculation always uses dist1 = 2 |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control; 2 = otherwise. It has no effect here: the information time calculation always uses dist2 = 1 |
A named list with the following components:
Power of the trial for the given control arm sample size
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
Reference for the negative binomial sample size formula
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_power_OF_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1, dist1 = 1, dist2 = 1)MSD_power_OF_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Normal distribution, Power calculation, O'Brien-Fleming boundary
MSD_power_OF_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )MSD_power_OF_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
mu_t |
Vector of means of the test arms, one entry per dose or endpoint |
mu_c |
Mean of the common control arm |
std_t |
Common standard deviation of the test arms; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Power of the trial for the given control arm sample size
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_power_OF_boundary_normal(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_power_OF_boundary_normal(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Poisson distribution, Power calculation, O'Brien-Fleming boundary
MSD_power_OF_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )MSD_power_OF_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
lambda_t |
Vector of means (rates) of the test arms, one entry per dose or endpoint |
lambda_c |
Mean (rate) of the common control arm |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Power of the trial for the given control arm sample size
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_power_OF_boundary_poisson(lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_power_OF_boundary_poisson(lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Survival analysis, Power calculation, O'Brien-Fleming boundary
MSD_power_OF_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, events_tot, rand_ratio, dist1, dist2 )MSD_power_OF_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, events_tot, rand_ratio, dist1, dist2 )
HR |
Vector of hazard ratios of the test arms vs. the control arm, one entry per dose or endpoint |
margin |
Non-inferiority margin on the hazard ratio scale; must be > 0, with margin = 1 for a superiority trial and margin != 1 for a non-inferiority trial (0 < margin < 1 if larger HR is better, margin > 1 if smaller HR is better) |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
events_tot |
Total number of events for the trial, used in place of the control arm sample size |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage; dist1 = 2 is the conservative choice |
dist2 |
Distribution indicator for the second stage; dist2 = 1 is the conservative choice |
A named list with the following components:
Power of the trial for the given total number of events
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
Reference for the Schoenfeld event number formula
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_power_OF_boundary_survival_Schoenfeld(HR = 0.7, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, events_tot = 100, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_power_OF_boundary_survival_Schoenfeld(HR = 0.7, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, events_tot = 100, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Simulations, Binary distribution, Type I error and power
msd_power_simu_binary( p_t, p_c, direction, s, bry_type, alpha_2, samsz, sim_num )msd_power_simu_binary( p_t, p_c, direction, s, bry_type, alpha_2, samsz, sim_num )
p_t |
Vector of true event rates of the test arms, one per dose; 0 < p_t < 1 |
p_c |
Event rate of the control arm; 0 < p_c < 1 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; when bry_type = 2 this is instead the vector of two-sided type I error levels spent at each look, which must sum to the total two-sided type I error |
samsz |
Sample size of the control arm used in the simulation; must be > 0 |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix with one row per look giving the look number, the information fraction, the critical boundary and the rejection rate by look
Matrix with one row per test arm giving the test-arm event rate, the control-arm event rate and the rejection rate by arm
Named numeric vector holding the rejection rate, the simulated type I error rate, the non-type I error rejection rate, the planned control and total sample sizes and the mean simulated control and total sample sizes
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
msd_power_simu_binary(p_t = c(0.4,0.4), p_c = 0.4, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)msd_power_simu_binary(p_t = c(0.4,0.4), p_c = 0.4, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Multiple comparisons Group sequential design: Simulations, Negative binomial distribution, Type I error and power
msd_power_simu_neg_binomial( r_t, r_c, kappa, direction, s, bry_type, alpha_2, samsz, sim_num )msd_power_simu_neg_binomial( r_t, r_c, kappa, direction, s, bry_type, alpha_2, samsz, sim_num )
r_t |
Vector of true mean rates of the test arms, one per dose; must be > 0 |
r_c |
Mean rate of the control arm; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; when bry_type = 2 this is instead the vector of two-sided type I error levels spent at each look, which must sum to the total two-sided type I error |
samsz |
Sample size of the control arm used in the simulation; must be > 0 |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix with one row per look giving the look number, the information fraction, the critical boundary and the rejection rate by look
Matrix with one row per test arm giving the test-arm mean rate, the control-arm mean rate and the rejection rate by arm
Named numeric vector holding the rejection rate, the simulated type I error rate, the non-type I error rejection rate, the planned control and total sample sizes and the mean simulated control and total sample sizes
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
msd_power_simu_neg_binomial(r_t = c(1.1,1.1,1.1), r_c = 1.1, kappa = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)msd_power_simu_neg_binomial(r_t = c(1.1,1.1,1.1), r_c = 1.1, kappa = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Multiple comparisons Group sequential design: Simulations, Normal distribution, Type I error and power
msd_power_simu_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, samsz, sim_num )msd_power_simu_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, samsz, sim_num )
mu_t |
Vector of true means of the test arms, one per dose |
mu_c |
Mean of the control arm |
sigma_t |
Standard deviation of the test arm; must be > 0 |
sigma_c |
Standard deviation of the control arm; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; when bry_type = 2 this is instead the vector of two-sided type I error levels spent at each look, which must sum to the total two-sided type I error |
samsz |
Sample size of the control arm used in the simulation; must be > 0 |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix with one row per look giving the look number, the information fraction, the critical boundary and the rejection rate by look
Matrix with one row per test arm giving the test-arm mean, the control-arm mean and the rejection rate by arm
Named numeric vector holding the rejection rate, the simulated type I error rate, the non-type I error rejection rate, the planned control and total sample sizes and the mean simulated control and total sample sizes
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
msd_power_simu_normal(mu_t = c(0,0,0), mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)msd_power_simu_normal(mu_t = c(0,0,0), mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Multiple comparisons Group sequential design: Simulations, Poisson distribution, Type I error and power
msd_power_simu_poisson( lambda_c, lambda_t, direction, s, bry_type, alpha_2, samsz, sim_num )msd_power_simu_poisson( lambda_c, lambda_t, direction, s, bry_type, alpha_2, samsz, sim_num )
lambda_c |
Mean event count of the control arm; must be > 0 |
lambda_t |
Vector of true mean event counts of the test arms, one per dose; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; when bry_type = 2 this is instead the vector of two-sided type I error levels spent at each look, which must sum to the total two-sided type I error |
samsz |
Sample size of the control arm used in the simulation; must be > 0 |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix with one row per look giving the look number, the information fraction, the critical boundary and the rejection rate by look
Matrix with one row per test arm giving the test-arm mean, the control-arm mean and the rejection rate by arm
Named numeric vector holding the rejection rate, the simulated type I error rate, the non-type I error rejection rate, the planned control and total sample sizes and the mean simulated control and total sample sizes
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
msd_power_simu_poisson(lambda_c = 0.8, lambda_t = c(0.8,0.8,0.8), direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)msd_power_simu_poisson(lambda_c = 0.8, lambda_t = c(0.8,0.8,0.8), direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Multiple comparisons Group sequential design: Simulations, Survival analysis, Type I error and power
msd_power_simu_survival(HR, s, bry_type, alpha_2, surv_rate, evt_num, sim_num)msd_power_simu_survival(HR, s, bry_type, alpha_2, surv_rate, evt_num, sim_num)
HR |
Vector of hazard ratios of each test arm vs. the control arm; HR < 1 favours the test arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = alpha-spending |
alpha_2 |
Type I error for the two-sided test; when bry_type = 2 this is instead the vector of two-sided type I error levels spent at each look, which must sum to the total two-sided type I error |
surv_rate |
Survival rate at the end of follow-up; 0 < surv_rate < 1 |
evt_num |
Planned number of events between one test arm and the control arm by the final analysis; must be > 0 |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix with one row per look giving the look number, the information fraction, the critical boundary and the rejection rate by look
Matrix with one row per test arm giving the hazard ratio and the rejection rate by arm
Named numeric vector holding the rejection rate, the simulated type I error rate, the non-type I error rejection rate, the planned two-arm and total event numbers and the mean simulated two-arm and total event numbers
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
msd_power_simu_survival(HR = c(0.8,0.8), s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, surv_rate = 0.5, evt_num = 800, sim_num = 20)msd_power_simu_survival(HR = c(0.8,0.8), s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, surv_rate = 0.5, evt_num = 800, sim_num = 20)
Multiple comparisons Group sequential design: Binary distribution, Sample size calculation, Alpha-spending boundary
MSD_Sample_size_alsp_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )MSD_Sample_size_alsp_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
p_t |
Vector of event rates of the test arms, one entry per dose or endpoint; 0 < p_t < 1, and the function returns NA if the largest effect size is below 1e-4 |
p_c |
Event rate of the common control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Vector of alpha-spending values for the one-sided test, one per analysis and of the same length as s, summing to the overall one-sided type I error; each value must satisfy 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Sample size required for a test arm; equal to rand_ratio times the control arm sample size
Sample size required for the common control arm
Vector with one test arm sample size per dose or endpoint, as if a separate two arm trial were conducted for that dose with the same critical boundary
Vector with the matching control arm sample size for each dose or endpoint
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_Sample_size_alsp_boundary_binary_diff( p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_Sample_size_alsp_boundary_binary_diff( p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Negative binomial distribution, Sample size calculation, Alpha-spending boundary
MSD_Sample_size_alsp_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa, dist1, dist2 )MSD_Sample_size_alsp_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa, dist1, dist2 )
r_t |
Vector of mean rates of the test arms, one entry per dose or endpoint; the function returns NA if the largest log rate ratio is below 1e-4 or if any adjusted test arm rate r_t +/- margin is below 1e-4 |
r_c |
Mean rate of the common control arm |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Vector of alpha-spending values for the one-sided test, one per analysis and of the same length as s, summing to the overall one-sided type I error; each value must satisfy 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
dist1 |
Distribution indicator for the first stage; it has no effect here, because the boundary calculation always uses dist1 = 2 |
dist2 |
Distribution indicator for the second stage; it has no effect here, because the information time calculation always uses dist2 = 1 |
A named list with the following components:
Sample size required for a test arm; equal to rand_ratio times the control arm sample size
Sample size required for the common control arm
Vector with one test arm sample size per dose or endpoint, as if a separate two arm trial were conducted for that dose with the same critical boundary
Vector with the matching control arm sample size for each dose or endpoint
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
Reference for the negative binomial sample size formula
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_Sample_size_alsp_boundary_neg_binomial( r_t = c(0.2,0.3,0.4), r_c = 0.6, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1, dist1 = 1, dist2 = 1)MSD_Sample_size_alsp_boundary_neg_binomial( r_t = c(0.2,0.3,0.4), r_c = 0.6, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Normal distribution, Sample size calculation, Alpha-spending boundary
MSD_Sample_size_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )MSD_Sample_size_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
mu_t |
Vector of means of the test arms, one entry per dose or endpoint; the function returns NA if the largest effect size is below 1e-4 |
mu_c |
Mean of the common control arm |
std_t |
Common standard deviation of the test arms; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Vector of alpha-spending values for the one-sided test, one per analysis and of the same length as s, summing to the overall one-sided type I error; each value must satisfy 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Sample size required for a test arm; equal to rand_ratio times the control arm sample size
Sample size required for the common control arm
Vector with one test arm sample size per dose or endpoint, as if a separate two arm trial were conducted for that dose with the same critical boundary
Vector with the matching control arm sample size for each dose or endpoint
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_Sample_size_alsp_boundary_normal( mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_Sample_size_alsp_boundary_normal( mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Poisson distribution, Sample size calculation, Alpha-spending boundary
MSD_Sample_size_alsp_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )MSD_Sample_size_alsp_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
lambda_t |
Vector of means (rates) of the test arms, one entry per dose or endpoint; the function returns NA if the largest effect size is below 1e-4 |
lambda_c |
Mean (rate) of the common control arm |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Vector of alpha-spending values for the one-sided test, one per analysis and of the same length as s, summing to the overall one-sided type I error; each value must satisfy 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Sample size required for a test arm; equal to rand_ratio times the control arm sample size
Sample size required for the common control arm
Vector with one test arm sample size per dose or endpoint, as if a separate two arm trial were conducted for that dose with the same critical boundary
Vector with the matching control arm sample size for each dose or endpoint
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_Sample_size_alsp_boundary_poisson( lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.11, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_Sample_size_alsp_boundary_poisson( lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.11, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Survival analysis, Sample size calculation, Alpha-spending boundary
MSD_Sample_size_alsp_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )MSD_Sample_size_alsp_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
HR |
Vector of hazard ratios of the test arms vs. the control arm, one entry per dose or endpoint; the function returns NA if the largest effect size is below 1e-4 |
margin |
Non-inferiority margin on the hazard ratio scale; must be > 0, with margin = 1 for a superiority trial and margin != 1 for a non-inferiority trial (0 < margin < 1 if larger HR is better, margin > 1 if smaller HR is better) |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Vector of alpha-spending values for the one-sided test, one per analysis and of the same length as s, summing to the overall one-sided type I error; each value must satisfy 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage; dist1 = 2 is the conservative choice |
dist2 |
Distribution indicator for the second stage; dist2 = 1 is the conservative choice |
A named list with the following components:
Total number of events, test arm and control arm combined, required for one dose vs. control comparison
Total number of events required for the whole trial, i.e. all doses and the common control arm
Vector with the combined number of events for each dose and its control, as if a separate two arm trial were conducted for that dose with the same critical boundary
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
Reference for the Schoenfeld event number formula
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_Sample_size_alsp_boundary_survival_Schoenfeld( HR = c(0.2,0.3,0.4), margin = 1.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_Sample_size_alsp_boundary_survival_Schoenfeld( HR = c(0.2,0.3,0.4), margin = 1.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Binary distribution, Sample size calculation, O'Brien-Fleming boundary
MSD_Sample_size_OF_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )MSD_Sample_size_OF_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
p_t |
Vector of event rates of the test arms, one entry per dose or endpoint; 0 < p_t < 1, and the function returns NA if the largest effect size is below 1e-4 |
p_c |
Event rate of the common control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Sample size required for a test arm; equal to rand_ratio times the control arm sample size
Sample size required for the common control arm
Vector with one test arm sample size per dose or endpoint, as if a separate two arm trial were conducted for that dose with the same critical boundary
Vector with the matching control arm sample size for each dose or endpoint
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_Sample_size_OF_boundary_binary_diff( p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_Sample_size_OF_boundary_binary_diff( p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Negative binomial distribution, Sample size calculation, O'Brien-Fleming boundary
MSD_Sample_size_OF_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa, dist1, dist2 )MSD_Sample_size_OF_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa, dist1, dist2 )
r_t |
Vector of mean rates of the test arms, one entry per dose or endpoint; the function returns NA if the largest log rate ratio is below 1e-4 or if any adjusted test arm rate r_t +/- margin is below 1e-4 |
r_c |
Mean rate of the common control arm |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control; 2 = otherwise. It has no effect here: the boundary calculation always uses dist1 = 2 |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control; 2 = otherwise. It has no effect here: the information time calculation always uses dist2 = 1 |
A named list with the following components:
Sample size required for a test arm; equal to rand_ratio times the control arm sample size
Sample size required for the common control arm
Vector with one test arm sample size per dose or endpoint, as if a separate two arm trial were conducted for that dose with the same critical boundary
Vector with the matching control arm sample size for each dose or endpoint
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
Reference for the negative binomial sample size formula
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_Sample_size_OF_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, nu_t = 6, kappa = 1, dist1 = 1, dist2 = 1)MSD_Sample_size_OF_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, nu_t = 6, kappa = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Normal distribution, Sample size calculation, O'Brien-Fleming boundary
MSD_Sample_size_OF_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )MSD_Sample_size_OF_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
mu_t |
Vector of means of the test arms, one entry per dose or endpoint; the function returns NA if the largest effect size is below 1e-4 |
mu_c |
Mean of the common control arm |
std_t |
Common standard deviation of the test arms; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Sample size required for a test arm; equal to rand_ratio times the control arm sample size
Sample size required for the common control arm
Vector with one test arm sample size per dose or endpoint, as if a separate two arm trial were conducted for that dose with the same critical boundary
Vector with the matching control arm sample size for each dose or endpoint
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_Sample_size_OF_boundary_normal( mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_Sample_size_OF_boundary_normal( mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Poisson distribution, Sample size calculation, O'Brien-Fleming boundary
MSD_Sample_size_OF_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )MSD_Sample_size_OF_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
lambda_t |
Vector of means (rates) of the test arms, one entry per dose or endpoint; the function returns NA if the largest effect size is below 1e-4 |
lambda_c |
Mean (rate) of the common control arm |
margin |
Non-inferiority margin; must be >= 0, set to 0 for a superiority trial and > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
dist2 |
Distribution indicator for the second stage: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions; 2 = otherwise |
A named list with the following components:
Sample size required for a test arm; equal to rand_ratio times the control arm sample size
Sample size required for the common control arm
Vector with one test arm sample size per dose or endpoint, as if a separate two arm trial were conducted for that dose with the same critical boundary
Vector with the matching control arm sample size for each dose or endpoint
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_Sample_size_OF_boundary_poisson( lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_Sample_size_OF_boundary_poisson( lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Multiple comparisons Group sequential design: Survival analysis, Sample size calculation, O'Brien-Fleming boundary
MSD_Sample_size_OF_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )MSD_Sample_size_OF_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
HR |
Vector of hazard ratios of the test arms vs. the control arm, one entry per dose or endpoint; the function returns NA if the largest effect size is below 1e-4 |
margin |
Non-inferiority margin on the hazard ratio scale; must be > 0, with margin = 1 for a superiority trial and margin != 1 for a non-inferiority trial (0 < margin < 1 if larger HR is better, margin > 1 if smaller HR is better) |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage; dist1 = 2 is the conservative choice |
dist2 |
Distribution indicator for the second stage; dist2 = 1 is the conservative choice |
A named list with the following components:
Total number of events, test arm and control arm combined, required for one dose vs. control comparison
Total number of events required for the whole trial, i.e. all doses and the common control arm
Vector with the combined number of events for each dose and its control, as if a separate two arm trial were conducted for that dose with the same critical boundary
Boundary output of the closed testing procedure: a list with one element per comparison level (the M-level boundary first, down to the 1-level boundary), each element a list containing the vectors 'critical bounadry', 'alpha spent at each interim analysis', 'cumulative alpha' and 'nominal p-value' for the analyses
Reference for the Schoenfeld event number formula
P. Gao & Y. Li (2024) Adaptive Multiple Comparison Sequential Design (AMCSD) for clinical trials, Journal of Biopharmaceutical Statistics, 34:3, 424-440, DOI:10.1080/10543406.2023.2233590. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2023.2233590
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
MSD_Sample_size_OF_boundary_survival_Schoenfeld( HR = 0.7, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)MSD_Sample_size_OF_boundary_survival_Schoenfeld( HR = 0.7, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Hybrid: New sample size estimation
new_sample_size_hybrid( N_planned, c_bry, t_prior, theta_prior, N_inter, t_inter, theta_hat_inter, gamma, beta )new_sample_size_hybrid( N_planned, c_bry, t_prior, theta_prior, N_inter, t_inter, theta_hat_inter, gamma, beta )
N_planned |
Planned sample size: the planned sample size at the final analysis |
c_bry |
Critical boundary or threshold for rejecting the null hypothesis at the final analysis |
t_prior |
Prior information time: the prior distribution of theta is assumed to have a normal distribution, and the Fisher's information time is the reciprocal of the variance of the prior distribution; set t_prior = 0 for a non-informative prior |
theta_prior |
Mean effect size of prior: the mean of the normal prior distribution of theta; set theta_prior = 0 for a non-informative prior |
N_inter |
Sample size at interim analysis: the number of subjects with data at the interim analysis |
t_inter |
Information time at interim analysis: the Fisher information at the interim analysis, the reciprocal of the variance of the unbiased maximum likelihood estimate theta_hat_inter |
theta_hat_inter |
Estimated effect size at interim analysis: the maximum likelihood estimate of theta at the interim analysis |
gamma |
Parameter chosen for the credible interval modified predictive power (CI_MPP); it defines the (1 - gamma) credible interval of the posterior for theta, e.g. gamma = 0.5 gives a 50 percent interval |
beta |
Conditional type II error; 1 - beta is the target conditional power used for the sample size re-estimation |
A named list with the following components:
A data frame with the columns "prior type", "SSR method" and "new sample size", giving the new sample size for every combination of prior type and sample size re-estimation (SSR) method. The prior types are the non-informative prior (mean of the prior zero and information time zero) and the informative prior entered by the user. Within each prior type the four SSR methods are the conditional power (cp), the predictive power (pp), the modified predictive power (mpp) and the credible interval modified predictive power (ci_mpp), each being the sample size that provides 1 - beta of that power; the sample size is NA when the target conditional power cannot be reached. For the informative prior the cp entry is based on the Bayesian posterior mean of theta
References for the methods
Gao, P. Improving the effectiveness of sample size re-estimation: an operating characteristic focused, hybrid frequentist-Bayesian approach, Statistics in Medicine, 2025; 44:e10310. To link to this article: https://onlinelibrary.wiley.com/doi/10.1002/sim.10310.
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796.
C. R. Mehta, S. J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Statistics in Medicine. Volume30, Issue28, 10 December 2011, Pages 3267-3284
new_sample_size_hybrid(N_planned = 100, c_bry = 1.96, t_prior = 45, theta_prior = 0.5, N_inter = 60, t_inter = 50, theta_hat_inter = 0.3, gamma = 0.5, beta = 0.1)new_sample_size_hybrid(N_planned = 100, c_bry = 1.96, t_prior = 45, theta_prior = 0.5, N_inter = 60, t_inter = 50, theta_hat_inter = 0.3, gamma = 0.5, beta = 0.1)
Single arm phase 2 oncology trial: Simulations: adaptive three stage design
one_arm_ad_3_stg_binary( p_1, p_0, n_1, r_1, n_2, r_2, r_2_e = NA, n_3, r_3, N_max, beta, sim_num )one_arm_ad_3_stg_binary( p_1, p_0, n_1, r_1, n_2, r_2, r_2_e = NA, n_3, r_3, N_max, beta, sim_num )
p_1 |
Assumed response rate |
p_0 |
Response rate indicating low activity with insufficient clinical benefit, i.e. the response rate under the null hypothesis |
n_1 |
Number of patients at first look |
r_1 |
If no more than r_1 responses are observed at the first look, then the trial would be stopped for futility |
n_2 |
Number of patients at second look |
r_2 |
If no more than r_2 responses are observed at the second look, then the trial would be stopped for futility |
r_2_e |
If at least r_2_e responses are observed at the second look, the trial would be stopped early for superiority; if r_2_e is not used in the design, enter NA |
n_3 |
Number of patients at third look |
r_3 |
If more than r_3 responses are observed at the third look, then the null hypothesis will be rejected |
N_max |
Maximum sample size for the trial |
beta |
Type II error rate, i.e. 1 - power |
sim_num |
Number of simulation replicates |
A named list with the following components:
Rate of rejection of the null hypothesis over the simulation replicates, using the response-count boundary without continuity correction
Rate of rejection of the null hypothesis over the simulation replicates when the continuity correction is applied to the response-count boundary
Average sample size over the simulation replicates, EN(p_1)
Rate of early termination for futility over the simulation replicates, PET(p_1)
R. Simon. Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10:1-10 (1989).
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
P. Gao, J.H. Ware, C. Mehta, (2008), Sample size re-estimation for adaptive sequential designs. Journal of Biopharmaceutical Statistics, 18: 1184–1196, 2008.
one_arm_ad_3_stg_binary(p_1=0.2,p_0=0.2,n_1=37,r_1=8,n_2=100,r_2=22,r_2_e=33, n_3=105,r_3=29,N_max=150,beta=0.1,sim_num=20)one_arm_ad_3_stg_binary(p_1=0.2,p_0=0.2,n_1=37,r_1=8,n_2=100,r_2=22,r_2_e=33, n_3=105,r_3=29,N_max=150,beta=0.1,sim_num=20)
Single arm phase 2 oncology trial, Simulations: adaptive two stage design
one_arm_ad_two_stg_binary( p_1, p_0, n_1, r_1, r_1_e = NA, n_2, r_2, N_max, beta, sim_num )one_arm_ad_two_stg_binary( p_1, p_0, n_1, r_1, r_1_e = NA, n_2, r_2, N_max, beta, sim_num )
p_1 |
Assumed response rate |
p_0 |
Response rate indicating low activity with insufficient clinical benefit, i.e. the response rate under the null hypothesis |
n_1 |
Number of patients at first look |
r_1 |
If no more than r_1 responses are observed at the first look, then the trial would be stopped for futility |
r_1_e |
If at least r_1_e responses are observed at the first look, the trial would be stopped early for superiority; if r_1_e is not used in the design, enter NA |
n_2 |
Number of patients at second look |
r_2 |
If more than r_2 responses are observed at the second look, then the null hypothesis will be rejected |
N_max |
Maximum sample size for the trial |
beta |
Type II error rate, i.e. 1 - power |
sim_num |
Number of simulation replicates |
A named list with the following components:
Rate of rejection of the null hypothesis over the simulation replicates, using the response-count boundary without continuity correction
Rate of rejection of the null hypothesis over the simulation replicates when the continuity correction is applied to the response-count boundary
Average sample size over the simulation replicates, EN(p_1)
Rate of early termination for futility over the simulation replicates, PET(p_1)
Matrix summarising the simulation results by case, with one column per case and one row per statistic: case rate, mean sample size, rejection rate, conditional power, proportion of replicates with p_hat > p_1, rejection rate using the critical boundary and corrected rejection rate
R. Simon. Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10:1-10 (1989).
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
P. Gao, J.H. Ware, C. Mehta, (2008), Sample size re-estimation for adaptive sequential designs. Journal of Biopharmaceutical Statistics, 18: 1184–1196, 2008.
set.seed(1) one_arm_ad_two_stg_binary(p_1 = 0.33, p_0 = 0.2, n_1 = 75, r_1 = 18, r_1_e = 26, n_2 = 80, r_2 = 23, N_max = 140, beta = 0.1, sim_num = 20)set.seed(1) one_arm_ad_two_stg_binary(p_1 = 0.33, p_0 = 0.2, n_1 = 75, r_1 = 18, r_1_e = 26, n_2 = 80, r_2 = 23, N_max = 140, beta = 0.1, sim_num = 20)
Single arm phase 2 oncology trial: Simulations, Two stage fixed expanded Simon's design: either hybrid or mid-point design
one_arm_rej_2_stg_binary(p_1, p_0, n_1, r_1_f, r_1_e = NA, n_2, r_2, sim_num)one_arm_rej_2_stg_binary(p_1, p_0, n_1, r_1_f, r_1_e = NA, n_2, r_2, sim_num)
p_1 |
Assumed response rate |
p_0 |
Response rate indicating low activity with insufficient clinical benefit, i.e. the response rate under the null hypothesis |
n_1 |
Number of patients at first look |
r_1_f |
If no more than r_1_f responses are observed at the first look, then the trial would be stopped for futility |
r_1_e |
If at least r_1_e responses are observed at the first look, the trial would be stopped early for superiority; if r_1_e is not used in the design, enter NA |
n_2 |
Number of patients at second look |
r_2 |
If more than r_2 responses are observed at the second look, then the null hypothesis will be rejected |
sim_num |
Number of simulation replicates |
A named list with the following components:
Rate of rejection of the null hypothesis over the simulation replicates
Average sample size over the simulation replicates, EN(p_1)
Rate of early termination for futility over the simulation replicates, PET(p_1)
R. Simon. Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10:1-10 (1989).
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
P. Gao, J.H. Ware, C. Mehta, (2008), Sample size re-estimation for adaptive sequential designs. Journal of Biopharmaceutical Statistics, 18: 1184–1196, 2008.
one_arm_rej_2_stg_binary(p_1=0.2,p_0=0.2,n_1=100,r_1_f=25,r_1_e=33, n_2=105,r_2=29,sim_num=20)one_arm_rej_2_stg_binary(p_1=0.2,p_0=0.2,n_1=100,r_1_f=25,r_1_e=33, n_2=105,r_2=29,sim_num=20)
Single arm phase 2 oncology trial: Simulations, Expanded Simon's design: the three stage design, no sample size change
one_arm_rej_3_stg_binary( p_1, p_0, n_1, r_1, n_2, r_2_f, r_2_e = NA, n_3, r_3, sim_num )one_arm_rej_3_stg_binary( p_1, p_0, n_1, r_1, n_2, r_2_f, r_2_e = NA, n_3, r_3, sim_num )
p_1 |
Assumed response rate |
p_0 |
Response rate indicating low activity with insufficient clinical benefit, i.e. the response rate under the null hypothesis |
n_1 |
Number of patients at first look |
r_1 |
If no more than r_1 responses are observed at the first look, then the trial would be stopped for futility |
n_2 |
Number of patients at second look |
r_2_f |
If no more than r_2_f responses are observed at the second look, then the trial would be stopped for futility |
r_2_e |
If at least r_2_e responses are observed at the second look, the trial would be stopped early for superiority; if r_2_e is not used in the design, enter NA |
n_3 |
Number of patients at third look |
r_3 |
If more than r_3 responses are observed at the third look, then the null hypothesis will be rejected |
sim_num |
Number of simulation replicates |
A named list with the following components:
Rate of rejection of the null hypothesis over the simulation replicates
Average sample size over the simulation replicates, EN(p_1)
Rate of early termination for futility over the simulation replicates, PET(p_1)
R. Simon. Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10:1-10 (1989).
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
P. Gao, J.H. Ware, C. Mehta, (2008), Sample size re-estimation for adaptive sequential designs. Journal of Biopharmaceutical Statistics, 18: 1184–1196, 2008.
one_arm_rej_3_stg_binary(p_1=0.2,p_0=0.2,n_1=37,r_1=8,n_2=100,r_2_f=22,r_2_e=33, n_3=105,r_3=29,sim_num=20)one_arm_rej_3_stg_binary(p_1=0.2,p_0=0.2,n_1=37,r_1=8,n_2=100,r_2_f=22,r_2_e=33, n_3=105,r_3=29,sim_num=20)
Single arm phase 2 oncology trial: Simulations: two stage fixed Simon's design
one_arm_rej_simon_binary(p_1, p_0, n_1, r_1, n_2, r_2, sim_num)one_arm_rej_simon_binary(p_1, p_0, n_1, r_1, n_2, r_2, sim_num)
p_1 |
Assumed response rate |
p_0 |
Response rate indicating low activity with insufficient clinical benefit, i.e. the response rate under the null hypothesis |
n_1 |
Number of patients at first look |
r_1 |
If no more than r_1 responses are observed at the first look, then the trial would be stopped for futility |
n_2 |
Number of patients at second look |
r_2 |
If more than r_2 responses are observed at the second look, then the null hypothesis will be rejected |
sim_num |
Number of simulation replicates |
A named list with the following components:
Rate of rejection of the null hypothesis over the simulation replicates, i.e. the power when p_1 > p_0 and the type I error rate when p_1 = p_0
Average sample size over the simulation replicates, EN(p_1)
Rate of early termination for futility over the simulation replicates, PET(p_1)
R. Simon. Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10:1-10 (1989).
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon’s design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
P. Gao, J.H. Ware, C. Mehta, (2008), Sample size re-estimation for adaptive sequential designs. Journal of Biopharmaceutical Statistics, 18: 1184–1196, 2008.
one_arm_rej_simon_binary(p_1=0.4,p_0=0.2,n_1=37,r_1=9,n_2=53,r_2=16,sim_num=20)one_arm_rej_simon_binary(p_1=0.4,p_0=0.2,n_1=37,r_1=9,n_2=53,r_2=16,sim_num=20)
Plot method for a Simon's two-stage phase II design
## S3 method for class 'ph2simon' plot(x, ...)## S3 method for class 'ph2simon' plot(x, ...)
x |
An object of class 'ph2simon', as returned by the internal 'ph2simon' helper |
... |
Further graphical arguments passed to 'plot' |
'NULL', invisibly. Called for the side effect of drawing the expected-sample-size curve.
Group sequential designs / Adaptive sequential designs: Binary distribution, Superiority trial, Power calculation, Alpha-spending boundary
power_alsp_boundary_binary_diff( p_t, p_c, s, alpha_2, sampsz_control, rand_ratio )power_alsp_boundary_binary_diff( p_t, p_c, s, alpha_2, sampsz_control, rand_ratio )
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test, entered for the alpha-spending boundary as the vector of the two-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha_2 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_alsp_boundary_binary_diff(p_t = 0.2, p_c = 0.6, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), sampsz_control = 27, rand_ratio = 1)power_alsp_boundary_binary_diff(p_t = 0.2, p_c = 0.6, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), sampsz_control = 27, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Binary distribution, Non-inferiority trial, Power calculation, Alpha-spending boundary
power_alsp_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio )power_alsp_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio )
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test, entered for the alpha-spending boundary as the vector of the one-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_alsp_boundary_binary_diff_NI(p_t = 0.2, p_c = 0.6, margin = 0.15, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 14, rand_ratio = 1)power_alsp_boundary_binary_diff_NI(p_t = 0.2, p_c = 0.6, margin = 0.15, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 14, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Superiority trial, Power calculation, Alpha-spending boundary
power_alsp_boundary_neg_binomial( r_t, r_c, s, alpha_2, sampsz_control, rand_ratio, nu_t, kappa )power_alsp_boundary_neg_binomial( r_t, r_c, s, alpha_2, sampsz_control, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test, entered for the alpha-spending boundary as the vector of the two-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha_2 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
power_alsp_boundary_neg_binomial(r_t = 1, r_c = 0.8, s = 1, alpha_2 = 0.05, sampsz_control = 897, rand_ratio = 1, nu_t = 1, kappa = 1)power_alsp_boundary_neg_binomial(r_t = 1, r_c = 0.8, s = 1, alpha_2 = 0.05, sampsz_control = 897, rand_ratio = 1, nu_t = 1, kappa = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Non-inferiority trial, Power calculation, Alpha-spending boundary
power_alsp_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa )power_alsp_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger r_t is better, otherwise smaller r_t is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test, entered for the alpha-spending boundary as the vector of the one-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
power_alsp_boundary_neg_binomial_NI(r_t = 1, r_c = 0.8, margin = 0, direction = 1, s = 1, alpha = 0.025, sampsz_control = 897, rand_ratio = 1, nu_t = 1, kappa = 1)power_alsp_boundary_neg_binomial_NI(r_t = 1, r_c = 0.8, margin = 0, direction = 1, s = 1, alpha = 0.025, sampsz_control = 897, rand_ratio = 1, nu_t = 1, kappa = 1)
Group sequential designs / Adaptive sequential designs: Normal distribution, Superiority trial, Power calculation, Alpha-spending boundary
power_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, s, sampsz_control, rand_ratio, alpha_2 )power_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, s, sampsz_control, rand_ratio, alpha_2 )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
alpha_2 |
Type I error for the two-sided test, entered for the alpha-spending boundary as the vector of the two-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha_2 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_alsp_boundary_normal(mu_t = 0.2, mu_c = 0, std_t = 1, std_c = 1, s = c(1/3,2/3,1), sampsz_control = 527, rand_ratio = 1, alpha_2 = c(0.001,0.001,0.048))power_alsp_boundary_normal(mu_t = 0.2, mu_c = 0, std_t = 1, std_c = 1, s = c(1/3,2/3,1), sampsz_control = 527, rand_ratio = 1, alpha_2 = c(0.001,0.001,0.048))
Group sequential designs / Adaptive sequential designs: Normal distribution, Non-inferiority trial, Power calculation, Alpha-spending boundary
power_alsp_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, sampsz_control, rand_ratio, alpha )power_alsp_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, sampsz_control, rand_ratio, alpha )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
alpha |
Type I error for the one-sided test, entered for the alpha-spending boundary as the vector of the one-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_alsp_boundary_normal_NI(mu_t = 0.2, mu_c = 0, std_t = 1, std_c = 1, margin = 0.15, direction = 1, s = c(1/3,2/3,1), sampsz_control = 534, rand_ratio = 1, alpha = c(0.001,0.001,0.023))power_alsp_boundary_normal_NI(mu_t = 0.2, mu_c = 0, std_t = 1, std_c = 1, margin = 0.15, direction = 1, s = c(1/3,2/3,1), sampsz_control = 534, rand_ratio = 1, alpha = c(0.001,0.001,0.023))
Group sequential designs / Adaptive sequential designs: Poisson distribution, Superiority trial, Power calculation, Alpha-spending boundary
power_alsp_boundary_poisson( lambda_t, lambda_c, s, alpha_2, sampsz_control, rand_ratio )power_alsp_boundary_poisson( lambda_t, lambda_c, s, alpha_2, sampsz_control, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test, entered for the alpha-spending boundary as the vector of the two-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha_2 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_alsp_boundary_poisson(lambda_t = 0.2, lambda_c = 0.4, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), sampsz_control = 158, rand_ratio = 1)power_alsp_boundary_poisson(lambda_t = 0.2, lambda_c = 0.4, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), sampsz_control = 158, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Poisson distribution, Non-inferiority trial, Power calculation, Alpha-spending boundary
power_alsp_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio )power_alsp_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test, entered for the alpha-spending boundary as the vector of the one-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_alsp_boundary_poisson_NI(lambda_t = 0.2, lambda_c = 0.4, margin = 0.15, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 52, rand_ratio = 1)power_alsp_boundary_poisson_NI(lambda_t = 0.2, lambda_c = 0.4, margin = 0.15, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 52, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Superiority trial, Power calculation, Alpha-spending boundary
power_alsp_boundary_survival_Schoenfeld(HR, s, events_tot, alpha_2, rand_ratio)power_alsp_boundary_survival_Schoenfeld(HR, s, events_tot, alpha_2, rand_ratio)
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
events_tot |
Total number of events across both the test arm and the control arm |
alpha_2 |
Type I error for the two-sided test, entered for the alpha-spending boundary as the vector of the two-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha_2 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
power_alsp_boundary_survival_Schoenfeld(HR = 0.8, s = c(0.5,1), events_tot = 841, alpha_2 = c(0.001,0.0499), rand_ratio = 1)power_alsp_boundary_survival_Schoenfeld(HR = 0.8, s = c(0.5,1), events_tot = 841, alpha_2 = c(0.001,0.0499), rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Non-inferiority trial, Power calculation, Alpha-spending boundary
power_alsp_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, events_tot, alpha, rand_ratio )power_alsp_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, events_tot, alpha, rand_ratio )
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
margin |
Non-inferiority margin on the hazard ratio scale; must be > 0, with HR > margin required when direction is 1 and HR < margin required otherwise |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
events_tot |
Total number of events across both the test arm and the control arm |
alpha |
Type I error for the one-sided test, entered for the alpha-spending boundary as the vector of the one-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
power_alsp_boundary_survival_Schoenfeld_NI(HR = 0.8, margin = 1.1, direction = 1, s = c(0.5,1), events_tot = 416, alpha = c(0.001,0.024), rand_ratio = 1)power_alsp_boundary_survival_Schoenfeld_NI(HR = 0.8, margin = 1.1, direction = 1, s = c(0.5,1), events_tot = 416, alpha = c(0.001,0.024), rand_ratio = 1)
Fixed sample designs: Binary distribution, Superiority trial, Power calculation
power_binary_diff(p_t, p_c, alpha_2, n_control, rand_ratio)power_binary_diff(p_t, p_c, alpha_2, n_control, rand_ratio)
p_t |
Event rate of the test arm; 0 < p_t < 1 |
p_c |
Event rate of the control arm; 0 < p_c < 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
n_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A numeric value giving the power.
power_binary_diff(p_t = 0.4, p_c = 0.3, alpha_2 = 0.05, n_control = 53, rand_ratio = 1)power_binary_diff(p_t = 0.4, p_c = 0.3, alpha_2 = 0.05, n_control = 53, rand_ratio = 1)
Fixed sample designs: Binary distribution, Non-inferiority trial, Power calculation
power_binary_diff_NI(p_t, p_c, margin, direction, alpha, n_control, rand_ratio)power_binary_diff_NI(p_t, p_c, margin, direction, alpha, n_control, rand_ratio)
p_t |
Event rate of the test arm; 0 < p_t < 1 |
p_c |
Event rate of the control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; set to 0 for a superiority trial |
direction |
Test direction: 1 = larger values of the endpoint parameter are better, otherwise smaller values are better |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
n_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A numeric value giving the power.
power_binary_diff_NI(p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 0, alpha = 0.025, n_control = 53, rand_ratio = 1)power_binary_diff_NI(p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 0, alpha = 0.025, n_control = 53, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Binary distribution, Superiority trial, Power calculation, O'Brien-Fleming boundary
power_OF_boundary_binary_diff(p_t, p_c, s, alpha_2, sampsz_control, rand_ratio)power_OF_boundary_binary_diff(p_t, p_c, s, alpha_2, sampsz_control, rand_ratio)
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_OF_boundary_binary_diff(p_t = 0.4, p_c = 0.3, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1)power_OF_boundary_binary_diff(p_t = 0.4, p_c = 0.3, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Binary distribution, Non-inferiority trial, Power calculation, O'Brien-Fleming boundary
power_OF_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio )power_OF_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio )
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_OF_boundary_binary_diff_NI(p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1)power_OF_boundary_binary_diff_NI(p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Superiority trial, Power calculation, O'Brien-Fleming boundary
power_OF_boundary_neg_binomial( r_t, r_c, s, alpha_2, sampsz_control, rand_ratio, nu_t, kappa )power_OF_boundary_neg_binomial( r_t, r_c, s, alpha_2, sampsz_control, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
power_OF_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1)power_OF_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Non-inferiority trial, Power calculation, O'Brien-Fleming boundary
power_OF_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa )power_OF_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger r_t is better, otherwise smaller r_t is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
power_OF_boundary_neg_binomial_NI(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1)power_OF_boundary_neg_binomial_NI(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1)
Group sequential designs / Adaptive sequential designs: Normal distribution, Superiority trial, Power calculation, O'Brien-Fleming boundary
power_OF_boundary_normal( mu_t, mu_c, std_t, std_c, s, sampsz_control, rand_ratio, alpha_2 )power_OF_boundary_normal( mu_t, mu_c, std_t, std_c, s, sampsz_control, rand_ratio, alpha_2 )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_OF_boundary_normal(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, s = c(1/3, 2/3, 1), sampsz_control = 100, rand_ratio = 1, alpha_2 = 0.05)power_OF_boundary_normal(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, s = c(1/3, 2/3, 1), sampsz_control = 100, rand_ratio = 1, alpha_2 = 0.05)
Group sequential designs / Adaptive sequential designs: Normal distribution, Non-inferiority trial, Power calculation, O'Brien-Fleming boundary
power_OF_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, sampsz_control, rand_ratio, alpha )power_OF_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, sampsz_control, rand_ratio, alpha )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_OF_boundary_normal_NI(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), sampsz_control = 100, rand_ratio = 1, alpha = 0.025)power_OF_boundary_normal_NI(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), sampsz_control = 100, rand_ratio = 1, alpha = 0.025)
Group sequential designs / Adaptive sequential designs: Poisson distribution, Superiority trial, Power calculation, O'Brien-Fleming boundary
power_OF_boundary_poisson( lambda_t, lambda_c, s, alpha_2, sampsz_control, rand_ratio )power_OF_boundary_poisson( lambda_t, lambda_c, s, alpha_2, sampsz_control, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_OF_boundary_poisson(lambda_t = 1.1, lambda_c = 1.0, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1)power_OF_boundary_poisson(lambda_t = 1.1, lambda_c = 1.0, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Poisson distribution, Non-inferiority trial, Power calculation, O'Brien-Fleming boundary
power_OF_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio )power_OF_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_OF_boundary_poisson_NI(lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1)power_OF_boundary_poisson_NI(lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Superiority trial, Power calculation, O'Brien-Fleming boundary
power_OF_boundary_survival_Schoenfeld(HR, s, events_tot, alpha_2, rand_ratio)power_OF_boundary_survival_Schoenfeld(HR, s, events_tot, alpha_2, rand_ratio)
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
events_tot |
Total number of events across both the test arm and the control arm |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
power_OF_boundary_survival_Schoenfeld(HR = 0.7, s = c(1/3, 2/3, 1), events_tot = 100, alpha_2 = 0.05, rand_ratio = 1)power_OF_boundary_survival_Schoenfeld(HR = 0.7, s = c(1/3, 2/3, 1), events_tot = 100, alpha_2 = 0.05, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Non-inferiority trial, Power calculation, O'Brien-Fleming boundary
power_OF_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, events_tot, alpha, rand_ratio )power_OF_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, events_tot, alpha, rand_ratio )
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
margin |
Non-inferiority margin on the hazard ratio scale; must be > 0, with HR > margin required when direction is 1 and HR < margin required otherwise |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
events_tot |
Total number of events across both the test arm and the control arm |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
power_OF_boundary_survival_Schoenfeld_NI(HR = 0.7, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), events_tot = 100, alpha = 0.025, rand_ratio = 1)power_OF_boundary_survival_Schoenfeld_NI(HR = 0.7, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), events_tot = 100, alpha = 0.025, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Binary distribution, Superiority trial, Power calculation, Pocock boundary
power_pocock_boundary_binary_diff( p_t, p_c, s, alpha_2, sampsz_control, rand_ratio )power_pocock_boundary_binary_diff( p_t, p_c, s, alpha_2, sampsz_control, rand_ratio )
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_pocock_boundary_binary_diff(p_t = 0.4, p_c = 0.3, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1)power_pocock_boundary_binary_diff(p_t = 0.4, p_c = 0.3, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Binary distribution, Non-inferiority trial, Power calculation, Pocock boundary
power_pocock_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio )power_pocock_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio )
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_pocock_boundary_binary_diff_NI(p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1)power_pocock_boundary_binary_diff_NI(p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Superiority trial, Power calculation, Pocock boundary
power_pocock_boundary_neg_binomial( r_t, r_c, s, alpha_2, sampsz_control, rand_ratio, nu_t, kappa )power_pocock_boundary_neg_binomial( r_t, r_c, s, alpha_2, sampsz_control, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
power_pocock_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1)power_pocock_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Non-inferiority trial, Power calculation, Pocock boundary
power_pocock_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa )power_pocock_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger r_t is better, otherwise smaller r_t is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
power_pocock_boundary_neg_binomial_NI(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1)power_pocock_boundary_neg_binomial_NI(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1)
Group sequential designs / Adaptive sequential designs: Normal distribution, Superiority trial, Power calculation, Pocock boundary
power_pocock_boundary_normal( mu_t, mu_c, std_t, std_c, s, sampsz_control, rand_ratio, alpha_2 )power_pocock_boundary_normal( mu_t, mu_c, std_t, std_c, s, sampsz_control, rand_ratio, alpha_2 )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_pocock_boundary_normal(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, s = c(1/3, 2/3, 1), sampsz_control = 100, rand_ratio = 1, alpha_2 = 0.05)power_pocock_boundary_normal(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, s = c(1/3, 2/3, 1), sampsz_control = 100, rand_ratio = 1, alpha_2 = 0.05)
Group sequential designs / Adaptive sequential designs: Normal distribution, Non-inferiority trial, Power calculation, Pocock boundary
power_pocock_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, sampsz_control, rand_ratio, alpha )power_pocock_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, sampsz_control, rand_ratio, alpha )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_pocock_boundary_normal_NI(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), sampsz_control = 100, rand_ratio = 1, alpha = 0.025)power_pocock_boundary_normal_NI(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), sampsz_control = 100, rand_ratio = 1, alpha = 0.025)
Group sequential designs / Adaptive sequential designs: Poisson distribution, Superiority trial, Power calculation, Pocock boundary
power_pocock_boundary_poisson( lambda_t, lambda_c, s, alpha_2, sampsz_control, rand_ratio )power_pocock_boundary_poisson( lambda_t, lambda_c, s, alpha_2, sampsz_control, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_pocock_boundary_poisson(lambda_t = 1.1, lambda_c = 1.0, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1)power_pocock_boundary_poisson(lambda_t = 1.1, lambda_c = 1.0, s = c(1/3, 2/3, 1), alpha_2 = 0.05, sampsz_control = 100, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Poisson distribution, Non-inferiority trial, Power calculation, Pocock boundary
power_pocock_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio )power_pocock_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
power_pocock_boundary_poisson_NI(lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1)power_pocock_boundary_poisson_NI(lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Superiority trial, Power calculation, Pocock boundary
power_pocock_boundary_survival_Schoenfeld( HR, s, events_tot, alpha_2, rand_ratio )power_pocock_boundary_survival_Schoenfeld( HR, s, events_tot, alpha_2, rand_ratio )
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
events_tot |
Total number of events across both the test arm and the control arm |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
power_pocock_boundary_survival_Schoenfeld(HR = 0.7, s = c(1/3, 2/3, 1), events_tot = 100, alpha_2 = 0.05, rand_ratio = 1)power_pocock_boundary_survival_Schoenfeld(HR = 0.7, s = c(1/3, 2/3, 1), events_tot = 100, alpha_2 = 0.05, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Non-inferiority trial, Power calculation, Pocock boundary
power_pocock_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, events_tot, alpha, rand_ratio )power_pocock_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, events_tot, alpha, rand_ratio )
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
margin |
Non-inferiority margin on the hazard ratio scale; must be > 0, with HR > margin required when direction is 1 and HR < margin required otherwise |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
events_tot |
Total number of events across both the test arm and the control arm |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
The achieved power
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
power_pocock_boundary_survival_Schoenfeld_NI(HR = 0.7, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), events_tot = 100, alpha = 0.025, rand_ratio = 1)power_pocock_boundary_survival_Schoenfeld_NI(HR = 0.7, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), events_tot = 100, alpha = 0.025, rand_ratio = 1)
Fixed sample designs: Negative binomial distribution, Superiority trial, Power calculation
power_two_sample_neg_binomial( r_t, r_c, alpha_2, n_control, rand_ratio, nu_t, kappa )power_two_sample_neg_binomial( r_t, r_c, alpha_2, n_control, rand_ratio, nu_t, kappa )
r_t |
Event rate of the test arm for the negative binomial endpoint; must be > 0 |
r_c |
Event rate of the control arm for the negative binomial endpoint; must be > 0 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
n_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time of the test arm; the Poisson part of the variance of the log rate ratio is (1 / nu_t) * (1 / r_c + 1 / (rand_ratio * r_t)), so nu_t must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; adds the over-dispersion term (1 + rand_ratio) * kappa / rand_ratio to the variance of the log rate ratio; must be > 0 |
A named list with the following components:
Power of the two-sided test of r_t = r_c
Character string giving the reference for the Zhu and Lakkis sample size formula
power_two_sample_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, alpha_2 = 0.05, n_control = 53, rand_ratio = 1, nu_t = 6, kappa = 1)power_two_sample_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, alpha_2 = 0.05, n_control = 53, rand_ratio = 1, nu_t = 6, kappa = 1)
Fixed sample designs: Negative binomial distribution, Non-inferiority trial, Power calculation
power_two_sample_neg_binomial_NI( r_t, r_c, margin, direction, alpha, n_control, rand_ratio, nu_t, kappa )power_two_sample_neg_binomial_NI( r_t, r_c, margin, direction, alpha, n_control, rand_ratio, nu_t, kappa )
r_t |
Event rate of the test arm for the negative binomial endpoint; must be > 0 |
r_c |
Event rate of the control arm for the negative binomial endpoint; must be > 0 |
margin |
Non-inferiority margin; set to 0 for a superiority trial |
direction |
Test direction: 1 = larger values of the endpoint parameter are better, otherwise smaller values are better |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
n_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time of the test arm; the Poisson part of the variance of the log rate ratio is (1 / nu_t) * (1 / r_c + 1 / (rand_ratio * r_t)), so nu_t must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; adds the over-dispersion term (1 + rand_ratio) * kappa / rand_ratio to the variance of the log rate ratio; must be > 0 |
A named list with the following components:
Power of the non-inferiority test comparing the event rates with the margin
Character string giving the reference for the Zhu and Lakkis sample size formula
power_two_sample_neg_binomial_NI(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, alpha = 0.025, n_control = 53, rand_ratio = 1, nu_t = 6, kappa = 1)power_two_sample_neg_binomial_NI(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, alpha = 0.025, n_control = 53, rand_ratio = 1, nu_t = 6, kappa = 1)
Fixed sample designs: Normal distribution, Superiority trial, Power calculation
power_two_sample_normal( mu_t, mu_c, sigma_t, sigma_c, alpha_2, n_control, rand_ratio )power_two_sample_normal( mu_t, mu_c, sigma_t, sigma_c, alpha_2, n_control, rand_ratio )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
sigma_t |
Standard deviation of the test arm; must be > 0 |
sigma_c |
Standard deviation of the control arm; must be > 0 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
n_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A numeric value giving the power.
power_two_sample_normal(mu_t = 2.6, mu_c = 0.6, sigma_t = 9.43, sigma_c = 9.86, alpha_2 = 0.05, n_control = 53, rand_ratio = 1)power_two_sample_normal(mu_t = 2.6, mu_c = 0.6, sigma_t = 9.43, sigma_c = 9.86, alpha_2 = 0.05, n_control = 53, rand_ratio = 1)
Fixed sample designs: Normal distribution, Non-inferiority trial, Power calculation
power_two_sample_normal_NI( mu_t, mu_c, sigma_t, sigma_c, margin, direction, alpha, n_control, rand_ratio )power_two_sample_normal_NI( mu_t, mu_c, sigma_t, sigma_c, margin, direction, alpha, n_control, rand_ratio )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
sigma_t |
Standard deviation of the test arm; must be > 0 |
sigma_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; set to 0 for a superiority trial |
direction |
Test direction: 1 = larger values of the endpoint parameter are better, otherwise smaller values are better |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
n_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A numeric value giving the power.
power_two_sample_normal_NI( mu_t = 2.6, mu_c = 0.6, sigma_t = 9.43, sigma_c = 9.86, margin = 0.1, direction = 1, alpha = 0.025, n_control = 53, rand_ratio = 1)power_two_sample_normal_NI( mu_t = 2.6, mu_c = 0.6, sigma_t = 9.43, sigma_c = 9.86, margin = 0.1, direction = 1, alpha = 0.025, n_control = 53, rand_ratio = 1)
Fixed sample designs: Poisson distribution, Superiority trial, Power calculation
power_two_sample_Poisson(lambda_t, lambda_c, alpha_2, n_control, rand_ratio)power_two_sample_Poisson(lambda_t, lambda_c, alpha_2, n_control, rand_ratio)
lambda_t |
Event rate of the test arm under the Poisson model; must be > 0 |
lambda_c |
Event rate of the control arm under the Poisson model; must be > 0 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
n_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A numeric value giving the power.
power_two_sample_Poisson(lambda_t = 1.1, lambda_c = 1.0, alpha_2 = 0.05, n_control = 53, rand_ratio = 1)power_two_sample_Poisson(lambda_t = 1.1, lambda_c = 1.0, alpha_2 = 0.05, n_control = 53, rand_ratio = 1)
Fixed sample designs: Poisson distribution, Non-inferiority trial, Power calculation
power_two_sample_Poisson_NI( lambda_t, lambda_c, margin, direction, alpha, n_control, rand_ratio )power_two_sample_Poisson_NI( lambda_t, lambda_c, margin, direction, alpha, n_control, rand_ratio )
lambda_t |
Event rate of the test arm under the Poisson model; must be > 0 |
lambda_c |
Event rate of the control arm under the Poisson model; must be > 0 |
margin |
Non-inferiority margin; set to 0 for a superiority trial |
direction |
Test direction: 1 = larger values of the endpoint parameter are better, otherwise smaller values are better |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
n_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A numeric value giving the power.
power_two_sample_Poisson_NI(lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 0, alpha = 0.025, n_control = 53, rand_ratio = 1)power_two_sample_Poisson_NI(lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 0, alpha = 0.025, n_control = 53, rand_ratio = 1)
Fixed sample designs: Survival analysis, Superiority trial, Power calculation
power_two_sample_surv_Schoenfeld(HR, alpha_2, event_tot, rand_ratio)power_two_sample_surv_Schoenfeld(HR, alpha_2, event_tot, rand_ratio)
HR |
Hazards ratio of test arm vs. the control arm; must be > 0 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
event_tot |
Total number of events across both arms |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Power of the two-sided test of HR = 1
Character string giving the reference for the Schoenfeld sample size formula
power_two_sample_surv_Schoenfeld(HR = 0.7, alpha_2 = 0.05, event_tot = 100, rand_ratio = 1)power_two_sample_surv_Schoenfeld(HR = 0.7, alpha_2 = 0.05, event_tot = 100, rand_ratio = 1)
Fixed sample designs: Survival analysis, Non-inferiority trial, Power calculation
power_two_sample_surv_Schoenfeld_NI( HR, margin, direction, alpha, event_tot, rand_ratio )power_two_sample_surv_Schoenfeld_NI( HR, margin, direction, alpha, event_tot, rand_ratio )
HR |
Hazards ratio of test arm vs. the control arm; must be > 0 |
margin |
Non-inferiority margin; set to 0 for a superiority trial |
direction |
Test direction: 1 = larger values of the endpoint parameter are better, otherwise smaller values are better |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
event_tot |
Total number of events across both arms |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Power of the non-inferiority test comparing the hazard ratio with the margin
Character string giving the reference for the Schoenfeld sample size formula
power_two_sample_surv_Schoenfeld_NI(HR = 0.7, margin = 0.1, direction = 0, alpha = 0.025, event_tot = 100, rand_ratio = 1)power_two_sample_surv_Schoenfeld_NI(HR = 0.7, margin = 0.1, direction = 0, alpha = 0.025, event_tot = 100, rand_ratio = 1)
Print method for a Simon's two-stage phase II design
## S3 method for class 'ph2simon' print(x, ...)## S3 method for class 'ph2simon' print(x, ...)
x |
An object of class 'ph2simon', as returned by the internal 'ph2simon' helper |
... |
Further arguments passed to 'print' |
'NULL', invisibly, or the warning message, invisibly, if the optimal sample size is too close to 'nmax'. Called for the side effect of printing the optimal and minimax designs.
Group sequential designs / Adaptive sequential designs: Binary distribution, Superiority trial, Sample size calculation, Alpha-spending boundary
Sample_size_alsp_boundary_binary_diff(p_t, p_c, s, alpha_2, beta, rand_ratio)Sample_size_alsp_boundary_binary_diff(p_t, p_c, s, alpha_2, beta, rand_ratio)
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test, entered for the alpha-spending boundary as the vector of the two-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha_2 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the two arms have the same endpoint parameter.
Sample_size_alsp_boundary_binary_diff(p_t = 0.2, p_c = 0.6, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), beta = 0.1, rand_ratio = 1)Sample_size_alsp_boundary_binary_diff(p_t = 0.2, p_c = 0.6, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Binary distribution, Non-inferiority trial, Sample size calculation, Alpha-spending boundary
Sample_size_alsp_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio )Sample_size_alsp_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio )
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test, entered for the alpha-spending boundary as the vector of the one-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the implied effect size is not positive.
Sample_size_alsp_boundary_binary_diff_NI( p_t = 0.2, p_c = 0.6, margin = 0.15, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1)Sample_size_alsp_boundary_binary_diff_NI( p_t = 0.2, p_c = 0.6, margin = 0.15, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Superiority trial, Sample size calculation, Alpha-spending boundary
Sample_size_alsp_boundary_neg_binomial( r_t, r_c, s, alpha_2, beta, rand_ratio, nu_t, kappa )Sample_size_alsp_boundary_neg_binomial( r_t, r_c, s, alpha_2, beta, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test, entered for the alpha-spending boundary as the vector of the two-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha_2 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
Returns NA when the two arms have the same endpoint parameter.
Sample_size_alsp_boundary_neg_binomial(r_t = 0.2, r_c = 0.4, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)Sample_size_alsp_boundary_neg_binomial(r_t = 0.2, r_c = 0.4, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Non-inferiority trial, Sample size calculation, Alpha-spending boundary
Sample_size_alsp_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa )Sample_size_alsp_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger r_t is better, otherwise smaller r_t is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test, entered for the alpha-spending boundary as the vector of the one-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
Returns NA when the implied effect size is not positive.
Sample_size_alsp_boundary_neg_binomial_NI( r_t = 0.2, r_c = 0.4, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)Sample_size_alsp_boundary_neg_binomial_NI( r_t = 0.2, r_c = 0.4, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)
Group sequential designs / Adaptive sequential designs: Normal distribution, Superiority trial, Sample size calculation, Alpha-spending boundary
Sample_size_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, s, alpha_2, beta, rand_ratio )Sample_size_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, s, alpha_2, beta, rand_ratio )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test, entered for the alpha-spending boundary as the vector of the two-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha_2 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the two arms have the same endpoint parameter.
Sample_size_alsp_boundary_normal(mu_t = 0.2, mu_c = 0, std_t = 1, std_c = 1, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), beta = 0.1, rand_ratio = 1)Sample_size_alsp_boundary_normal(mu_t = 0.2, mu_c = 0, std_t = 1, std_c = 1, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Normal distribution, Non-inferiority trial, Sample size calculation, Alpha-spending boundary
Sample_size_alsp_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio )Sample_size_alsp_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test, entered for the alpha-spending boundary as the vector of the one-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the implied effect size is not positive.
Sample_size_alsp_boundary_normal_NI( mu_t = 0.2, mu_c = 0, std_t = 1, std_c = 1, margin = 0.15, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1)Sample_size_alsp_boundary_normal_NI( mu_t = 0.2, mu_c = 0, std_t = 1, std_c = 1, margin = 0.15, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Poisson distribution, Superiority trial, Sample size calculation, Alpha-spending boundary
Sample_size_alsp_boundary_poisson( lambda_t, lambda_c, s, alpha_2, beta, rand_ratio )Sample_size_alsp_boundary_poisson( lambda_t, lambda_c, s, alpha_2, beta, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test, entered for the alpha-spending boundary as the vector of the two-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha_2 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the two arms have the same endpoint parameter.
Sample_size_alsp_boundary_poisson(lambda_t = 0.2, lambda_c = 0.4, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), beta = 0.1, rand_ratio = 1)Sample_size_alsp_boundary_poisson(lambda_t = 0.2, lambda_c = 0.4, s = c(1/3,2/3,1), alpha_2 = c(0.001,0.001,0.048), beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Poisson distribution, Non-inferiority trial, Sample size calculation, Alpha-spending boundary
Sample_size_alsp_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio )Sample_size_alsp_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test, entered for the alpha-spending boundary as the vector of the one-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the implied effect size is not positive.
Sample_size_alsp_boundary_poisson_NI( lambda_t = 0.2, lambda_c = 0.4, margin = 0.15, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1)Sample_size_alsp_boundary_poisson_NI( lambda_t = 0.2, lambda_c = 0.4, margin = 0.15, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Superiority trial, Sample size calculation, Alpha-spending boundary
Sample_size_alsp_boundary_survival_Schoenfeld(HR, s, alpha_2, beta, rand_ratio)Sample_size_alsp_boundary_survival_Schoenfeld(HR, s, alpha_2, beta, rand_ratio)
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test, entered for the alpha-spending boundary as the vector of the two-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha_2 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Total number of events required across both the test arm and the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
Returns NA when the two arms have the same endpoint parameter.
Sample_size_alsp_boundary_survival_Schoenfeld(HR = 0.59, s = c(0.596,1), alpha_2 = c(0.001,0.0249), beta = 0.1, rand_ratio = 1)Sample_size_alsp_boundary_survival_Schoenfeld(HR = 0.59, s = c(0.596,1), alpha_2 = c(0.001,0.0249), beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Non-inferiority trial, Sample size calculation, Alpha-spending boundary
Sample_size_alsp_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, alpha, beta, rand_ratio )Sample_size_alsp_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, alpha, beta, rand_ratio )
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
margin |
Non-inferiority margin on the hazard ratio scale; must be > 0, with HR > margin required when direction is 1 and HR < margin required otherwise |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test, entered for the alpha-spending boundary as the vector of the one-sided alpha-spending values alpha_1, ..., alpha_K at the K analyses; each element must be > 0 and the elements must sum to alpha |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Total number of events required across both the test arm and the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
Returns NA when HR is not on the required side of margin for the specified direction.
Sample_size_alsp_boundary_survival_Schoenfeld_NI( HR = 0.59, margin = 1, direction = 2, s = c(0.596,1), alpha = c(0.005,0.02), beta = 0.1, rand_ratio = 1)Sample_size_alsp_boundary_survival_Schoenfeld_NI( HR = 0.59, margin = 1, direction = 2, s = c(0.596,1), alpha = c(0.005,0.02), beta = 0.1, rand_ratio = 1)
Fixed sample designs: Binary distribution, Superiority trial, Sample size calculation
sample_size_binary_diff(p_t, p_c, alpha_2, beta, rand_ratio)sample_size_binary_diff(p_t, p_c, alpha_2, beta, rand_ratio)
p_t |
Event rate of the test arm; 0 < p_t < 1 |
p_c |
Event rate of the control arm; 0 < p_c < 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the control arm
Sample size required for the experimental (test) arm
sample_size_binary_diff(p_t = 0.2, p_c = 0.4, alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)sample_size_binary_diff(p_t = 0.2, p_c = 0.4, alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Fixed sample designs: Binary distribution, Non-inferiority trial, Sample size calculation
sample_size_binary_diff_NI( p_t, p_c, margin, direction, alpha, beta, rand_ratio )sample_size_binary_diff_NI( p_t, p_c, margin, direction, alpha, beta, rand_ratio )
p_t |
Event rate of the test arm; 0 < p_t < 1 |
p_c |
Event rate of the control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; set to 0 for a superiority trial |
direction |
Test direction: 1 = larger values of the endpoint parameter are better, otherwise smaller values are better |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the control arm
Sample size required for the experimental (test) arm
sample_size_binary_diff_NI( p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 1, alpha = 0.025, beta = 0.1, rand_ratio = 1)sample_size_binary_diff_NI( p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 1, alpha = 0.025, beta = 0.1, rand_ratio = 1)
Fixed sample designs: Negative binomial distribution, Superiority trial, Sample size calculation
sample_size_neg_binomial(r_t, r_c, alpha_2, beta, rand_ratio, nu_t, kappa)sample_size_neg_binomial(r_t, r_c, alpha_2, beta, rand_ratio, nu_t, kappa)
r_t |
Event rate of the test arm for the negative binomial endpoint; must be > 0 |
r_c |
Event rate of the control arm for the negative binomial endpoint; must be > 0 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time of the test arm; the Poisson part of the variance of the log rate ratio is (1 / nu_t) * (1 / r_c + 1 / (rand_ratio * r_t)), so nu_t must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; adds the over-dispersion term (1 + rand_ratio) * kappa / rand_ratio to the variance of the log rate ratio; must be > 0 |
A named list with the following components:
Standardized effect size, i.e. the absolute log rate ratio divided by the square root of the variance
Sample size required for the test (experimental) arm
Sample size required for the control arm
Character string giving the reference for the Zhu and Lakkis sample size formula
sample_size_neg_binomial(r_t = 0.85, r_c = 1.25, alpha_2 = 0.05, beta = 0.2, rand_ratio = 1, nu_t = 1, kappa = 1)sample_size_neg_binomial(r_t = 0.85, r_c = 1.25, alpha_2 = 0.05, beta = 0.2, rand_ratio = 1, nu_t = 1, kappa = 1)
Fixed sample designs: Negative binomial distribution, Non-inferiority trial, Sample size calculation
sample_size_neg_binomial_NI( r_t, r_c, margin, direction, alpha, beta, rand_ratio, nu_t, kappa )sample_size_neg_binomial_NI( r_t, r_c, margin, direction, alpha, beta, rand_ratio, nu_t, kappa )
r_t |
Event rate of the test arm for the negative binomial endpoint; must be > 0 |
r_c |
Event rate of the control arm for the negative binomial endpoint; must be > 0 |
margin |
Non-inferiority margin; set to 0 for a superiority trial |
direction |
Test direction: 1 = larger values of the endpoint parameter are better, otherwise smaller values are better |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time of the test arm; the Poisson part of the variance of the log rate ratio is (1 / nu_t) * (1 / r_c + 1 / (rand_ratio * r_t)), so nu_t must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; adds the over-dispersion term (1 + rand_ratio) * kappa / rand_ratio to the variance of the log rate ratio; must be > 0 |
A named list with the following components:
Standardized effect size, i.e. the absolute log rate ratio divided by the square root of the variance
Sample size required for the test (experimental) arm
Sample size required for the control arm
Character string giving the reference for the Zhu and Lakkis sample size formula
sample_size_neg_binomial_NI(r_t = 0.9, r_c = 1, margin = 0.3, direction = 1, alpha = 0.025, beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)sample_size_neg_binomial_NI(r_t = 0.9, r_c = 1, margin = 0.3, direction = 1, alpha = 0.025, beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)
Group sequential designs / Adaptive sequential designs: Binary distribution, Superiority trial, Sample size calculation, O'Brien-Fleming boundary
Sample_size_OF_boundary_binary_diff(p_t, p_c, s, alpha_2, beta, rand_ratio)Sample_size_OF_boundary_binary_diff(p_t, p_c, s, alpha_2, beta, rand_ratio)
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the two arms have the same endpoint parameter.
Sample_size_OF_boundary_binary_diff(p_t = 0.4, p_c = 0.3, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)Sample_size_OF_boundary_binary_diff(p_t = 0.4, p_c = 0.3, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Binary distribution, Non-inferiority trial, Sample size calculation, O'Brien-Fleming boundary
Sample_size_OF_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio )Sample_size_OF_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio )
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the implied effect size is not positive.
Sample_size_OF_boundary_binary_diff_NI( p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)Sample_size_OF_boundary_binary_diff_NI( p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Superiority trial, Sample size calculation, O'Brien-Fleming boundary
Sample_size_OF_boundary_neg_binomial( r_t, r_c, s, alpha_2, beta, rand_ratio, nu_t, kappa )Sample_size_OF_boundary_neg_binomial( r_t, r_c, s, alpha_2, beta, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
Returns NA when the two arms have the same endpoint parameter.
Sample_size_OF_boundary_neg_binomial(r_t = 0.2, r_c = 0.4, s = c(1/3,2/3,1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)Sample_size_OF_boundary_neg_binomial(r_t = 0.2, r_c = 0.4, s = c(1/3,2/3,1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Non-inferiority trial, Sample size calculation, O'Brien-Fleming boundary
Sample_size_OF_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa )Sample_size_OF_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger r_t is better, otherwise smaller r_t is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
Returns NA when the implied effect size is not positive.
Sample_size_OF_boundary_neg_binomial_NI( r_t = 0.2, r_c = 0.4, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)Sample_size_OF_boundary_neg_binomial_NI( r_t = 0.2, r_c = 0.4, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)
Group sequential designs / Adaptive sequential designs: Normal distribution, Superiority trial, Sample size calculation, O'Brien-Fleming boundary
Sample_size_OF_boundary_normal( mu_t, mu_c, std_t, std_c, s, alpha_2, beta, rand_ratio )Sample_size_OF_boundary_normal( mu_t, mu_c, std_t, std_c, s, alpha_2, beta, rand_ratio )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the two arms have the same endpoint parameter.
Sample_size_OF_boundary_normal(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)Sample_size_OF_boundary_normal(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Normal distribution, Non-inferiority trial, Sample size calculation, O'Brien-Fleming boundary
Sample_size_OF_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio )Sample_size_OF_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the implied effect size is not positive.
Sample_size_OF_boundary_normal_NI( mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)Sample_size_OF_boundary_normal_NI( mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Poisson distribution, Superiority trial, Sample size calculation, O'Brien-Fleming boundary
Sample_size_OF_boundary_poisson( lambda_t, lambda_c, s, alpha_2, beta, rand_ratio )Sample_size_OF_boundary_poisson( lambda_t, lambda_c, s, alpha_2, beta, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the two arms have the same endpoint parameter.
Sample_size_OF_boundary_poisson(lambda_t = 1.1, lambda_c = 1.0, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)Sample_size_OF_boundary_poisson(lambda_t = 1.1, lambda_c = 1.0, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Poisson distribution, Non-inferiority trial, Sample size calculation, O'Brien-Fleming boundary
Sample_size_OF_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio )Sample_size_OF_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the implied effect size is not positive.
Sample_size_OF_boundary_poisson_NI( lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)Sample_size_OF_boundary_poisson_NI( lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Superiority trial, Sample size calculation, O'Brien-Fleming boundary
Sample_size_OF_boundary_survival_Schoenfeld(HR, s, alpha_2, beta, rand_ratio)Sample_size_OF_boundary_survival_Schoenfeld(HR, s, alpha_2, beta, rand_ratio)
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Total number of events required across both the test arm and the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
Returns NA when the two arms have the same endpoint parameter.
Sample_size_OF_boundary_survival_Schoenfeld(HR = 0.7, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)Sample_size_OF_boundary_survival_Schoenfeld(HR = 0.7, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Non-inferiority trial, Sample size calculation, O'Brien-Fleming boundary
Sample_size_OF_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, alpha, beta, rand_ratio )Sample_size_OF_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, alpha, beta, rand_ratio )
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
margin |
Non-inferiority margin on the hazard ratio scale; must be > 0, with HR > margin required when direction is 1 and HR < margin required otherwise |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Total number of events required across both the test arm and the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
Returns NA when HR is not on the required side of margin for the specified direction.
Sample_size_OF_boundary_survival_Schoenfeld_NI( HR = 0.7, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)Sample_size_OF_boundary_survival_Schoenfeld_NI( HR = 0.7, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Binary distribution, Superiority trial, Sample size calculation, Pocock boundary
Sample_size_pocock_boundary_binary_diff(p_t, p_c, s, alpha_2, beta, rand_ratio)Sample_size_pocock_boundary_binary_diff(p_t, p_c, s, alpha_2, beta, rand_ratio)
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the two arms have the same endpoint parameter.
Sample_size_pocock_boundary_binary_diff(p_t = 0.4, p_c = 0.3, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)Sample_size_pocock_boundary_binary_diff(p_t = 0.4, p_c = 0.3, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Binary distribution, Non-inferiority trial, Sample size calculation, Pocock boundary
Sample_size_pocock_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio )Sample_size_pocock_boundary_binary_diff_NI( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio )
p_t |
Response rate (probability of response) in the test arm; 0 < p_t < 1 |
p_c |
Response rate (probability of response) in the control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the implied effect size is not positive.
Sample_size_pocock_boundary_binary_diff_NI( p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)Sample_size_pocock_boundary_binary_diff_NI( p_t = 0.4, p_c = 0.3, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Superiority trial, Sample size calculation, Pocock boundary
Sample_size_pocock_boundary_neg_binomial( r_t, r_c, s, alpha_2, beta, rand_ratio, nu_t, kappa )Sample_size_pocock_boundary_neg_binomial( r_t, r_c, s, alpha_2, beta, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
Returns NA when the two arms have the same endpoint parameter.
Sample_size_pocock_boundary_neg_binomial(r_t = 0.2, r_c = 0.4, s = c(1/3,2/3,1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)Sample_size_pocock_boundary_neg_binomial(r_t = 0.2, r_c = 0.4, s = c(1/3,2/3,1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)
Group sequential designs / Adaptive sequential designs: Negative binomial distribution, Non-inferiority trial, Sample size calculation, Pocock boundary
Sample_size_pocock_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa )Sample_size_pocock_boundary_neg_binomial_NI( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa )
r_t |
Event rate (mean number of events per unit of exposure time) in the test arm; must be > 0 |
r_c |
Event rate (mean number of events per unit of exposure time) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger r_t is better, otherwise smaller r_t is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time (follow-up time) of each subject, used to convert the number of events into a rate; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Zhu and Lakkis, 2014)
Returns NA when the implied effect size is not positive.
Sample_size_pocock_boundary_neg_binomial_NI( r_t = 0.2, r_c = 0.4, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)Sample_size_pocock_boundary_neg_binomial_NI( r_t = 0.2, r_c = 0.4, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1)
Group sequential designs / Adaptive sequential designs: Normal distribution, Superiority trial, Sample size calculation, Pocock boundary
Sample_size_pocock_boundary_normal( mu_t, mu_c, std_t, std_c, s, alpha_2, beta, rand_ratio )Sample_size_pocock_boundary_normal( mu_t, mu_c, std_t, std_c, s, alpha_2, beta, rand_ratio )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the two arms have the same endpoint parameter.
Sample_size_pocock_boundary_normal(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)Sample_size_pocock_boundary_normal(mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Normal distribution, Non-inferiority trial, Sample size calculation, Pocock boundary
Sample_size_pocock_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio )Sample_size_pocock_boundary_normal_NI( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
std_t |
Standard deviation of the test arm; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the implied effect size is not positive.
Sample_size_pocock_boundary_normal_NI( mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)Sample_size_pocock_boundary_normal_NI( mu_t = 2.6, mu_c = 0.6, std_t = 9.43, std_c = 9.86, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Poisson distribution, Superiority trial, Sample size calculation, Pocock boundary
Sample_size_pocock_boundary_poisson( lambda_t, lambda_c, s, alpha_2, beta, rand_ratio )Sample_size_pocock_boundary_poisson( lambda_t, lambda_c, s, alpha_2, beta, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the two arms have the same endpoint parameter.
Sample_size_pocock_boundary_poisson(lambda_t = 0.2, lambda_c = 0.4, s = c(1/3,2/3,1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)Sample_size_pocock_boundary_poisson(lambda_t = 0.2, lambda_c = 0.4, s = c(1/3,2/3,1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Poisson distribution, Non-inferiority trial, Sample size calculation, Pocock boundary
Sample_size_pocock_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio )Sample_size_pocock_boundary_poisson_NI( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio )
lambda_t |
Event rate (mean number of events) in the test arm; must be > 0 |
lambda_c |
Event rate (mean number of events) in the control arm; must be > 0 |
margin |
Non-inferiority margin; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Returns NA when the implied effect size is not positive.
Sample_size_pocock_boundary_poisson_NI( lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)Sample_size_pocock_boundary_poisson_NI( lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Superiority trial, Sample size calculation, Pocock boundary
Sample_size_pocock_boundary_survival_Schoenfeld( HR, s, alpha_2, beta, rand_ratio )Sample_size_pocock_boundary_survival_Schoenfeld( HR, s, alpha_2, beta, rand_ratio )
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Total number of events required across both the test arm and the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
Returns NA when the two arms have the same endpoint parameter.
Sample_size_pocock_boundary_survival_Schoenfeld(HR = 0.7, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)Sample_size_pocock_boundary_survival_Schoenfeld(HR = 0.7, s = c(1/3, 2/3, 1), alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Group sequential designs / Adaptive sequential designs: Survival analysis, Non-inferiority trial, Sample size calculation, Pocock boundary
Sample_size_pocock_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, alpha, beta, rand_ratio )Sample_size_pocock_boundary_survival_Schoenfeld_NI( HR, margin, direction, s, alpha, beta, rand_ratio )
HR |
Hazard ratio of the test arm vs. the control arm; must be > 0 |
margin |
Non-inferiority margin on the hazard ratio scale; must be > 0, with HR > margin required when direction is 1 and HR < margin required otherwise |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Total number of events required across both the test arm and the control arm
Critical boundaries c_1, ..., c_K at the K analyses
One-sided nominal alpha at each analysis
One-sided alpha spending at each analysis
One-sided cumulative alpha spending at each analysis
Sum of the one-sided nominal alphas
Two-sided nominal alpha at each analysis
Two-sided alpha spending at each analysis
Two-sided cumulative alpha spending at each analysis
Sum of the two-sided nominal alphas
Reference for the sample size formula (Schoenfeld, 1983)
Returns NA when HR is not on the required side of margin for the specified direction.
Sample_size_pocock_boundary_survival_Schoenfeld_NI( HR = 0.7, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)Sample_size_pocock_boundary_survival_Schoenfeld_NI( HR = 0.7, margin = 0.1, direction = 1, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1)
Fixed sample designs: Normal distribution, Superiority trial, Sample size calculation
sample_size_two_sample_normal( mu_t, mu_c, sigma_t, sigma_c, alpha_2, beta, rand_ratio )sample_size_two_sample_normal( mu_t, mu_c, sigma_t, sigma_c, alpha_2, beta, rand_ratio )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
sigma_t |
Standard deviation of the test arm; must be > 0 |
sigma_c |
Standard deviation of the control arm; must be > 0 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the control arm
Sample size required for the test arm
sample_size_two_sample_normal(mu_t = 2, mu_c = 0, sigma_t = 9.79, sigma_c = 9.79, alpha_2 = 0.05, beta = 0.2, rand_ratio = 1)sample_size_two_sample_normal(mu_t = 2, mu_c = 0, sigma_t = 9.79, sigma_c = 9.79, alpha_2 = 0.05, beta = 0.2, rand_ratio = 1)
Fixed sample designs: Normal distribution, Non-inferiority trial, Sample size calculation
sample_size_two_sample_normal_NI( mu_t, mu_c, sigma_t, sigma_c, margin, direction, alpha, beta, rand_ratio )sample_size_two_sample_normal_NI( mu_t, mu_c, sigma_t, sigma_c, margin, direction, alpha, beta, rand_ratio )
mu_t |
Mean of the test arm |
mu_c |
Mean of the control arm |
sigma_t |
Standard deviation of the test arm; must be > 0 |
sigma_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; set to 0 for a superiority trial |
direction |
Test direction: 1 = larger values of the endpoint parameter are better, otherwise smaller values are better |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the control arm
Sample size required for the test arm
sample_size_two_sample_normal_NI(mu_t = 2, mu_c = 2.5, sigma_t = 9.79, sigma_c = 9.79, margin = 0.2, direction = 2, alpha = 0.05, beta = 0.2, rand_ratio = 1)sample_size_two_sample_normal_NI(mu_t = 2, mu_c = 2.5, sigma_t = 9.79, sigma_c = 9.79, margin = 0.2, direction = 2, alpha = 0.05, beta = 0.2, rand_ratio = 1)
Fixed sample designs: Poisson distribution, Superiority trial, Sample size calculation
sample_size_two_sample_Poisson(lambda_t, lambda_c, alpha_2, beta, rand_ratio)sample_size_two_sample_Poisson(lambda_t, lambda_c, alpha_2, beta, rand_ratio)
lambda_t |
Event rate of the test arm under the Poisson model; must be > 0 |
lambda_c |
Event rate of the control arm under the Poisson model; must be > 0 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the experimental (test) arm
Sample size required for the control arm
sample_size_two_sample_Poisson(lambda_t = 1.1, lambda_c = 1.0, alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)sample_size_two_sample_Poisson(lambda_t = 1.1, lambda_c = 1.0, alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Fixed sample designs: Poisson distribution, Non-inferiority trial, Sample size calculation
sample_size_two_sample_Poisson_NI( lambda_t, lambda_c, margin, direction, alpha, beta, rand_ratio )sample_size_two_sample_Poisson_NI( lambda_t, lambda_c, margin, direction, alpha, beta, rand_ratio )
lambda_t |
Event rate of the test arm under the Poisson model; must be > 0 |
lambda_c |
Event rate of the control arm under the Poisson model; must be > 0 |
margin |
Non-inferiority margin; set to 0 for a superiority trial |
direction |
Test direction: 1 = larger values of the endpoint parameter are better, otherwise smaller values are better |
alpha |
Type I error for the one-sided test; 0 < alpha < 1 |
beta |
Type II error rate, i.e. 1 - power |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Sample size required for the experimental (test) arm
Sample size required for the control arm
sample_size_two_sample_Poisson_NI( lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 1, alpha = 0.025, beta = 0.1, rand_ratio = 1)sample_size_two_sample_Poisson_NI( lambda_t = 1.1, lambda_c = 1.0, margin = 0.1, direction = 1, alpha = 0.025, beta = 0.1, rand_ratio = 1)
Fixed sample designs: Survival analysis, Superiority trial, Sample size calculation
sample_size_two_sample_surv_Schoenfeld(HR, alpha_2, beta, rand_ratio)sample_size_two_sample_surv_Schoenfeld(HR, alpha_2, beta, rand_ratio)
HR |
Hazards ratio of test arm vs. the control arm; must be > 0 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Total number of events required for the two-arm comparison
Character string giving the reference for the Schoenfeld sample size formula
sample_size_two_sample_surv_Schoenfeld(HR = 0.7, alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)sample_size_two_sample_surv_Schoenfeld(HR = 0.7, alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Fixed sample designs: Survival analysis, Non-inferiority trial, Sample size calculation
sample_size_two_sample_surv_Schoenfeld_NI( HR, margin, direction, alpha_2, beta, rand_ratio )sample_size_two_sample_surv_Schoenfeld_NI( HR, margin, direction, alpha_2, beta, rand_ratio )
HR |
Hazards ratio of test arm vs. the control arm; must be > 0 |
margin |
Non-inferiority margin; set to 0 for a superiority trial |
direction |
Test direction: 1 = larger values of the endpoint parameter are better, otherwise smaller values are better |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
beta |
Type II error rate, i.e. 1 - power |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
A named list with the following components:
Total number of events required for the two-arm comparison
Character string giving the reference for the Schoenfeld sample size formula
sample_size_two_sample_surv_Schoenfeld_NI( HR = 0.7, margin = 0.1, direction = 1, alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)sample_size_two_sample_surv_Schoenfeld_NI( HR = 0.7, margin = 0.1, direction = 1, alpha_2 = 0.05, beta = 0.1, rand_ratio = 1)
Single arm phase 2 oncology trial: Two-stage design, Simon's design
simon_ph2_sz(alpha, beta, p_0, p_1)simon_ph2_sz(alpha, beta, p_0, p_1)
alpha |
Type I error for the one-sided test |
beta |
Type II error rate, i.e. 1 - power |
p_0 |
Response rate indicating low activity with insufficient clinical benefit |
p_1 |
Assumed response rate |
A matrix with one row for each of the minimax design (smallest n_2), the optimal design (smallest EN(p_0)) and the average design (an augmentation of the original Simon's design, Gao and Zhang 2024), and one column per design quantity:
If no more than r1 responses are observed at the first look, then the trial would be stopped for futility
Number of patients at first look
The null hypothesis will be rejected if at least r_2 responses are observed at the end of the trial
Number of patients at second look; n_2 is also the total sample size
Expected sample size under the null hypothesis
Probability of early termination under the null hypothesis
Type I error, the probability of rejecting the null hypothesis under p <= p_0
Probability of early termination under p = p_1
Power under p = p_1
R. Simon. Optimal Two-Stage Designs for Phase II Clinical Trials. Controlled Clinical Trials 10:1-10 (1989).
P. Gao & W. Zhang (15 Apr 2024): Adaptive sequential design for phase II single-arm oncology trials: an expansion of Simon's design, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2341673. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2341673.
simon_ph2_sz(alpha=0.025,beta=0.1,p_0=0.2,p_1=0.33)simon_ph2_sz(alpha=0.025,beta=0.1,p_0=0.2,p_1=0.33)
Phase 2/3 seamless combination Group sequential design: Simulations, Binary distribution, Optimizing adaptive sequential design
Two_stage_asd_simu_OC_binary( groups, p_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )Two_stage_asd_simu_OC_binary( groups, p_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
groups |
Matrix of assumed event rates for the test arms under study: one row per candidate test-arm scenario and one column per dose, so that the number of doses is the number of columns; each event rate must be > 0 and < 1 |
p_c |
Event rate of the control arm; must be > 0 and < 1 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s1 |
Vector of information fractions of the first interim analysis schedule; the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule; the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
alpha_2_alps_s1 |
Vector of two-sided alpha to be spent at each analysis of the alpha-spending schedule s1; each element must be > 0 and the elements must sum to alpha_2 |
alpha_2_alps_s2 |
Vector of two-sided alpha to be spent at each analysis of the alpha-spending schedule s2; each element must be > 0 and the elements must sum to alpha_2 |
theta_cut |
Lack-of-efficacy threshold: an effect size below which the test drug is not considered to provide clinically meaningful efficacy |
samsz_1 |
Planned sample size of the control arm for the first sample size scenario |
samsz_2 |
Planned sample size of the control arm for the second sample size scenario |
N_max_1 |
Maximum sample size (control arm) allowed after sample size re-estimation for the first maximum sample size scenario |
N_max_2 |
Maximum sample size (control arm) allowed after sample size re-estimation for the second maximum sample size scenario |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
List holding the group sequential design and the operating characteristics simulated under the schedule s1 with O'Brien-Fleming boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s2 with O'Brien-Fleming boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s1 with alpha-spending boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s2 with alpha-spending boundaries
Each of these four lists has the components "Group sequential design" (data frame with the columns "Looks", "Information fractions" and the critical boundaries, plus "Alpha-spending" and "Alpha-spending boundaries" for the alpha-spending schedules) and the "Simulation results: Information fractions ..., Sample size ..., N_max ..." operating characteristics of the corresponding adaptive design, one data frame per combination of interim analysis schedule, planned sample size and maximum sample size, with one row per assumed event-rate scenario.
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
set.seed(1) Two_stage_asd_simu_OC_binary( groups = rbind(c(0.3,0.4,0.5),c(0.1,0.2,0.3),c(0.1,0.2,0.3),c(0.1,0.2,0.3)), p_c = 0.2, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)set.seed(1) Two_stage_asd_simu_OC_binary( groups = rbind(c(0.3,0.4,0.5),c(0.1,0.2,0.3),c(0.1,0.2,0.3),c(0.1,0.2,0.3)), p_c = 0.2, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)
Phase 2/3 seamless combination Group sequential design: Simulations, Negative binomial distribution, Optimizing adaptive sequential design
Two_stage_asd_simu_OC_neg_binomial( groups, r_c, kappa, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )Two_stage_asd_simu_OC_neg_binomial( groups, r_c, kappa, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
groups |
Matrix of assumed mean event rates for the test arms under study: one row per candidate test-arm scenario and one column per dose, so that the number of doses is the number of columns; each mean event rate must be > 0 |
r_c |
Mean event rate of the control arm; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s1 |
Vector of information fractions of the first interim analysis schedule; the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule; the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
alpha_2_alps_s1 |
Vector of two-sided alpha to be spent at each analysis of the alpha-spending schedule s1; each element must be > 0 and the elements must sum to alpha_2 |
alpha_2_alps_s2 |
Vector of two-sided alpha to be spent at each analysis of the alpha-spending schedule s2; each element must be > 0 and the elements must sum to alpha_2 |
theta_cut |
Lack-of-efficacy threshold: an effect size below which the test drug is not considered to provide clinically meaningful efficacy |
samsz_1 |
Planned sample size of the control arm for the first sample size scenario |
samsz_2 |
Planned sample size of the control arm for the second sample size scenario |
N_max_1 |
Maximum sample size (control arm) allowed after sample size re-estimation for the first maximum sample size scenario |
N_max_2 |
Maximum sample size (control arm) allowed after sample size re-estimation for the second maximum sample size scenario |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
List holding the group sequential design and the operating characteristics simulated under the schedule s1 with O'Brien-Fleming boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s2 with O'Brien-Fleming boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s1 with alpha-spending boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s2 with alpha-spending boundaries
Each of these four lists has the components "Group sequential design" (data frame with the columns "Looks", "Information fractions" and the critical boundaries, plus "Alpha-spending" and "Alpha-spending boundaries" for the alpha-spending schedules) and the "Simulation results: Information fractions ..., Sample size ..., N_max ..." operating characteristics of the corresponding adaptive design, one data frame per combination of interim analysis schedule, planned sample size and maximum sample size, with one row per assumed mean event-rate scenario.
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
set.seed(1) Two_stage_asd_simu_OC_neg_binomial( groups = rbind(c(0.3,0.4,0.5),c(0.2,0.2,0.5),c(0.2,0.2,0.3),c(0.2,0.3,0.4)), r_c = 0.2, kappa = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)set.seed(1) Two_stage_asd_simu_OC_neg_binomial( groups = rbind(c(0.3,0.4,0.5),c(0.2,0.2,0.5),c(0.2,0.2,0.3),c(0.2,0.3,0.4)), r_c = 0.2, kappa = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)
Phase 2/3 seamless combination Group sequential design: Simulations, Normal distribution, Optimizing adaptive sequential design
Two_stage_asd_simu_OC_normal( groups, mu_c, sigma_t, sigma_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )Two_stage_asd_simu_OC_normal( groups, mu_c, sigma_t, sigma_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
groups |
Matrix of assumed effect sizes for the test arms under study: one row per candidate test-arm scenario and one column per dose, so that the number of doses is the number of columns |
mu_c |
Mean of the control arm |
sigma_t |
Standard deviation of the test arms |
sigma_c |
Standard deviation of the control arm |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s1 |
Vector of information fractions of the first interim analysis schedule; the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule; the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
alpha_2_alps_s1 |
Vector of two-sided alpha to be spent at each analysis of the alpha-spending schedule s1; each element must be > 0 and the elements must sum to alpha_2 |
alpha_2_alps_s2 |
Vector of two-sided alpha to be spent at each analysis of the alpha-spending schedule s2; each element must be > 0 and the elements must sum to alpha_2 |
theta_cut |
Lack-of-efficacy threshold: an effect size below which the test drug is not considered to provide clinically meaningful efficacy |
samsz_1 |
Planned sample size of the control arm for the first sample size scenario |
samsz_2 |
Planned sample size of the control arm for the second sample size scenario |
N_max_1 |
Maximum sample size (control arm) allowed after sample size re-estimation for the first maximum sample size scenario |
N_max_2 |
Maximum sample size (control arm) allowed after sample size re-estimation for the second maximum sample size scenario |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list of four elements, one per boundary type and interim analysis schedule, with the following components:
List holding the group sequential design and the operating characteristics simulated under the schedule s1 with O'Brien-Fleming boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s2 with O'Brien-Fleming boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s1 with alpha-spending boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s2 with alpha-spending boundaries
Each of these four lists has the components "Group sequential design" (data frame with the columns "Looks", "Information fractions" and the critical boundaries, plus "Alpha-spending" and "Alpha-spending boundaries" for the alpha-spending schedules) and "Simulation results: Information fractions ..., Sample size ..., N_max ..." (the operating characteristics of the corresponding adaptive design).
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
set.seed(1) Two_stage_asd_simu_OC_normal(groups = rbind(c(0.2, 0.3, 0.4), c(0.3, 0.4, 0.5)), mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)set.seed(1) Two_stage_asd_simu_OC_normal(groups = rbind(c(0.2, 0.3, 0.4), c(0.3, 0.4, 0.5)), mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)
Phase 2/3 seamless combination Group sequential design: Simulations, Poisson distribution, Optimizing adaptive sequential design
Two_stage_asd_simu_OC_poisson( groups, lambda_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )Two_stage_asd_simu_OC_poisson( groups, lambda_c, direction, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, theta_cut, samsz_1, samsz_2, N_max_1, N_max_2, beta, sim_num )
groups |
Matrix of assumed means for the test arms under study: one row per candidate test-arm scenario and one column per dose, so that the number of doses is the number of columns; each mean must be > 0 |
lambda_c |
Mean of the control arm; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s1 |
Vector of information fractions of the first interim analysis schedule; the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule; the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
alpha_2_alps_s1 |
Vector of two-sided alpha to be spent at each analysis of the alpha-spending schedule s1; each element must be > 0 and the elements must sum to alpha_2 |
alpha_2_alps_s2 |
Vector of two-sided alpha to be spent at each analysis of the alpha-spending schedule s2; each element must be > 0 and the elements must sum to alpha_2 |
theta_cut |
Lack-of-efficacy threshold: an effect size below which the test drug is not considered to provide clinically meaningful efficacy |
samsz_1 |
Planned sample size of the control arm for the first sample size scenario |
samsz_2 |
Planned sample size of the control arm for the second sample size scenario |
N_max_1 |
Maximum sample size (control arm) allowed after sample size re-estimation for the first maximum sample size scenario |
N_max_2 |
Maximum sample size (control arm) allowed after sample size re-estimation for the second maximum sample size scenario |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
List holding the group sequential design and the operating characteristics simulated under the schedule s1 with O'Brien-Fleming boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s2 with O'Brien-Fleming boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s1 with alpha-spending boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s2 with alpha-spending boundaries
Each of these four lists has the components "Group sequential design" (data frame with the columns "Looks", "Information fractions" and the critical boundaries, plus "Alpha-spending" and "Alpha-spending boundaries" for the alpha-spending schedules) and the "Simulation results: Information fractions ..., Sample size ..., N_max ..." operating characteristics of the corresponding adaptive design, one data frame per combination of interim analysis schedule, planned sample size and maximum sample size, with one row per assumed mean scenario.
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
set.seed(1) Two_stage_asd_simu_OC_poisson( groups = rbind(c(0.3,0.4,0.5),c(0.2,0.2,0.5),c(0.2,0.2,0.3),c(0.2,0.3,0.4)), lambda_c = 0.2, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)set.seed(1) Two_stage_asd_simu_OC_poisson( groups = rbind(c(0.3,0.4,0.5),c(0.2,0.2,0.5),c(0.2,0.2,0.3),c(0.2,0.3,0.4)), lambda_c = 0.2, direction = 1, s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.001,0.049), theta_cut = 0.01, samsz_1 = 100, samsz_2 = 150, N_max_1 = 1000, N_max_2 = 1500, beta = 0.1, sim_num = 2)
Phase 2/3 seamless combination Group sequential design: Simulations, Survival analysis, Optimizing adaptive sequential design
Two_stage_asd_simu_OC_survival( groups, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, surv_rate, evt_num_1, evt_num_2, evt_max_1, evt_max_2, HR_cut, beta, sim_num )Two_stage_asd_simu_OC_survival( groups, s1, s2, alpha_2, alpha_2_alps_s1, alpha_2_alps_s2, surv_rate, evt_num_1, evt_num_2, evt_max_1, evt_max_2, HR_cut, beta, sim_num )
groups |
Matrix of assumed hazard ratios for the test arms under study: one row per candidate test-arm scenario and one column per dose, so that the number of doses is the number of columns; each hazard ratio is relative to the control arm, must be > 0, with HR < 1 indicating a beneficial effect and HR = 1 the null hypothesis |
s1 |
Vector of information fractions of the first interim analysis schedule; the last element must be 1 |
s2 |
Vector of information fractions of the second interim analysis schedule; the last element must be 1 |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
alpha_2_alps_s1 |
Vector of two-sided alpha to be spent at each analysis of the alpha-spending schedule s1; each element must be > 0 and the elements must sum to alpha_2 |
alpha_2_alps_s2 |
Vector of two-sided alpha to be spent at each analysis of the alpha-spending schedule s2; each element must be > 0 and the elements must sum to alpha_2 |
surv_rate |
Survival rate at the end of follow-up, used to translate the number of events into a sample size |
evt_num_1 |
Planned number of events between a test arm and the control arm for the first event number scenario |
evt_num_2 |
Planned number of events between a test arm and the control arm for the second event number scenario |
evt_max_1 |
Maximum number of events between a test arm and the control arm allowed after sample size re-estimation for the first maximum event number scenario |
evt_max_2 |
Maximum number of events between a test arm and the control arm allowed after sample size re-estimation for the second maximum event number scenario |
HR_cut |
Hazard-ratio cut-off for lack of efficacy at the interim analysis: the test drug is not considered to provide clinically meaningful efficacy when the observed hazard ratio is greater than HR_cut |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
List holding the group sequential design and the operating characteristics simulated under the schedule s1 with O'Brien-Fleming boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s2 with O'Brien-Fleming boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s1 with alpha-spending boundaries
List holding the group sequential design and the operating characteristics simulated under the schedule s2 with alpha-spending boundaries
Each of these four lists has the components "Group sequential design" (data frame with the columns "Looks", "Information fractions" and the critical boundaries, plus "Alpha-spending" and "Alpha-spending boundaries" for the alpha-spending schedules) and the "Simulation results: Information fractions ..., Sample size ..., evt_max ..." operating characteristics of the corresponding adaptive design, one data frame per combination of interim analysis schedule, planned event number and maximum number of events, with one row per assumed hazard-ratio scenario.
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
set.seed(1) Two_stage_asd_simu_OC_survival( groups = rbind(c(0.8, 0.83, 0.85), c(0.78, 0.8, 0.82), c(0.75, 0.78, 0.8)), s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.01,0.04), surv_rate = 0.5, evt_num_1 = 200, evt_num_2 = 300, evt_max_1 = 2000, evt_max_2 = 3000, HR_cut = 0.95, beta = 0.1, sim_num = 2)set.seed(1) Two_stage_asd_simu_OC_survival( groups = rbind(c(0.8, 0.83, 0.85), c(0.78, 0.8, 0.82), c(0.75, 0.78, 0.8)), s1 = c(1/3,2/3,1), s2 = c(0.5,1), alpha_2 = 0.05, alpha_2_alps_s1 = c(0.001,0.001,0.048), alpha_2_alps_s2 = c(0.01,0.04), surv_rate = 0.5, evt_num_1 = 200, evt_num_2 = 300, evt_max_1 = 2000, evt_max_2 = 3000, HR_cut = 0.95, beta = 0.1, sim_num = 2)
Phase 2/3 seamless combination Group sequential design: Final analysis, Without sample size change, last_vt == 1
two_stage_ci_0(s, theta_hat_0, theta_hat_0_sd, alpha_2, dist)two_stage_ci_0(s, theta_hat_0, theta_hat_0_sd, alpha_2, dist)
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
theta_hat_0 |
Vector of naive point estimates of the treatment effect for the comparisons at the end of the phase 2 stage (the first interim analysis); its length is the number of comparisons |
theta_hat_0_sd |
Common standard error of 'theta_hat_0' at the end of the phase 2 stage, shared by all comparisons; must be > 0 |
alpha_2 |
Type I error for the two-sided test |
dist |
Distribution indicator of the endpoint: 1 = the endpoint is normally distributed and the comparison is between multiple doses and a common control, otherwise the non-normal branch is used |
A named list with the following components:
The p-value for the hypothesis test
Numeric vector of length 3 giving the lower confidence limit, the median unbiased point estimate and the upper confidence limit; the lower limit is consistent with the p-value
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_ci_0(s = c(1/3, 2/3, 1), theta_hat_0 = 0.2, theta_hat_0_sd = 0.5, alpha_2 = 0.05, dist = 1)two_stage_ci_0(s = c(1/3, 2/3, 1), theta_hat_0 = 0.2, theta_hat_0_sd = 0.5, alpha_2 = 0.05, dist = 1)
Phase 2/3 seamless combination Group sequential design: Final analysis, Without sample size change, last_vt > 1
two_stage_ci_1( s, c_bry_two_stg, c_bry_p3, comp, theta_hat_p3, theta_hat_p3_sd, theta_hat_last, theta_hat_last_sd, last_vt, alpha_2, dist )two_stage_ci_1( s, c_bry_two_stg, c_bry_p3, comp, theta_hat_p3, theta_hat_p3_sd, theta_hat_last, theta_hat_last_sd, last_vt, alpha_2, dist )
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1; here the length of 's' must be 2 or 3, i.e. the phase 2 stage plus one or two phase 3 analyses |
c_bry_two_stg |
Vector of critical boundaries of the two-stage design for the overall test, one per analysis; must have the same length as 's' |
c_bry_p3 |
Vector of critical boundaries of the prior (phase 3) study, one per phase 3 analysis; must have 'length(s) - 1' elements |
comp |
Number of comparisons in the phase 2 stage, e.g. the number of doses or the number of endpoints; must be at least 2 |
theta_hat_p3 |
Treatment effect estimate observed at the analysis at which the trial stopped, using phase 3 data only |
theta_hat_p3_sd |
Standard error of 'theta_hat_p3'; must be > 0 |
theta_hat_last |
Treatment effect estimate observed at the analysis at which the trial stopped, based on all data accumulated by then |
theta_hat_last_sd |
Standard error of 'theta_hat_last'; must be > 0 |
last_vt |
Index of the analysis at which the trial stopped; must be > 1 because the trial stopped in the phase 3 stage |
alpha_2 |
Type I error for the two-sided test |
dist |
Distribution indicator of the endpoint: 1 = the endpoint is normally distributed and the comparison is between multiple doses and a common control, otherwise the non-normal branch is used |
A named list with the following components:
The p-value for the hypothesis test
Numeric vector of length 3 giving the lower confidence limit, the median unbiased point estimate and the upper confidence limit; the lower limit is consistent with the p-value, while the point estimate and the upper limit use phase 3 data only
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_ci_1(s = c(1/3,2/3,1), c_bry_two_stg = c(3.938325, 2.784816,2.273793), c_bry_p3 = c(3.954862,1.977431), comp = 3, theta_hat_p3 = 0.45, theta_hat_p3_sd = 0.6, theta_hat_last = 2.784816*0.12, theta_hat_last_sd = 0.12, last_vt = 2, alpha_2 = 0.05, dist = 1)two_stage_ci_1(s = c(1/3,2/3,1), c_bry_two_stg = c(3.938325, 2.784816,2.273793), c_bry_p3 = c(3.954862,1.977431), comp = 3, theta_hat_p3 = 0.45, theta_hat_p3_sd = 0.6, theta_hat_last = 2.784816*0.12, theta_hat_last_sd = 0.12, last_vt = 2, alpha_2 = 0.05, dist = 1)
Phase 2/3 seamless combination Group sequential design: Final analysis, With one sample size change
two_stage_est_back( s, c_bry_two_stg, c_bry_p3, comp, snew, c_bry_new, c_bry_new_p3, theta_hat_inter, theta_hat_inter_sd, inter_vt, theta_hat_last_ad, theta_hat_last_ad_sd, last_vt_ad, theta_hat_inter_p3, theta_hat_inter_p3_sd, theta_hat_last_ad_p3, theta_hat_last_ad_p3_sd, alpha_2, dist )two_stage_est_back( s, c_bry_two_stg, c_bry_p3, comp, snew, c_bry_new, c_bry_new_p3, theta_hat_inter, theta_hat_inter_sd, inter_vt, theta_hat_last_ad, theta_hat_last_ad_sd, last_vt_ad, theta_hat_inter_p3, theta_hat_inter_p3_sd, theta_hat_last_ad_p3, theta_hat_last_ad_p3_sd, alpha_2, dist )
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
c_bry_two_stg |
Vector of critical boundaries of the two-stage design for the overall test, one per analysis; must have the same length as 's' |
c_bry_p3 |
Vector of critical boundaries of the prior (phase 3) study, one per phase 3 analysis; must have 'length(s) - 1' elements |
comp |
Number of comparisons in the phase 2 stage, e.g. the number of doses or the number of endpoints; must be at least 2 |
snew |
Vector of new information fractions after the sample size is changed, one per analysis after the change; must have at least 'last_vt_ad' elements |
c_bry_new |
Vector of critical boundaries after the sample size is changed, one per analysis after the change; must have the same length as 'snew' |
c_bry_new_p3 |
Vector of critical boundaries after the sample size is changed on the prior (phase 3) study scale, one per analysis after the change; must have the same length as 'snew' |
theta_hat_inter |
Treatment effect estimate observed at the interim analysis at which the sample size is changed |
theta_hat_inter_sd |
Standard error of 'theta_hat_inter'; must be > 0 |
inter_vt |
Index of the interim analysis at which the sample size is re-estimated (1-based); must be less than 'length(s)', so that at least one further analysis remains |
theta_hat_last_ad |
Treatment effect estimate observed at the last analysis after the sample size change |
theta_hat_last_ad_sd |
Standard error of 'theta_hat_last_ad'; must be > 0 |
last_vt_ad |
Index of the last analysis after the sample size change |
theta_hat_inter_p3 |
Treatment effect estimate observed at the interim analysis at which the sample size is changed, using phase 3 data only |
theta_hat_inter_p3_sd |
Standard error of 'theta_hat_inter_p3'; must be > 0 |
theta_hat_last_ad_p3 |
Treatment effect estimate observed at the last analysis after the sample size change, using phase 3 data only |
theta_hat_last_ad_p3_sd |
Standard error of 'theta_hat_last_ad_p3'; must be > 0 |
alpha_2 |
Type I error for the two-sided test |
dist |
Distribution indicator of the endpoint: 1 = the endpoint is normally distributed and the comparison is between multiple doses and a common control, otherwise the non-normal branch is used |
A named list with the following components:
The p-value for the hypothesis test
Numeric vector of length 3 giving the lower confidence limit, the median unbiased point estimate and the upper confidence limit; the lower limit is consistent with the p-value, while the point estimate and the upper limit use phase 3 data only
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_est_back(s=c(1/3,2/3,1), c_bry_two_stg=c(3.776606,2.670463,2.180424), c_bry_p3=c(3.954862,1.977431), comp=2, snew=c(0.5,1), c_bry_new=c(3,2.2), c_bry_new_p3=c(3.2,2.4), theta_hat_inter=0.4, theta_hat_inter_sd=0.35, inter_vt=1, theta_hat_last_ad=0.5, theta_hat_last_ad_sd=0.23, last_vt_ad=1, theta_hat_inter_p3=0.2, theta_hat_inter_p3_sd=0.4, theta_hat_last_ad_p3=0.32, theta_hat_last_ad_p3_sd=0.4, alpha_2=0.05, dist=1)two_stage_est_back(s=c(1/3,2/3,1), c_bry_two_stg=c(3.776606,2.670463,2.180424), c_bry_p3=c(3.954862,1.977431), comp=2, snew=c(0.5,1), c_bry_new=c(3,2.2), c_bry_new_p3=c(3.2,2.4), theta_hat_inter=0.4, theta_hat_inter_sd=0.35, inter_vt=1, theta_hat_last_ad=0.5, theta_hat_last_ad_sd=0.23, last_vt_ad=1, theta_hat_inter_p3=0.2, theta_hat_inter_p3_sd=0.4, theta_hat_last_ad_p3=0.32, theta_hat_last_ad_p3_sd=0.4, alpha_2=0.05, dist=1)
Phase 2/3 seamless combination Group sequential design: Binary distribution, Power calculation, Alpha-spending boundary
two_stage_power_alsp_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )two_stage_power_alsp_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
p_t |
Event rates of the test arms; a vector of length >= 2 with 0 < p_t < 1 |
p_c |
Event rate of the common control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; for the alpha-spending boundary this is the vector of alpha-spending values at each analysis, with one value per element of s and the values summing to the overall one-sided type I error, e.g. c(0.001, 0.001, 0.023) |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Nested list with the total power, the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_power_alsp_boundary_binary_diff(p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0.15, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 82, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_power_alsp_boundary_binary_diff(p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0.15, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 82, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Negative binomial distribution, Power calculation, Alpha-spending boundary
two_stage_power_alsp_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa, dist1, dist2 )two_stage_power_alsp_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa, dist1, dist2 )
r_t |
Mean event rates of the test arms; a vector of length >= 2 |
r_c |
Mean event rate of the common control arm |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; for the alpha-spending boundary this is the vector of alpha-spending values at each analysis, with one value per element of s and the values summing to the overall one-sided type I error, e.g. c(0.001, 0.001, 0.048) |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, 2 = otherwise |
A named list with the following components:
Nested list with the total power, the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Reference for the negative binomial sample size formula used
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_power_alsp_boundary_neg_binomial(r_t = c(0.2,0.3,0.4), r_c = 0.6, margin = 0.1, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.048), sampsz_control = 111, rand_ratio = 1, nu_t = 1, kappa = 1, dist1 = 1, dist2 = 1)two_stage_power_alsp_boundary_neg_binomial(r_t = c(0.2,0.3,0.4), r_c = 0.6, margin = 0.1, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.048), sampsz_control = 111, rand_ratio = 1, nu_t = 1, kappa = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Normal distribution, Power calculation, Alpha-spending boundary
two_stage_power_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )two_stage_power_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
mu_t |
Means of the test arms; a vector of length >= 2 |
mu_c |
Mean of the common control arm |
std_t |
Common standard deviation of the test arms; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; for the alpha-spending boundary this is the vector of alpha-spending values at each analysis, with one value per element of s and the values summing to the overall one-sided type I error, e.g. c(0.001, 0.001, 0.023) |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Nested list with the total power, the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_power_alsp_boundary_normal(mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0.01, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 180, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_power_alsp_boundary_normal(mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0.01, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 180, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Poisson distribution, Power calculation, Alpha-spending boundary
two_stage_power_alsp_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )two_stage_power_alsp_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
lambda_t |
Mean event rates of the test arms; a vector of length >= 2 |
lambda_c |
Mean event rate of the common control arm |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; for the alpha-spending boundary this is the vector of alpha-spending values at each analysis, with one value per element of s and the values summing to the overall one-sided type I error, e.g. c(0.001, 0.001, 0.023) |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Nested list with the total power, the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_power_alsp_boundary_poisson(lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.12, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 150, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_power_alsp_boundary_poisson(lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.12, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), sampsz_control = 150, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Survival analysis, Power calculation, Alpha-spending boundary
two_stage_power_alsp_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, events_tot, rand_ratio, dist1, dist2 )two_stage_power_alsp_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, events_tot, rand_ratio, dist1, dist2 )
HR |
Hazard ratios of the test arms vs. the control arm; a vector of length >= 2 |
margin |
Non-inferiority margin for the hazard ratio; must be > 0, with margin = 1 for a superiority trial and margin != 1 for a non-inferiority trial |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; for the alpha-spending boundary this is the vector of alpha-spending values at each analysis, with one value per element of s and the values summing to the overall one-sided type I error, e.g. c(0.001, 0.001, 0.023) |
events_tot |
Total number of events available for the trial |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 2 for this survival endpoint setting (this is conservative) |
dist2 |
Distribution indicator for the second stage: 1 for this survival endpoint setting (this is conservative) |
A named list with the following components:
Nested list with the total power, the selection probability for each HR, the exit probability for each HR, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected HR
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Reference for the Schoenfeld sample size formula used
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_power_alsp_boundary_survival_Schoenfeld(HR = c(0.2,0.3,0.4), margin = 1.1, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), events_tot = 20, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_power_alsp_boundary_survival_Schoenfeld(HR = c(0.2,0.3,0.4), margin = 1.1, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), events_tot = 20, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Binary distribution, Power calculation, O'Brien-Fleming boundary
two_stage_power_OF_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )two_stage_power_OF_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
p_t |
Event rates of the test arms; a vector of length >= 2 with 0 < p_t < 1 |
p_c |
Event rate of the common control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1, typically 0.025 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Nested list with the total power, the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_power_OF_boundary_binary_diff(p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0.15, direction = 1, s = c(1/3,2/3,1), alpha = 0.025, sampsz_control = 82, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_power_OF_boundary_binary_diff(p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0.15, direction = 1, s = c(1/3,2/3,1), alpha = 0.025, sampsz_control = 82, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Negative binomial distribution, Power calculation, O'Brien-Fleming boundary
two_stage_power_OF_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa, dist1, dist2 )two_stage_power_OF_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, sampsz_control, rand_ratio, nu_t, kappa, dist1, dist2 )
r_t |
Mean event rates of the test arms; a vector of length >= 2 |
r_c |
Mean event rate of the common control arm |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1, typically 0.025 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, 2 = otherwise |
A named list with the following components:
Nested list with the total power, the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Reference for the negative binomial sample size formula used
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_power_OF_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1, dist1 = 1, dist2 = 1)two_stage_power_OF_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, sampsz_control = 100, rand_ratio = 1, nu_t = 6, kappa = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Normal distribution, Power calculation, O'Brien-Fleming boundary
two_stage_power_OF_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )two_stage_power_OF_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
mu_t |
Means of the test arms; a vector of length >= 2 |
mu_c |
Mean of the common control arm |
std_t |
Common standard deviation of the test arms; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1, typically 0.025 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Nested list with the total power, the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_power_OF_boundary_normal(mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0.01, direction = 1, s = c(1/3,2/3,1), alpha = 0.025, sampsz_control = 180, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_power_OF_boundary_normal(mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0.01, direction = 1, s = c(1/3,2/3,1), alpha = 0.025, sampsz_control = 180, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Poisson distribution, Power calculation, O'Brien-Fleming boundary
two_stage_power_OF_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )two_stage_power_OF_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, sampsz_control, rand_ratio, dist1, dist2 )
lambda_t |
Mean event rates of the test arms; a vector of length >= 2 |
lambda_c |
Mean event rate of the common control arm |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1, typically 0.025 |
sampsz_control |
Sample size of the control arm |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Nested list with the total power, the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_power_OF_boundary_poisson(lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.12, direction = 1, s = c(1/3,2/3,1), alpha = 0.025, sampsz_control = 150, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_power_OF_boundary_poisson(lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.12, direction = 1, s = c(1/3,2/3,1), alpha = 0.025, sampsz_control = 150, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Survival analysis, Power calculation, O'Brien-Fleming boundary
two_stage_power_OF_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, events_tot, rand_ratio, dist1, dist2 )two_stage_power_OF_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, events_tot, rand_ratio, dist1, dist2 )
HR |
Hazard ratios of the test arms vs. the control arm; a vector of length >= 2 |
margin |
Non-inferiority margin for the hazard ratio; must be > 0, with margin = 1 for a superiority trial and margin != 1 for a non-inferiority trial |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1, typically 0.025 |
events_tot |
Total number of events available for the trial |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 2 for this survival endpoint setting (this is conservative) |
dist2 |
Distribution indicator for the second stage: 1 for this survival endpoint setting (this is conservative) |
A named list with the following components:
Nested list with the total power, the selection probability for each HR, the exit probability for each HR, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected HR
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Reference for the Schoenfeld sample size formula used
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_power_OF_boundary_survival_Schoenfeld(HR = c(0.55,0.55), margin = 1, direction = 2, s = c(1/3,2/3,1), alpha = 0.025, events_tot = 137, rand_ratio = 1, dist1 = 2, dist2 = 1)two_stage_power_OF_boundary_survival_Schoenfeld(HR = c(0.55,0.55), margin = 1, direction = 2, s = c(1/3,2/3,1), alpha = 0.025, events_tot = 137, rand_ratio = 1, dist1 = 2, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Binary distribution, Sample size calculation, Alpha-spending boundary
two_stage_Sample_size_alsp_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )two_stage_Sample_size_alsp_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
p_t |
Event rates of the test arms; a vector of length >= 2 with 0 < p_t < 1 |
p_c |
Event rate of the common control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; for the alpha-spending boundary this is the vector of alpha-spending values at each analysis, with one value per element of s and the values summing to the overall one-sided type I error, e.g. c(0.001, 0.001, 0.023) |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Sample size for each dose in a two-arm study using the same critical boundary
Sample size for the control arm in a two-arm study using the same critical boundary
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Nested list with the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_Sample_size_alsp_boundary_binary_diff( p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0.15, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_Sample_size_alsp_boundary_binary_diff( p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0.15, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Negative binomial distribution, Sample size calculation, Alpha-spending boundary
two_stage_Sample_size_alsp_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa, dist1, dist2 )two_stage_Sample_size_alsp_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa, dist1, dist2 )
r_t |
Mean event rates of the test arms; a vector of length >= 2 |
r_c |
Mean event rate of the common control arm |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; for the alpha-spending boundary this is the vector of alpha-spending values at each analysis, with one value per element of s and the values summing to the overall one-sided type I error, e.g. c(0.001, 0.001, 0.023) |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, 2 = otherwise |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Sample size for each dose in a two-arm study using the same critical boundary
Sample size for the control arm in a two-arm study using the same critical boundary
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Nested list with the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
Reference for the negative binomial sample size formula used
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_Sample_size_alsp_boundary_neg_binomial( r_t = c(0.2,0.3,0.4), r_c = 0.6, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1, dist1 = 1, dist2 = 1)two_stage_Sample_size_alsp_boundary_neg_binomial( r_t = c(0.2,0.3,0.4), r_c = 0.6, margin = 0.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, nu_t = 1, kappa = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Normal distribution, Sample size calculation, Alpha-spending boundary
two_stage_Sample_size_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )two_stage_Sample_size_alsp_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
mu_t |
Means of the test arms; a vector of length >= 2. The largest implied mean effect must be > 1e-4, otherwise the function returns NA |
mu_c |
Mean of the common control arm |
std_t |
Common standard deviation of the test arms; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; for the alpha-spending boundary this is the vector of alpha-spending values at each analysis, with one value per element of s and the values summing to the overall one-sided type I error, e.g. c(0.001, 0.001, 0.023) |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Sample size for each dose in a two-arm study using the same critical boundary
Sample size for the control arm in a two-arm study using the same critical boundary
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Nested list with the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_Sample_size_alsp_boundary_normal( mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0.1, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_Sample_size_alsp_boundary_normal( mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0.1, direction = 1, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Poisson distribution, Sample size calculation, Alpha-spending boundary
two_stage_Sample_size_alsp_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )two_stage_Sample_size_alsp_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
lambda_t |
Mean event rates of the test arms; a vector of length >= 2 |
lambda_c |
Mean event rate of the common control arm |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; for the alpha-spending boundary this is the vector of alpha-spending values at each analysis, with one value per element of s and the values summing to the overall one-sided type I error, e.g. c(0.001, 0.001, 0.023) |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Sample size for each dose in a two-arm study using the same critical boundary
Sample size for the control arm in a two-arm study using the same critical boundary
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Nested list with the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_Sample_size_alsp_boundary_poisson( lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.3, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_Sample_size_alsp_boundary_poisson( lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.3, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Survival analysis, Sample size calculation, Alpha-spending boundary
two_stage_Sample_size_alsp_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )two_stage_Sample_size_alsp_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
HR |
Hazard ratios of the test arms vs. the control arm; a vector of length >= 2 |
margin |
Non-inferiority margin for the hazard ratio; must be > 0, with margin = 1 for a superiority trial and margin != 1 for a non-inferiority trial |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; for the alpha-spending boundary this is the vector of alpha-spending values at each analysis, with one value per element of s and the values summing to the overall one-sided type I error, e.g. c(0.001, 0.001, 0.023) |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 2 for this survival endpoint setting (this is conservative) |
dist2 |
Distribution indicator for the second stage: 1 for this survival endpoint setting (this is conservative) |
A named list with the following components:
Number of events required for a single dose compared with the control
Total number of events required for all doses and the control
Number of events for each dose and the control in a two-arm study using the same critical boundary
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Nested list with the selection probability for each HR, the exit probability for each HR, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected HR
Reference for the Schoenfeld sample size formula used
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_Sample_size_alsp_boundary_survival_Schoenfeld( HR = c(0.2,0.3,0.4), margin = 1.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_Sample_size_alsp_boundary_survival_Schoenfeld( HR = c(0.2,0.3,0.4), margin = 1.1, direction = 2, s = c(1/3,2/3,1), alpha = c(0.001,0.001,0.023), beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Binary distribution, Sample size calculation, O'Brien-Fleming boundary
two_stage_Sample_size_OF_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )two_stage_Sample_size_OF_boundary_binary_diff( p_t, p_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
p_t |
Event rates of the test arms; a vector of length >= 2 with 0 < p_t < 1 |
p_c |
Event rate of the common control arm; 0 < p_c < 1 |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1, typically 0.025 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Sample size for each dose in a two-arm study using the same critical boundary
Sample size for the control arm in a two-arm study using the same critical boundary
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Nested list with the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_Sample_size_OF_boundary_binary_diff( p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0, direction = 2, s = c(1/3,2/3,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_Sample_size_OF_boundary_binary_diff( p_t = c(0.2,0.3,0.4), p_c = 0.5, margin = 0, direction = 2, s = c(1/3,2/3,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Negative binomial distribution, Sample size calculation, O'Brien-Fleming boundary
two_stage_Sample_size_OF_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa, dist1, dist2 )two_stage_Sample_size_OF_boundary_neg_binomial( r_t, r_c, margin, direction, s, alpha, beta, rand_ratio, nu_t, kappa, dist1, dist2 )
r_t |
Mean event rates of the test arms; a vector of length >= 2 |
r_c |
Mean event rate of the common control arm |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1, typically 0.025 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
nu_t |
Exposure time; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, 2 = otherwise |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Sample size for each dose in a two-arm study using the same critical boundary
Sample size for the control arm in a two-arm study using the same critical boundary
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Nested list with the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
Reference for the negative binomial sample size formula used
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_Sample_size_OF_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, nu_t = 6, kappa = 1, dist1 = 1, dist2 = 1)two_stage_Sample_size_OF_boundary_neg_binomial(r_t = c(1.1, 1.1, 1.1), r_c = 1.1, margin = 0.1, direction = 0, s = c(1/3, 2/3, 1), alpha = 0.025, beta = 0.1, rand_ratio = 1, nu_t = 6, kappa = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Normal distribution, Sample size calculation, O'Brien-Fleming boundary
two_stage_Sample_size_OF_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )two_stage_Sample_size_OF_boundary_normal( mu_t, mu_c, std_t, std_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
mu_t |
Means of the test arms; a vector of length >= 2. The largest implied mean effect must be > 1e-4, otherwise the function returns NA |
mu_c |
Mean of the common control arm |
std_t |
Common standard deviation of the test arms; must be > 0 |
std_c |
Standard deviation of the control arm; must be > 0 |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1, typically 0.025 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Sample size for each dose in a two-arm study using the same critical boundary
Sample size for the control arm in a two-arm study using the same critical boundary
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Nested list with the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_Sample_size_OF_boundary_normal( mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0.1, direction = 1, s = c(1/3,2/3,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_Sample_size_OF_boundary_normal( mu_t = c(0.2,0.3,0.4), mu_c = 0, std_t = 1, std_c = 1, margin = 0.1, direction = 1, s = c(1/3,2/3,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Poisson distribution, Sample size calculation, O'Brien-Fleming boundary
two_stage_Sample_size_OF_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )two_stage_Sample_size_OF_boundary_poisson( lambda_t, lambda_c, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
lambda_t |
Mean event rates of the test arms; a vector of length >= 2 |
lambda_c |
Mean event rate of the common control arm |
margin |
Non-inferiority margin; margin >= 0, with margin = 0 for a superiority trial and margin > 0 for a non-inferiority trial |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1, typically 0.025 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
First-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
dist2 |
Second-stage distribution indicator: 1 = the sample size calculation is for multiple dose vs. a common control, with normal distributions, 2 = otherwise |
A named list with the following components:
Sample size required for the test arm
Sample size required for the control arm
Sample size for each dose in a two-arm study using the same critical boundary
Sample size for the control arm in a two-arm study using the same critical boundary
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Nested list with the selection probability for each mu, the exit probability for each mu, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected mu
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_Sample_size_OF_boundary_poisson( lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.15, direction = 2, s = c(1/3,2/3,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)two_stage_Sample_size_OF_boundary_poisson( lambda_t = c(0.2,0.3,0.4), lambda_c = 0.5, margin = 0.15, direction = 2, s = c(1/3,2/3,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 1, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Survival analysis, Sample size calculation, O'Brien-Fleming boundary
two_stage_Sample_size_OF_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )two_stage_Sample_size_OF_boundary_survival_Schoenfeld( HR, margin, direction, s, alpha, beta, rand_ratio, dist1, dist2 )
HR |
Hazard ratios of the test arms vs. the control arm; a vector of length >= 2 |
margin |
Non-inferiority margin for the hazard ratio; must be > 0, with margin = 1 for a superiority trial and margin != 1 for a non-inferiority trial |
direction |
Test direction: 1 = larger HR is better, otherwise smaller HR is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
alpha |
Type I error for the one-sided test; 0 < alpha < 1, typically 0.025 |
beta |
Type II error rate, i.e. 1 - power; 0 < beta < 1 |
rand_ratio |
Randomization ratio: number of subjects in the test arm vs. the number of subjects in the control arm; must be > 0 |
dist1 |
Distribution indicator for the first stage: 2 for this survival endpoint setting (this is conservative) |
dist2 |
Distribution indicator for the second stage: 1 for this survival endpoint setting (this is conservative) |
A named list with the following components:
Number of events required for a single dose compared with the control
Total number of events required for all doses and the control
Number of events for each dose and the control in a two-arm study using the same critical boundary
Nested list with the two-stage critical boundaries, the one-sided alpha spending, nominal alpha and cumulative alpha at each interim analysis, and the phase 3 information fraction, phase 3 critical boundary and nominal p-value for phase 3
Nested list with the selection probability for each HR, the exit probability for each HR, the total phase 2 probability, the total exit probability and the conditional exit probability given the selected HR
Reference for the Schoenfeld sample size formula used
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stage_Sample_size_OF_boundary_survival_Schoenfeld( HR = c(1.1,1.2,1.25), margin = 1, direction = 1, s = c(0.3,0.7,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 2, dist2 = 1)two_stage_Sample_size_OF_boundary_survival_Schoenfeld( HR = c(1.1,1.2,1.25), margin = 1, direction = 1, s = c(0.3,0.7,1), alpha = 0.025, beta = 0.1, rand_ratio = 1, dist1 = 2, dist2 = 1)
Phase 2/3 seamless combination Group sequential design: Simulations, Binary distribution, Operating characteristics
two_stg_adapt_power_simu_binary( p_t, p_c, direction, s, bry_type, alpha_2, theta_cut, N_max, samsz, beta, sim_num )two_stg_adapt_power_simu_binary( p_t, p_c, direction, s, bry_type, alpha_2, theta_cut, N_max, samsz, beta, sim_num )
p_t |
Event rate of the test arm; each element must be > 0 and < 1 |
p_c |
Event rate of the control arm; must be > 0 and < 1 |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock; any other value leaves the boundaries undefined |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
theta_cut |
Lack-of-efficacy threshold: an effect size below which the test drug is not considered to provide clinically meaningful efficacy |
N_max |
Maximum sample size (control arm) allowed after sample size re-estimation |
samsz |
Planned sample size of the control arm used in the simulation |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix of the planned adaptive group sequential design, with one row per analysis and the columns "Look", "Information fraction", "Critical boundaries", "rejection by look" and "Sample size increase"
Matrix with one row per dose and the columns "p<sub>t1</sub>,...", "p<sub>c</sub>", "selection rate", "rejection rate by each p_t" and "conditional rejection rate"; the selection rate is the probability that the dose is selected to proceed to phase 3, the rejection rate is the probability that the dose is selected and the null hypothesis is rejected, and the conditional rejection rate is the rejection rate divided by the selection rate
Named vector of the "Rejection rate without SSR", the "Rejection rate with SSR", the "rate of sample size increase", the "Rejection rate without SSR/(meeting sample size incerase criteria)", the "Rejection rate with SSR/(sample size incerase)", the "Type I error", the "Non-Type I error rejection rate", the "Futility rate", the "Planned control sample size", the "Maximum control sample size", the "Mean control sample size", the "Planned total sample size", the "Mean total sample size" and the "mean control sample size conditional on increase"
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stg_adapt_power_simu_binary(p_t = c(0.4,0.5), p_c = 0.3, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, theta_cut = 0.05, N_max = 1000, samsz = 100, beta = 0.1, sim_num = 20)two_stg_adapt_power_simu_binary(p_t = c(0.4,0.5), p_c = 0.3, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, theta_cut = 0.05, N_max = 1000, samsz = 100, beta = 0.1, sim_num = 20)
Phase 2/3 seamless combination Group sequential design: Simulations, Negative binomial distribution, Operating characteristics
two_stg_adapt_power_simu_neg_binomial( r_t, r_c, kappa, direction, s, bry_type, alpha_2, theta_cut, samsz, N_max, beta, sim_num )two_stg_adapt_power_simu_neg_binomial( r_t, r_c, kappa, direction, s, bry_type, alpha_2, theta_cut, samsz, N_max, beta, sim_num )
r_t |
Mean event rates of the test arms; each element must be > 0 |
r_c |
Mean event rate of the control arm; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock; any other value leaves the boundaries undefined |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
theta_cut |
Lack-of-efficacy threshold: an effect size below which the test drug is not considered to provide clinically meaningful efficacy |
samsz |
Planned sample size of the control arm used in the simulation |
N_max |
Maximum sample size (control arm) allowed after sample size re-estimation |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix of the planned adaptive group sequential design, with one row per analysis and the columns "Look", "Information fraction", "Critical boundaries", "rejection by look" and "Sample size increase"
Matrix with one row per dose and the columns "r<sub>t1</sub>,...", "r<sub>c</sub>", "selection rate", "rejection rate by each r_t" and "conditional rejection rate"; the selection rate is the probability that the dose is selected to proceed to phase 3, the rejection rate is the probability that the dose is selected and the null hypothesis is rejected, and the conditional rejection rate is the rejection rate divided by the selection rate
Named vector of the "Rejection rate without SSR", the "Rejection rate with SSR", the "rate of sample size increase", the "Rejection rate without SSR/(meeting sample size incerase criteria)", the "Rejection rate with SSR/(sample size incerase)", the "Type I error", the "Non-Type I error rejection rate", the "Futility rate", the "Planned control sample size", the "Maximum control sample size", the "Mean control sample size", the "Planned total sample size", the "Mean total sample size" and the "mean control sample size conditional on increase"
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stg_adapt_power_simu_neg_binomial( r_t = c(1.1,1.1,1.1), r_c = 1.1, kappa = 1, direction = 1, s = c(1/3,1), bry_type = 1, alpha_2 = 0.05, theta_cut = 0.05, samsz = 100, N_max = 1000, beta = 0.1, sim_num = 20)two_stg_adapt_power_simu_neg_binomial( r_t = c(1.1,1.1,1.1), r_c = 1.1, kappa = 1, direction = 1, s = c(1/3,1), bry_type = 1, alpha_2 = 0.05, theta_cut = 0.05, samsz = 100, N_max = 1000, beta = 0.1, sim_num = 20)
Phase 2/3 seamless combination Group sequential design: Simulations, Normal distribution, Operating characteristics
two_stg_adapt_power_simu_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, theta_cut, samsz, N_max, beta, sim_num )two_stg_adapt_power_simu_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, theta_cut, samsz, N_max, beta, sim_num )
mu_t |
Vector of means of the test arms, one entry per dose |
mu_c |
Mean of the control arm |
sigma_t |
Standard deviation of the test arms |
sigma_c |
Standard deviation of the control arm |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock; any other value leaves the boundaries undefined |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
theta_cut |
Lack-of-efficacy threshold: an effect size below which the test drug is not considered to provide clinically meaningful efficacy |
samsz |
Planned sample size of the control arm used in the simulation |
N_max |
Maximum sample size (control arm) allowed after sample size re-estimation |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named numeric vector with the following components:
Rejection rate of the trial if the sample size is not re-estimated
Rejection rate of the trial under the adaptive procedure with sample size re-estimation
Rate of simulations in which the sample size is increased
Rejection rate without sample size re-estimation among the simulations that meet the sample size increase criteria
Rejection rate with sample size re-estimation among the simulations in which the sample size is increased
Simulated one-sided type I error rate
Simulated rejection rate when the true effect is non-zero
Rate of simulations stopped for lack of efficacy, i.e. for which the observed interim effect size is below theta_cut
Planned sample size of the control arm
Maximum permitted sample size of the control arm
Mean simulated sample size of the control arm
Planned total sample size of the trial
Mean simulated total sample size of the trial
Mean simulated control sample size among the simulations in which the sample size is increased
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stg_adapt_power_simu_normal(mu_t = c(0,0,0), mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 2, alpha_2 = c(0.001,0.001,0.048), theta_cut = 0.05, samsz = 100, N_max = 500, beta = 0.1, sim_num = 20)two_stg_adapt_power_simu_normal(mu_t = c(0,0,0), mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/3,2/3,1), bry_type = 2, alpha_2 = c(0.001,0.001,0.048), theta_cut = 0.05, samsz = 100, N_max = 500, beta = 0.1, sim_num = 20)
Phase 2/3 seamless combination Group sequential design: Simulations, Poisson distribution, Operating characteristics
two_stg_adapt_power_simu_poisson( lambda_t, lambda_c, direction, s, bry_type, alpha_2, theta_cut, N_max, samsz, beta, sim_num )two_stg_adapt_power_simu_poisson( lambda_t, lambda_c, direction, s, bry_type, alpha_2, theta_cut, N_max, samsz, beta, sim_num )
lambda_t |
Mean of the test arm; each element must be > 0 |
lambda_c |
Mean of the control arm; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock; any other value leaves the boundaries undefined |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
theta_cut |
Lack-of-efficacy threshold: an effect size below which the test drug is not considered to provide clinically meaningful efficacy |
N_max |
Maximum sample size (control arm) allowed after sample size re-estimation |
samsz |
Planned sample size of the control arm used in the simulation |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix of the planned adaptive group sequential design, with one row per analysis and the columns "Look", "Information fraction", "Critical boundaries", "rejection by look" and "Sample size increase"
Matrix with one row per dose and the columns "lambda_t1,...", "lambda_c", "selection rate", "rejection rate by each lambda_t" and "conditional rejection rate"; the selection rate is the probability that the dose is selected to proceed to phase 3, the rejection rate is the probability that the dose is selected and the null hypothesis is rejected, and the conditional rejection rate is the rejection rate divided by the selection rate
Named vector of the "Rejection rate without SSR", the "Rejection rate with SSR", the "rate of sample size increase", the "Rejection rate without SSR/(meeting sample size incerase criteria)", the "Rejection rate with SSR/(sample size incerase)", the "Type I error", the "Non-Type I error rejection rate", the "Futility rate", the "Planned control sample size", the "Maximum control sample size", the "Mean control sample size", the "Planned total sample size", the "Mean total sample size" and the "mean control sample size conditional on increase"
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stg_adapt_power_simu_poisson(lambda_t = c(1.1,1.1,1.1), lambda_c = 1.1, direction = 1, s = c(1/3,1), bry_type = 1, alpha_2 = 0.05, theta_cut = 0.05, N_max = 1000, samsz = 100, beta = 0.1, sim_num = 20)two_stg_adapt_power_simu_poisson(lambda_t = c(1.1,1.1,1.1), lambda_c = 1.1, direction = 1, s = c(1/3,1), bry_type = 1, alpha_2 = 0.05, theta_cut = 0.05, N_max = 1000, samsz = 100, beta = 0.1, sim_num = 20)
Phase 2/3 seamless combination Group sequential design: Simulations, Survival analysis, Operating characteristics
two_stg_adapt_power_simu_survival( HR, s, bry_type, alpha_2, evt_num, evt_max, HR_cut, surv_rate, beta, sim_num )two_stg_adapt_power_simu_survival( HR, s, bry_type, alpha_2, evt_num, evt_max, HR_cut, surv_rate, beta, sim_num )
HR |
Vector of hazard ratios of each test arm vs. the control arm, one entry per dose; must be > 0, with HR < 1 indicating a beneficial effect and HR = 1 the null hypothesis |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock; any other value leaves the boundaries undefined |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
evt_num |
Planned number of events between a test arm and the control arm at the final analysis |
evt_max |
Maximum number of events between a test arm and the control arm allowed after sample size re-estimation |
HR_cut |
Hazard-ratio cut-off for lack of efficacy at the interim analysis: the test drug is not considered to provide clinically meaningful efficacy when the observed hazard ratio is greater than HR_cut |
surv_rate |
Survival rate at the end of follow-up, used to translate the number of events into a sample size |
beta |
Conditional type II error; 1 - beta is the target conditional power for the sample size re-estimation; 0 < beta < 1 |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix of the planned adaptive group sequential design, with one row per analysis and the columns "Look", "Information fraction", "Critical boundaries", "rejection by look" and "Sample size increase"
Matrix with one row per dose and the columns "HR", "selection rate", "rejection rate by each HR" and "conditional rejection rate"; the selection rate is the probability that the dose is selected to proceed to phase 3, the rejection rate is the probability that the dose is selected and the null hypothesis is rejected, and the conditional rejection rate is the rejection rate divided by the selection rate
Named vector of the "rejection rate without SSR", the "rejection rate with SSR", the "Rate of sample size increase", the "Rejection rate without SSR/(meeting sample size incerase criteria)", the "Rejection rate with SSR/(sample size incerase)", the "Type I error", the "Non-Type I error rejection rate", the "Futility rate", the "Planned event number for two arms", the "Maximum number of events for two arms", the "Mean event number for two arms", the "Planned total number of events", the "total number of events" and the "Mean event number for two arms conditional on increase"
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stg_adapt_power_simu_survival(HR = rep(1,2), s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, evt_num = 150, evt_max = 1500, HR_cut = 0.95, surv_rate = 0.5, beta = 0.1, sim_num = 20)two_stg_adapt_power_simu_survival(HR = rep(1,2), s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, evt_num = 150, evt_max = 1500, HR_cut = 0.95, surv_rate = 0.5, beta = 0.1, sim_num = 20)
Phase 2/3 seamless combination Group sequential design: Simulations, Binary distribution, Type I error and power
two_stg_power_simu_binary( p_t, p_c, direction, s, bry_type, alpha_2, samsz, sim_num )two_stg_power_simu_binary( p_t, p_c, direction, s, bry_type, alpha_2, samsz, sim_num )
p_t |
Event rate of the test arm |
p_c |
Event rate of the control arm |
direction |
Test direction: 1 = larger p is better, otherwise smaller p is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock; any other value leaves the boundaries undefined |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
samsz |
Planned sample size of the control arm used in the simulation |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix of the planned group sequential design, with one row per analysis and the columns "Look", "Information fraction", "Critical boundaries" and "rejection by look"
Matrix with one row per dose and the columns "p<sub>t1</sub>,...", "p<sub>c</sub>", "selection rate", " rejection rate by each p_t" and "conditional rejection rate"; the selection rate is the probability that the dose is selected to proceed to phase 3, the rejection rate is the probability that the dose is selected and the null hypothesis is rejected, and the conditional rejection rate is the rejection rate divided by the selection rate
Named vector of the overall "rejection rate", the "Type I error", the "Non-Type I error rejection rate", the "Planned control sample size", the "Mean control sample size", the "Planned total sample size" and the "Mean total sample size"
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stg_power_simu_binary(p_t = c(0.4,0.4,0.4), p_c = 0.4, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)two_stg_power_simu_binary(p_t = c(0.4,0.4,0.4), p_c = 0.4, direction = 1, s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Phase 2/3 seamless combination Group sequential design: Simulations, Negative binomial distribution, Type I error and power
two_stg_power_simu_neg_binomial( r_t, r_c, kappa, direction, s, bry_type, alpha_2, samsz, sim_num )two_stg_power_simu_neg_binomial( r_t, r_c, kappa, direction, s, bry_type, alpha_2, samsz, sim_num )
r_t |
Mean event rates of the test arms; each element must be > 0 |
r_c |
Mean event rate of the control arm; must be > 0 |
kappa |
Dispersion parameter of the negative binomial model; must be > 0 |
direction |
Test direction: 1 = larger r is better, otherwise smaller r is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock; any other value leaves the boundaries undefined |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
samsz |
Planned sample size of the control arm used in the simulation |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix of the planned group sequential design, with one row per analysis and the columns "Look", "Information fraction", "Critical boundaries" and "rejection by look"
Matrix with one row per dose and the columns "r<sub>t1</sub>,...", "r<sub>c</sub>", "selection rate", "rejection rate by each r_t" and "conditional rejection rate"; the selection rate is the probability that the dose is selected to proceed to phase 3, the rejection rate is the probability that the dose is selected and the null hypothesis is rejected, and the conditional rejection rate is the rejection rate divided by the selection rate
Named vector of the overall "rejection rate", the "Type I error", the "Non-Type I error rejection rate", the "Planned control sample size", the "Mean control sample size", the "Planned total sample size" and the "Mean total sample size"
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stg_power_simu_neg_binomial(r_t = c(1.1,1.1,1.1), r_c = 1.1, kappa = 1, direction = 1, s = c(1/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)two_stg_power_simu_neg_binomial(r_t = c(1.1,1.1,1.1), r_c = 1.1, kappa = 1, direction = 1, s = c(1/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Phase 2/3 seamless combination Group sequential design: Simulations, Normal distribution, Type I error and power
two_stg_power_simu_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, samsz, sim_num )two_stg_power_simu_normal( mu_t, mu_c, sigma_t, sigma_c, direction, s, bry_type, alpha_2, samsz, sim_num )
mu_t |
Vector of means of the test arms, one entry per dose |
mu_c |
Mean of the control arm |
sigma_t |
Standard deviation of the test arms |
sigma_c |
Standard deviation of the control arm |
direction |
Test direction: 1 = larger mu is better, otherwise smaller mu is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock; any other value leaves the boundaries undefined |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
samsz |
Planned sample size of the control arm used in the simulation |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix of the planned group sequential design, with one row per analysis and the columns "Look", "Information fraction", "Critical boundaries" and "rejection by look"
Matrix with one row per dose and the columns "mu_t1,...", "mu_c", "selection rate", " rejection rate by each mu_t" and "conditional rejection rate"; the selection rate is the probability that the dose is selected to proceed to phase 3, the rejection rate is the probability that the dose is selected and the null hypothesis is rejected, and the conditional rejection rate is the rejection rate divided by the selection rate
Named vector of the overall "rejection rate", the "Type I error", the "Non-Type I error rejection rate", the "Planned control sample size", the "Mean control sample size", the "Planned total sample size" and the "Mean total sample size"
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stg_power_simu_normal(mu_t = c(0,0,0), mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)two_stg_power_simu_normal(mu_t = c(0,0,0), mu_c = 0, sigma_t = 1, sigma_c = 1, direction = 1, s = c(1/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Phase 2/3 seamless combination Group sequential design: Simulations, Poisson distribution, Type I error and power
two_stg_power_simu_poisson( lambda_t, lambda_c, direction, s, bry_type, alpha_2, samsz, sim_num )two_stg_power_simu_poisson( lambda_t, lambda_c, direction, s, bry_type, alpha_2, samsz, sim_num )
lambda_t |
Mean of the test arm; each element must be > 0 |
lambda_c |
Mean of the control arm; must be > 0 |
direction |
Test direction: 1 = larger lambda is better, otherwise smaller lambda is better |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock; any other value leaves the boundaries undefined |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
samsz |
Planned sample size of the control arm used in the simulation |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix of the planned group sequential design, with one row per analysis and the columns "Look", "Information fraction", "Critical boundaries" and "rejection by look"
Matrix with one row per dose and the columns "lambda_t1,...", "lambda_c", "selection rate", "rejection rate by each lambda_t" and "conditional rejection rate"; the selection rate is the probability that the dose is selected to proceed to phase 3, the rejection rate is the probability that the dose is selected and the null hypothesis is rejected, and the conditional rejection rate is the rejection rate divided by the selection rate
Named vector of the overall "rejection rate", the "Type I error", the "Non-Type I error rejection rate", the "Planned control sample size", the "Mean control sample size", the "Planned total sample size" and the "Mean total sample size"
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stg_power_simu_poisson(lambda_t = c(1.1,1.3,1.4), lambda_c = 1.1, direction = 1, s = c(1/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)two_stg_power_simu_poisson(lambda_t = c(1.1,1.3,1.4), lambda_c = 1.1, direction = 1, s = c(1/3,1), bry_type = 1, alpha_2 = 0.05, samsz = 100, sim_num = 20)
Phase 2/3 seamless combination Group sequential design: Simulations, Survival analysis, Type I error and power
two_stg_power_simu_survival( HR, s, bry_type, alpha_2, surv_rate, evt_num, sim_num )two_stg_power_simu_survival( HR, s, bry_type, alpha_2, surv_rate, evt_num, sim_num )
HR |
Vector of hazard ratios of each test arm vs. the control arm, one entry per dose; HR < 1 indicates a beneficial effect |
s |
Vector of information fractions of each analysis, e.g. c(1/3, 2/3, 1); the last element must be 1 |
bry_type |
Critical boundary type: 1 = O'Brien-Fleming, 2 = Pocock; any other value leaves the boundaries undefined |
alpha_2 |
Type I error for the two-sided test; 0 < alpha_2 < 1 |
surv_rate |
Survival rate at the end of follow-up, used to translate the number of events into a sample size |
evt_num |
Number of events between a test arm and the control arm at the final analysis, used as the planned number of events |
sim_num |
Number of simulation replicates |
A named list with the following components:
Matrix of the planned group sequential design, with one row per analysis and the columns "Look", "Information fraction", "Critical boundaries" and "rejection by look"
Matrix with one row per dose and the columns "HR", "selection rate", "rejection rate by each HR" and "conditional rejection rate"; the selection rate is the probability that the dose is selected to proceed to phase 3, the rejection rate is the probability that the dose is selected and the null hypothesis is rejected, and the conditional rejection rate is the rejection rate divided by the selection rate
Named vector of the overall "rejection rate", the "Type I error", the "Non-Type I error rejection rate", the "Planned event number for two arms", the "Mean event number for two arms", the "Planned total number of events" and the "total number of events"
P. Gao, Y. Li (05 May 2024): Adaptive two-stage seamless sequential design for clinical trials, Journal of Biopharmaceutical Statistics, DOI: 10.1080/10543406.2024.2342518. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2342518
Gao, P., Zhang, W. (2024). A systematic approach to adaptive sequential design for clinical trials: using simulations to select a design with desired operating characteristics. Journal of Biopharmaceutical Statistics,2024 Aug;34(5):737-752. doi:10.1080/10543406.2024.2358796. Epub 2024 May 30. To link to this article: https://www.tandfonline.com/doi/full/10.1080/10543406.2024.2358796
H.H. Müller, H. Schäfer. Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches. BIOMETRICs 57, 886-891. September 2001.
P. Gao, J. H. Ware, C. Mehta. Sample size re-estimation for adaptive sequential design in clinical trials. J Biopharm Stat. 2008;18(6):1184-96.
two_stg_power_simu_survival(HR = c(0.8,0.8), s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, surv_rate = 0.5, evt_num = 800, sim_num = 20)two_stg_power_simu_survival(HR = c(0.8,0.8), s = c(1/3,2/3,1), bry_type = 1, alpha_2 = 0.05, surv_rate = 0.5, evt_num = 800, sim_num = 20)