| Title: | Single Arm Phase 2 Oncology Trial R Code Library |
|---|---|
| Description: | Single arm phase 2 oncology trial. |
| Authors: | Ping Gao [aut, cre] |
| Maintainer: | Ping Gao <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 0.1 |
| Built: | 2026-10-15 09:35:03 UTC |
| Source: | https://github.com/innovatiostat/rcode |
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. |
r_1_e |
The trial would be stopped for superiority if at least r_1_e responses are observed at the first look. |
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. |
x_1 |
Number of responses at the first look. |
x_new |
Number of responses at the final visit. |
alpha |
One-sided type I error. |
UL: upper limit of confidence interval.
LL: lower limit of confidence interval.
The results with continuity correction are not necessarily different from those with continuity correction.
The need for continuity correction can be determined through simulations on type I error (see Gao, Zhang, 2004).
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)
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 |
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. |
r_2 |
If no more than r_2 responses are observed at the second look, then the trial would be stopped for futility. |
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 (at the second look). |
x_2 |
Number of responses at the second look. |
x_new |
Number of responses at the final visit. |
alpha |
One-sided type I error. |
UL: upper limit of confidence interval.
LL: lower limit of confidence interval.
The results with continuity correction are not necessarily different from those with continuity correction.
The need for continuity correction can be determined through simulations on type I error (see Gao, Zhang, 2004).
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)
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 |
One-sided type I error. |
beta |
Type II error. |
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 |
Min(n3-n2) sets the minimal difference between n3 and n2 for the design. |
n1: Number of patients at first look. n1 for the three-stage design is the from Simon’s design with p=p_1.
n_2: Number of patients at second look. n_2 for the three-stage design is the n1 from Simon’s design with p=p_low.
r1: If no more than r1 responses are observed at the first look, then the trial would be stopped for futility.
r_2: The null hypothesis will be rejected if at least r_2 responses are observed at the end of the trial.
EN(p0) is the expected sample size under the null hypothesis.
PET(p0) is the probability of early termination under the null hypothesis.
PET_p is the probability of early termination under p=p_1.
PET_p_low is the probability of early termination under p=p_low.
power_p_1 is the power under p=p_1.
power_p_low is the power under p=p_low.
Type I error is the probability of rejecting the null hypothesis under p<=p_0.
The minimax design has the smallest n_2 among all possible choices of (n1,r1,n_2,r_2).
The optimal design has the smallest EN(p0) among all possible choices of (n1,r1,n_2,r_2).
The n1 and n_2 for the average design is the average of n1's and n_2's from the minimax and the optimal designs.
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.
## Not run: EXP_3_stg_sz(alpha=c(0.001,0.024),beta=0.1,p_0=0.2,p_low=0.33,p_1=0.4,d_23_cut=5) ## End(Not run)## Not run: EXP_3_stg_sz(alpha=c(0.001,0.024),beta=0.1,p_0=0.2,p_low=0.33,p_1=0.4,d_23_cut=5) ## End(Not run)
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 |
The trial would be stopped for superiority if at least r2e responses are observed at the second look. |
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 |
|
alpha |
One-sided type I error. |
UL: upper limit of confidence interval.
LL: lower limit of confidence interval.
The results with continuity correction are not necessarily different from those with continuity correction.
The need for continuity correction can be determined through simulations on type I error (see Gao, Zhang, 2004).
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)
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. |
n_3 |
Number of patients at third look. |
r_2_e |
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. |
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 |
|
alpha |
One-sided type I error. |
UL: upper limit of confidence interval.
LL: lower limit of confidence interval.
The results with continuity correction are not necessarily different from those with continuity correction.
The need for continuity correction can be determined through simulations on type I error (see Gao, Zhang, 2004).
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)
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 |
One-sided type I error. |
beta |
Type II error. |
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 |
Min(n2-n1) sets the minimal difference between n2 and n1 for the design. |
n1: Number of patients at first look.
n_2: Number of patients at second look. n_2 is also the total sample size.
r1: If no more than r1 responses are observed at the first look, then the trial would be stopped for futility.
r_2: The null hypothesis will be rejected if at least r_2 responses are observed at the end of the trial.
EN(p0) is the expected sample size under the null hypothesis.
PET(p0) is the probability of early termination under the null hypothesis.
PET_p is the probability of early termination under p=p_1.
PET_p_low is the probability of early termination under p=p_low.
power_p is the power under p=p_1.
power_p_low is the power under p=p_low.
Type I error is the probability of rejecting the null hypothesis under p<=p_0.
The minimax design has the smallest n2 among all possible choices of (n1,r1,n_2,r_2).
The optimal design has the smallest EN(p0) ) among all possible choices of (n1,r1,n_2,r_2).
The n1 for the average design is the average of n1's from the minimax and the optimal designs.
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)
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 |
For two-stage designs, r_final=r2. For the three-stage design, r_final=r3. |
n_inter |
Number of patients at the interim look. For two-stage designs, n_inter=n1. For the three-stage design, n_inter=n2. |
n_final |
For two-stage designs, n_inter=n2. For the three-stage design, n_inter=n3. |
x_inter |
Number of responses observed at the interim look. |
beta |
The conditional type II error. 1-beta is the desired conditional power. |
N_max |
Pre-selected maximum sample size. |
n_new: is the new sample size.
r_new (without continuity correction) and r_new (with continuity correction) are thresholds such that the null hypothesis will be rejected at n_new if at least r_new responses are observed. Simulations on type I error will help to determine if continuity correction is needed.
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)
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 |
One-sided type I error. |
beta |
Type II error. |
p_0 |
Response rate indicating low activity with insufficient clinical benefit. |
p_low |
Minimally clinically beneficial response rate. |
p_1 |
Assumed response rate. |
q |
0<q<1. |
d_12_cut |
Min(n2-n1) sets the minimal difference between n2 and n1 for the design. |
n1: Number of patients at first look.
n_2: Number of patients at second look. n_2 is also the total sample size.
r1: If no more than r1 responses are observed at the first look, then the trial would be stopped for futility.
r_2: The null hypothesis will be rejected if at least r_2 responses are observed at the end of the trial.
EN(p0) is the expected sample size under the null hypothesis.
PET(p0) is the probability of early termination under the null hypothesis.
PET_p is the probability of early termination under p=p_1.
PET_p_low is the probability of early termination under p=p_low.
power_p is the power under p=p_1.
power_p_low is the power under p=p_low.
Type I error is the probability of rejecting the null hypothesis under p<=p_0.
The minimax design has the smallest n_2 among all possible choices of (n1,r1,n2,r2).
The optimal design has the smallest EN(p_0) ) among all possible choices of (n1,r1,n_2,r_2).
The n1 for the average design is the average of n1's from the minimax and the optimal designs.
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=0.4,q=0.5,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=0.4,q=0.5,d_12_cut=5)
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 |
The assumed response rate. |
p_0 |
Response rate indicating low activity with insufficient clinical benefit. |
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 |
The trial would be stopped for superiority if at least r_2_e responses are observed at the second look. If r_2_e is not used in the design, enter NA. |
n_3 |
Number of patients at third 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_max |
Maximum sample size for the trial. |
beta |
Type II error. The target power is 1−beta. |
sim_num |
Purpose of simulation: To verify type I error control by setting p_1=p_0, To evaluate the operating characteristics of the adaptive design and to determine if continuity correction is necessary. |
Average sample size is EN(p_1).
Rate of termination is PET(p_1)
If rejection rate without continuity correction does not exceed α, the continuity correction is not needed for this group of parameters p_0,n_1,r_1,n_2,r_2,r_2_e,n_3,r_3. Otherwise, continuity correction should be applied.
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.
## Not run: 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=100000) ## End(Not run)## Not run: 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=100000) ## End(Not run)
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 |
The assumed response rate. |
p_0 |
Response rate indicating low activity with insufficient clinical benefit. |
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 |
The trial would be stopped for superiority if at least r_1_e responses are observed at the first look. If r_1_e is not used in the design, enter NA. |
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 null hypothesis will be rejected. |
N_max |
Maximum sample size for the trial. |
beta |
Type II error. The target power is 1−beta. |
sim_num |
Purpose of simulation: To verify type I error control by setting p_1=p_0, To evaluate the operating characteristics of the adaptive design and to determine if continuity correction is necessary. |
Average sample size is EN(p_1).
Rate of termination is PET(p_1)
If rejection rate without continuity correction does not exceed α, the continuity correction is not needed for this group of parameters p_0,n_1,r_1,r_1_e,n_2,r_2. Otherwise, continuity correction should be applied.
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.
## Not run: one_arm_ad_two_stg_binary(p_1=0.2,p_0=0.2,n_1=37,r_1=9,r_1_e=NA,n_2=53,r_2=16, N_max=140,beta=0.1,sim_num=100000) ## End(Not run)## Not run: one_arm_ad_two_stg_binary(p_1=0.2,p_0=0.2,n_1=37,r_1=9,r_1_e=NA,n_2=53,r_2=16, N_max=140,beta=0.1,sim_num=100000) ## End(Not run)
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 |
The assumed response rate. |
p_0 |
Response rate indicating low activity with insufficient clinical benefit. |
n_1 |
Number of patients at first look. |
r_1_f |
If no more than r_1 responses are observed at the first look, then the trial would be stopped for futility. |
r_1_e |
The trial would be stopped for superiority if at least r_1_e responses are observed at the first look. If r_1_e is not used in the design, enter NA. |
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 null hypothesis will be rejected. |
sim_num |
Purpose of simulation: To verify type I error control by setting p_1=p_0, To verify the parameters n_1,r_1_f,r_1_e,n_2,r_2 are correctly chosen, such that the rejection rate matches that from the design table from either the hybrid design or the mid-point design. |
Average sample size is EN(p_1).
Rate of termination is 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.
## Not run: 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=100000) ## End(Not run)## Not run: 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=100000) ## End(Not run)
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 |
The assumed response rate. |
p_0 |
Response rate indicating low activity with insufficient clinical benefit. |
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 responses are observed at the second look, then the trial would be stopped for futility. |
r_2_e |
The trial would be stopped for superiority if at least r_2_e responses are observed at the first look. If r_2_e is not used in the design, enter NA. |
n_3 |
Number of patients at third look. |
r_3 |
If at least r_2 responses are observed at the third look, then the null hypothesis will be rejected. |
sim_num |
Purpose of simulation: To verify type I error control by setting p_1=p_0, To verify the parameters n_1,r_1,n_2,r_2_f,r_2_e,n_3,r_3 are correctly chosen, such that the rejection rate matches that from the design table of the three-stage design. |
Average sample size is EN(p_1).
Rate of termination is 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.
## Not run: 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=100000) ## End(Not run)## Not run: 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=100000) ## End(Not run)
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 |
The assumed response rate. |
p_0 |
Response rate indicating low activity with insufficient clinical benefit. |
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 null hypothesis will be rejected. |
sim_num |
Purpose of simulation: To verify type I error control by setting p_1=p_0, To verify the parameters n_1,r_1,n_2,r_2 are correctly chosen, such that the trial has 1−β power if the true response rate is p_1. |
Average sample size is EN(p_1).
Rate of termination is 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.
## Not run: 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=100000) ## End(Not run)## Not run: 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=100000) ## End(Not run)
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 |
One-sided type I error. |
beta |
Type II error. |
p_0 |
Response rate indicating low activity with insufficient clinical benefit. |
p_1 |
Assumed response rate. |
r1: If no more than r1 responses are observed at the first look, then the trial would be stopped for futility.
n1: Number of patients at first look.
r_2: The null hypothesis will be rejected if at least r_2 responses are observed at the end of the trial.
n_2: Number of patients at second look. n_2 is also the total sample size.
EN(p_0) is the expected sample size under the null hypothesis.
PET(p_0) is the probability of early termination under the null hypothesis.
PET_p_1 is the probability of early termination under p=p_1.
power_p_1 is the power under p=p_1.
Type I error is the probability of rejecting the null hypothesis under p<=p_0.
The minimax design has the smallest n_2 among all possible choices of (n1,r1,n_2,r_2).
The optimal design has the smallest EN(p_0) among all possible choices of (n1,r1,n_2,r_2).
The average design is an augmentation of the original Simon’s design (Gao, Zhang. 2024). The n1 for the average design is the average of n1's from the minimax and the optimal designs.
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)