Modification of Sample Size in Group Sequential Clinical Trials

1. Modification of Sample Size in Group Sequential Clinical Trials MadanGopalKundu PhD (Biostatistics) student, IUPUI

2. Sample Size Less expensive More expensive LESS SAMPLE SIZE MORE Less Statistical Power More Statistical Power Sample size (N) = No. of patients Estimation of Sample size depends on - Type I error, Power and Expected effect size Reasonable power at reasonable cost (Best deal!!) Sample size is determined at the beginning of trial.

3. Group Sequential Design Determine N Conduct of Clinical Trial Final Analysis Interim analysis 1 Interim analysis 2 Interim analysis 3 • Scope of Early termination of trial • Overwhelming efficacy • futility of the drug

4. Outline Introduction A case study Group sequential Z test A sequential test procedure with sample size modification (based on Z test) Generalization: Brownian motion

5. Introduction From Interim analyses we have…. Sample size is planned based on…. Less than Expected Effect Size Observed Effect Size Planned sample size is NOT sufficient There is scope to modify sample size when trial is ongoing Q: Does it increase overall type I error? Q: Is there any testing strategy to modify sample size without increasing overall type I error?

6. Motivation: A case study Planning Expected Effect size = 0.30 Interim Analysis • After evaluation of 300 patients • Incidence rate in Placebo ~ 22% • Incidence rate in New Drug = 16.5% • Need to increase the sample size Observed Effect size = 0.14 Phase III, comparative, placebo-controlled trial for prevention of myocardial infarction. Assumption: Incidence rate in Placebo: 22% Incidence rate in New Drug: 11% Planned sample size = 600 (power>95%)

7. Motivation: A case study Concern Finally… • It was decided not to increase the sample size • Trial eventually failed to show a statistically significant effect Does it inflate overall type I error? No valid testing procedure was available to account for such an outcome –dependent adjustment of sample size

8. Solution to this Dilemma… To have accurate estimate of treatment effect size at the beginning of trial - Less likely!! Implementation of valid inferential procedure that allows adjustment of sample size in the mid-course of trial - Cui, Hung and Wang method

9. Theoretical set-up Population I N (µ1, σ2=1) Population II N (µ2, σ2=1) x1, x2, …. , xN y1, y2, …. , yN Effect size, ∆ = µ1 - µ2 Our interest is to test (using two sample Z-test) Ho : ∆ = 0 vs Ha : ∆ > 0 Assuming ∆ = δ, total sample size (N) per population

10. Group-sequential structure Trial Initiation (K-1) Interim Analyses Final Analysis 0 1 2 L-1 L K-1 K Additional Subjects n1 n2 nL-1 nL nK-1 nK Cumulative Subjects N1 N2 NL-1 NL NK-1 NK=N Information Time Observed Effect size ∆ 1 ∆ 2 ∆ L-1 ∆ L ∆ K-1 ∆ K 2-sample Z Test Statistic T1 T2 TL-1 TL TK-1 TK Critical values C1 C2 CL-1 CL CK-1 CK Reject Ho & Stop trial if: T1>C1 T2>C2 TL-1>CL-1 TK-1>CK-1 TK>CK TL>CL

11. Conditional Power • Sample size may be modified based on conditional power. • { } is the Rejection Region. The conditional power evaluated at the Lth interim analysis

12. Sample Size Modification • This adjustment of sample size preserves the unconditional power at 1-β when • If is smaller than δ then it gives large M. Calculate and Decide two positive constants ≤1 ≤ If or , N should be modified to

13. Does Sample Size adjustment inflates Type I error?? Simulation studies Increase in sample size Substantial inflation in Type I error rate Decrease in sample size Mild effect on Type I error rate and power

14. Sample size modification Trial Initiation (K-1) Interim Analyses Final Analysis 0 1 2 L L+1 L+j K-1 K Cumulative sample size N1 N2 NL NL+1 NL+j NK-1 N Cumulative sample size (with adjust) N1 N2 NL ML+1 ML+j M MK-1 Modify sample size in Lth interim analysis: N→M

15. Effect on Test Statistic because of sample size modification Test statistic at (L+j)th interim analysis, … (Eq. 1) Where, When sample size is NOT allowed to increase

16. Effect on Test Statistic because of sample size modification … (Eq. 2) Where, (Eq. 1) versus (Eq. 2) Note: Here is replaced by Note: Weights are also changed and become random as is a function of When sample size is allowed to increase

17. A different group sequential test procedure (CHW) … (Eq. 3) Note: UL+jreduces to TL+j, when ML+j = NL+j Trial Initiation Final Analysis Test procedure (K-1) Interim Analyses 0 1 2 L L+1 L+j K-1 K Test Statistic T1 T2 TL UL+1 UL+j UK-1 UK Critical values C1 C2 CL CL+1 CL+j CK-1 CK Reject Ho & Stop trial if: T1>C1 T2>C2 TL>CL UL+1>CL+1 UL+j>CL+j UK-1>CK-1 UK>CK Here weights are kept fixed but are replaced by

18. Distribution of UL+j Under H0 : µ1 - µ2 =0

19. Distribution of UL+j

20. Impact on Type-I error D So, D Overall Type I error of the new test Procedure = = Overall Type I error of the Original test Procedure = α Monte Carlo Simulation: New test has its type I error rate maintained at α. Conclusion: New test procedure allows to modify sample size without increase in overall type I error.

21. Generalization… Scope Repeated significance test with Brownian motion process and Independent increment Steps • B(t) be such repeated significance test at information time t. • T(t) = B(t)/t1/2 • Let at t=tL sample size increased to M • w = N/M • b =(w – tL)/(1-tL) Test Statistic U(t) = T(t) , if t≤tL = , if t>tL

22. Brownian Motion B(t|tЄT) is known as Brownian motion process if Multivariate Normal Distribution Mean = 0 Var {B(t)}= t Var {B(t2) – B(t1)} = t2 – t1 Cov {B(t2), B(t1)} = min(t2, t1)

23. Brownian Motion: Z test statistic

24. Brownian Motion: Other Test statistic T-test Approx. Brownian motion (see Pocock 1977) Log-rank test Wilcoxon test Approx. Brownian motion (see Slud & Wei 1982) Approx. Brownian motion (see Tsiatis 1982, Sellke & Siegmund 1983, Slud 1984)

25. Reference Cui L, Hung H J and Wang S J (1999). Modification of sample size in Group Sequential Clinical Trials. Biometrics55: 853-857. Pocock S J (1977). Group sequential methods in the design and analysis of clinical trials. Biometrika64: 191-199. Lan K K G and Wittes J (1988). The B-value: A tool for monitoring data. Biometrics 44: 579-585. Lan K K G and Zucker D M (1993). Sequential monitoring of clinical trials: the role of information and brownian motion. Statistics in Medicine 12: 753-765. Reboussin D M, DeMets D L, Kim K M and Lan K K G (2000). Computation for group sequential boundaries using the Lan-DeMets spending function method. Controlled Clinical Trials 21: 190-207. Lan K K G and DeMets D L (1983). Discrete sequential boundaries for clinical trials. Biometrika70: 659-663.

26. Reference Shih W J (2003). Group Sequential Methods. Encyclopedia of Biopharm. Statistics1:1 423-432. Tsiatis A A (1982). Repeated significance testing for a general class of statistics used in censored survival analysis. JASA77: 855-861. Sellke T and Siegmund D (1983). Sequential analysis of the proportional hazard model. Biometrika70: 315-326 Slud E V (1984). Sequential linear rank tests for two sample censored survival data. Annals of Statistics12: 551-571. Slud E and Wei L J (1982). Two-sample repeated significance tests based on the modified Wilcoxon test statistic . JASA 77(380): 862-867.

27. Thank You…!

28. BACK-UP Slides…

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