1. A Flexible Two Stage Design in Active Control Non-inferiority Trials Gang Chen, Yong-Cheng Wang, and George Chi † Division of Biometrics I, CDER, FDA Qing Liu* J & J PRD MCP 2002, Bethesda MD August 5-7, 2002 †: The views expressed in this presentation do not necessarily represent those of the U.S. Food and Drug Administration. *: This is a continuing research based on the work initiated while Dr. Qing Liu was affiliated with FDA.

2. Outline An Example Non-inferiority trial: –objectives, hypotheses, tests, type I error, sample size Two stage adaptive designs for sample size re-estimation A flexible design for sample size calculation –superiority trials –non-inferiority trials Summary and issues

3. An Example An example of a flexible design for sample size calculation in a non-inferiority trial: T is an approved dose for a treatment. Investigator (sponsor) wants to lower the dose to improve the toxicity profile without compromising much loss in treatment effect.

4. An Example Trial design: A randomized, active-control trial Primary efficacy endpoint: response rate Two arms: –T: approved dose –C: a low dose never studied Non-inferiority hypothesis: low treatment dose can preserve at least 75% effect of the approved treatment dose

5. An Example Since there is no information on the efficacy for this low dose treatment, a two stage non-inferiority design is proposed –stage 1: recruit 100 patient/arm to evaluate the treatment effect of low dose and calculate sample size –stage 2: recruit n patients (calculated based on stage 1 data) for the non-inferiority trial. Sponsor’s Question: Can we include stage 1 data in the final analysis?

6. Non-inferiority trial - objectives Brief introduction of objectives, hypotheses, tests, type I error control, sample size determination in the design of active control non-inferiority trials: Objectives: –To establish efficacy through testing a fraction retention of control effect –To establish non-inferiority or equivalence

7. Non-inferiority trial - hypotheses Some notations: T, C and P denote the treatment, control and placebo respectively. µ tp =T-P: treatment effect relative to the placebo P µ cp =C-P: control effect relative to the placebo P µ tc =T-C: the treatment effect relative to C. The proportion of the active control effect:  =µ tp /µ cp.

8. Non-inferiority trial - hypotheses Non-inferiority hypotheses: –hypotheses with a pre-selected fixed margin –hypotheses with a fixed fraction retention The detailed discussion on those hypotheses is given in [1]. When testing whether the treatment maintains a proportion  0 (<= 1) of active control effect, hypotheses are: H 0 : µ tp  0 µ cp or (under constancy assumption for the control effect) H 0 : µ tc -(1-  0 )µ cp [1]: Chen et al (2001), Active control trials - hypotheses and issues. ASA Proceedings

9. Non-inferiority trial - test statistic The test statistic for the above hypotheses:

10. Non-inferiority trial - type I error Asymptotic alpha of the test [2]: [2]: Rothmann et al (2001), Non-inferiority methods for mortality trials. ASA Proceedings.

11. Non-inferiority trial - sample size The sample size n (under H a : T=C) for a binary endpoint:

12. Non-inferiority trial - sample size The sample size determination in a non-inferiority trial depends on the following factors –control effect size and a proportion retention –standard errors from current and historical trials –alpha and power

13. Non-inferiority trial - sample size At the design stage: –Known: alpha, beta, fraction retention and control effect size (estimate) and its associated variation, –Unknown: treatment effect and its associated variation (relative to control). A two stage flexible design can be used for sample size determination. The purpose for the 1 st stage is only for evaluation of treatment effect and its associated variation –Question: Can we include stage 1 data in the final analysis?

14. Non-inferiority trial - sample size Answer: Yes, but the overall type I error should be controlled at the desired level.

15. Two stage adaptive designs for sample size re-estimation Existing methods for two stage adaptive designs, example: those methods proposed by Bauer & Kieser, Proschan and Hunsbarger, Liu & Chi, Cui et al: –choosing a conditional error function to control type I error –down-weight data collected from second stage to preserve the overall type I error. –base sample size on conditional power –other In non-inferiority trials, those methods above can not apply directly and need to be modified.

16. A flexible two stage design To calculate sample size: Stage 1 sample size n 1 is selected arbitrarily, and stage 1 data provides information on treatment effect size, variation and futility. Total sample size can be estimated based on stage 1 data. Final analysis includes the pooled data of both stages.

17. A flexible two stage design Major issues: Two sources for the type I error inflation: –arbitrary selection of the size of stage 1 (n 1 ) –total sample size (n) calculation based on the first stage data Distribution of the final test statistic: –test: T=T(n, control information, stage 1 & 2 data) –sample size: n = n(stage 1data, control information)

18. Type I error inflation due to arbitrary selection of stage 1 size superiority test, alpha=.05, power=.8

19. Type I error inflation due to arbitrary selection of stage 1 size non-inferiority test, alpha=.05, power=.8, Nc = 300

20. A flexible two stage superiority design The procedure : At stage 1 –Effect size  n1 and its associated variation are estimated –Overall sample size n can be calculated based on  n1.. At the final –Let T n (  n1 ) be a test statistic –Conditional type I error to reject the null is  (  n1 ) = Pr (T n (  n1 ) >C  /2 ), where C  /2 is the critical value –Overall type I error becomes:  =E  (  n1 ).

21. A flexible two stage superiority design Simulation results for type I error inflation  *=E  (  n1 ) due to the inclusion of stage 1 trial data.

22. A flexible two stage non-inferiority design Similarly, the procedure : At stage 1 –Treatment effect size  TC n1 and its associated variation SE TC n1 are estimated based on stage 1 data and control information –Overall sample size n is calculated At the final –Let T n (  TC n1, control info) be a test statistic –Conditional type I error to reject the null is  (  TC n1 ) = Pr (T n (  TC n1 ) >C  /2 ), where C  /2 is the critical value –Overall type I error becomes:  =E  (  TC n1 ).

23. A flexible two stage non-inferiority design Simulation results  * = E  (  TC n1 ) due to the inclusion of stage 1 trial data.

24. Summary and issues Question: Can we include stage 1 data in the final analysis? Response: Yes, but the inclusion of stage 1 data needs to control both sources for overall Type I error inflation (under research) –due to arbitrary selection of the sample size of stage 1 –due to the inclusion of stage 1 data in the final test to assess the distribution of of final test statistic: (under research) –test: T=T(n, control information, stage 1 & 2 data) –sample size: n = n(stage 1data, control information)