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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
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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.
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
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.
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
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 npatients (calculated based on stage 1 data) for the non-inferiority trial. • Sponsor’s Question: Can we include stage 1 data in the final analysis?
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
Non-inferiority trial - hypotheses Some notations: • T, C and P denote the treatment, control and placebo respectively. • µtp=T-P: treatment effect relative to the placebo P µcp=C-P: control effect relative to the placebo P µtc=T-C: the treatment effect relative to C. • The proportion of the active control effect: =µtp/µcp.
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: H0: µtp< 0 µcp vs. H1: µtp> 0 µcp or (under constancy assumption for the control effect) H0: µtc< -(1-0)µcp vs. H1: µtc> -(1-0)µcp [1]: Chen et al (2001), Active control trials - hypotheses and issues. ASA Proceedings
Non-inferiority trial - test statistic • The test statistic for the above hypotheses:
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.
Non-inferiority trial - sample size • The sample size n (under Ha: T=C) for a binary endpoint:
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
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 1st stage is only for evaluation of treatment effect and its associated variation • Question: Can we include stage 1 data in the final analysis?
Non-inferiority trial - sample size Answer: Yes, but • the overall type I error should be controlled at the desired level.
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.
A flexible two stage design To calculate sample size: • Stage 1 sample size n1 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.
A flexible two stage design Major issues: • Two sources for the type I error inflation: • arbitrary selection of the size of stage 1 (n1) • 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)
Type I error inflation due to arbitrary selection of stage 1 size superiority test, alpha=.05, power=.8
Type I error inflation due to arbitrary selection of stage 1 size non-inferiority test, alpha=.05, power=.8, Nc = 300
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 Tn( n1) be a test statistic • Conditional type I error to reject the null is ( n1) = Pr (Tn( n1) >C /2), where C /2 is the critical value • Overall type I error becomes:=E ( n1).
A flexible two stage superiority design • Simulation results for type I error inflation *=E ( n1) due to the inclusion of stage 1 trial data.
A flexible two stage non-inferiority design Similarly, the procedure : • At stage 1 • Treatment effect size TCn1 and its associated variation SETC n1 are estimated based on stage 1 data and control information • Overall sample size n is calculated • At the final • Let Tn(TCn1, control info) be a test statistic • Conditional type I error to reject the null is (TCn1) = Pr (Tn(TCn1) >C /2), where C /2 is the critical value • Overall type I error becomes:=E ( TCn1).
A flexible two stage non-inferiority design • Simulation results * = E(TCn1) due to the inclusion of stage 1 trial data.
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)