Testing and Estimation Procedures in Multi-Armed Designs with Treatment Selection Gernot Wassmer, PhD Institut für Medizinische Statistik, Informatik und Epidemiologie Universität zu Köln ADDPLAN GmbH Adaptive Design KOL Lecture Series, August 14th, 2009

Introduction  Confirmatory adaptive designs are a generalization of group sequential designs, where - in interim analyses - confirmatory analysis is performed under control of the Type I error rate and data dependent changes of design are allowed.  Three particular applications –Sample size reassessment –Treatment arm selection –Subset selection (“enrichment designs”)  This talk shows –how to reach a test decision in an adaptive multi-armed trial with treatment selection at interim –how to calculate confidence intervals and overall p-values

Confirmatory adaptive designs can be based on –the combination testing principle –the conditional error approach Combination testing principle Combination of p-values with a specific combination function (Bauer, 1989; Bauer & Köhne, 1994) Inverse normal method: The test decision is based on Lehmacher & Wassmer, 1999 where the weights w k are prefixed

The conditional error approach Plan a trial with reasonable (optimum) design, including sample size calculation and timing of interim analyses. Calculate the conditional Type I error rate  (x 1,…,x k ) at any time during the course of the trial  (x 1,…,x k ) = conditional probability, under H 0, of rejecting H 0 in one of the subsequent stages, given x 1,…,x k x 1,…,x k : data up to stage k Remainder of the trial can be defined as a test at level  (x 1,…,x k ) where the design of this test is arbitrary. Müller & Schäfer (2001): “CRP principle” Brannath, Posch & Bauer (2002): “Recursice testing principle”

The situation  Consider many-to-one comparisons, e.g., G treatment arms and one control, normal case.  Throughout this talk, we consider one-sided testing.  In an interim stage a treatment arm is selected based on data observed so far.  Not only selection procedures, but also other adaptive strategies (e.g., sample size reassessment) can be performed.  Application within “Adaptive seamless designs” using the combination testing principle

Sources for alpha inflation  Interim analyses  Sample size reassessment  Multiple arms The proposed adaptive procedure fulfils the regulatory requirements for the analysis of adaptive trials in that it strongly controls the prespecified Type I error rate. This procedure will be based on the application of the closed test procedure together with combination tests (e.g., Bauer & Kieser, 1999; Hellmich, 2001; Posch et al., 2005, Bretz et al., 2009). Other approaches: Thall et al., 1988; Follmann et al, 1994; Stallard and Todd, 2003; Stallard and Friede, 2008;

Stage II … Simple “trick”: Test of intersection hypotheses are formally performed as tests for ? ? Stage I Closed testing procedure

At the first interim analysis, consider a test statistic for e.g., the test statistic Closed testing procedure That is, compute Dunnett’s adjusted p-value for each intersection hypothesis, critical values are according to Or compute the p-value using Dunnett’s t distribution.

where p J is the p-value of the Dunnett test for testing q S is the second stage test statistic for the selected treatment arm, and u 2 is the critical value for the second stage. Test decision for the second stage: is rejected if This is the use of the inverse normal method for the Dunnett test situation.

can be rejected if all combination tests exceed the critical value u 2. Stage I Stage II … Example S = 3 Simple shortcut: If the treatment arm with the largest test statistic is selected, it suffices to combine the test for H 0 : with the test for H 0 :

Properties of the Procedure  Choice of tests for intersection hypotheses is free, i.e., you might select, e.g., Dunnett‘s test, Bonferroni-, Simes or Sidak‘s test.  The procedure may become inconsonant and, hence, conservative. I.e., you can reject the global hypothesis, but no single hypothesis (Friede and Stallard, 2008).  A hypothesis can be rejected at a later stage even it was not selected for the current stage (and not rejected before). This can happen if, e.g., the test statistic for the global hypothesis exceeds u 2 in the second stage but not u 1 in the first stage, and the test statistic for the de-selected hypothesis exceeds u 1 in the first stage. 12

An alternative procedure (König et al., 2008) Compute conditional error at first stage: In the second stage, perform a  Conditional second-stage Dunnett test  Separate second-stage Dunnett test This is the application of the CRP principle (Müller & Schäfer, 2001). It assumes the variance to be known

A comparison shows that  the conditional second-stage Dunnett test performs best but is hardly better if a treatment arm selection was performed (cf., Friede and Stallard, 2008)  it is identical with the conventional Dunnett test if no adaptations were performed  becomes complicated if, e.g., –allocation is not constant –variance is unknown  the inverse normal technique is not optimum but enables early stopping and more general adaptations  is straightforward if, e.g., –allocation is not constant –variance is unknown

A comparison shows that  the conditional second-stage Dunnett test performs best but is hardly better if a treatment arm selection was performed (cf., Friede and Stallard, 2008)  it is identical with the conventional Dunnett test if no adaptations were performed  becomes complicated if, e.g., –allocation is not constant –variance is unknown  the inverse normal technique is not optimum but enables early stopping and more general adaptations  is straightforward if, e.g., –allocation is not constant –variance is unknown

Overall p-values  Defined as smallest p-value for which the test results yield rejection of the considered (single) hypothesis  Repeated overall p-value can be calculated at any stage of the trial.  That is,  p-values account for the step-down nature of the closed testing principle and are completely consistent with the test decision.

Overall confidence intervals  Confidence intervals based on stepwise testing are difficult to construct. This is a specific feature of multiple testing procedures and not of adaptive testing.  Posch et al. (2005) proposed to construct confidence intervals based on the single step adjusted overall p-values. These can also be applied for the conditional Dunnett test.  The RCIs are not, in general, consistent with the test decision. It might happen that, e.g., a hypothesis is rejected but the lower bound of the CI is smaller 0.  They can be provided for each step of the trial.  In general, they may fail to become narrower for increasing sample size (e.g., if Bonferroni or Simes intersection tests are used).

Illustration  Two-stage design with G treatment arms  Selection of treatment arm with highest respone, no efficacy stop at interim  Bonferroni (or Simes) correction is used for first stage  Lower bound lb j of 95% confidence intervals for effect  j =  j -  0 of selected treatment arm at second stage is calculated through 21 It is easy to see that

Summary  The adaptive procedures fulfil the regulatory requirements for the analysis of adaptive trials in that they control the prespecified Type I error rate. For regulatory purposes, the class of envisaged decisions after stage 1 should be stated in the protocol.  The “rules” for adaptation and stopping for futility –not need to be pre-specified –Adaptations may depend on all interim data including secondary and safety endpoints. –can make use of Bayesian principles integrating all information available, also external to the study –should be evaluated (e.g. via simulations) and preferred version recommended, e.g., in DMC charter  Software ADDPLAN MC is available for designing and analyzing these trials 22

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