STATISTICS IN MEDICINE Statist Med 2000 00 1 6 Prepared using simauth cls Version 2002 09 18 v1 11 Efficient group sequential designs when there are several effect sizes under consideration Christopher Jennison1 and Bruce W Turnbull2 2 1 Department of Mathematical Sciences University of Bath Bath BA2 7AY U K School of Operations Research and Industrial Engineering Cornell University Ithaca NY 14853 U S A SUMMARY We consider the construction of efficient group sequential designs where the goal is a low expected sample size not only at the null hypothesis and the alternative taken to be the minimal clinically meaningful effect size but also at more optimistic anticipated effect sizes Pre specified Type I error rate and power requirements can be achieved both by standard group sequential tests and by more recently proposed adaptive procedures We investigate four nested classes of designs A Group sequential tests with equal group sizes and stopping boundaries determined by a monomial error spending function the family B As A but the initial group size is allowed to be different from the others C Group sequential tests with arbitrary group sizes and arbitrary boundaries fixed in advance D Adaptive tests as C but at each analysis future group sizes and critical values are updated depending on the current value of the test statistic By examining the performance of optimal procedures within each class we conclude that class B provides simple and efficient designs with efficiency close to that of the more complex designs of classes C and D We provide tables and figures illustrating the performances of optimal designs within each class and defining the optimal c 2000 John Wiley Sons Ltd procedures of classes A and B Copyright KEY WORDS clinical trial group sequential test sample size re estimation adaptive design flexible design optimal design error spending function 1 INTRODUCTION Along with practical considerations the sample size for a clinical trial is determined by setting up null and alternate hypotheses concerning a primary parameter of interest and then specifying a Type I error rate and power 1 to be controlled at a given treatment effect size Usually traditional values of and are used e g 0 025 0 05 0 05 0 1 0 2 however there can be much debate over the choice of Some textbooks advocate that should be chosen to represent the minimum clinically relevant or commercially viable effect Correspondence to School of Operations Research and Industrial Engineering Cornell University Ithaca NY 14853 U S A Contract grant sponsor National Institutes of Health contract grant number R01 CA66218 c 2000 John Wiley Sons Ltd Copyright Received 19 April 2004 Revised 14 December 2004 2 C JENNISON AND B W TURNBULL size see for example Senn 1 p 170 and Piantadosi 2 p 149 Others such as Shun et al 3 say that can be taken to be the anticipated effect size a value based on expectations from prior experimental observational and theoretical evidence Pocock 4 suggests that either approach might be taken on pages 125 and 132 is to be a realistic value while in the example on page 128 it is to be a clinically relevant difference that is important to detect In Section 3 5 of the ICH Guidance E9 5 it is also stated that is to be based on a judgement concerning either the minimal clinically relevant effect size or the anticipated effect The choice of is crucial because for example a halving in the chosen effect size will lead to a quadrupling in the sample size for a fixed sample test and in the maximum sample size for a group sequential test Using the lower sample size appropriate to a high treatment effect will leave the trial underpowered to detect a smaller but still important effect Because of this Shun et al 3 and others have proposed that the trial be designed using the higher effect size and corresponding lower sample size but that sample size be re estimated at an interim analysis based on the emerging observed treatment difference This has been termed the start small then ask for more strategy 6 Liu and Chi 7 present formal two stage designs in which the first stage sample size is sufficient to provide specified power at an expected effect size but additional observations in the second stage increase power at smaller effect sizes and guarantee an overall power requirement at a minimal clinically significant treatment effect There have been several accounts in the literature of studies in which sample size has been adapted in order to increase power at lower effect sizes Cui et al 8 report on a placebo controlled myocardial infarction prevention trial with a sample size of 600 subjects per treatment arm this number being based on a planned effect size of a 50 reduction in incidence and 95 power However midway through the trial only about a 25 reduction in incidence was observed a reduction which was still of clinical and commercial importance Because of the low conditional power at this stage the sponsor of the trial submitted a proposal to expand the sample size In recent years classes of procedures termed flexible adaptive self designing or variance spending have been developed which enable such sample size re estimation to be done while preserving the Type I error rate See Bauer 9 Proschan and Hunsberger 10 Fisher 11 Cui et al 8 Wassmer 12 Li et al 13 and Posch et al 14 among others Remarks by some authors e g Shen and Fisher 15 and Shun et al 3 suggest a desire to set a specific power 1 at whatever is the true value of the effect size parameter This aim may lead to adaptive designs with a power curve rising sharply from at 0 then remaining almost flat at 1 In consequence significant risk of a negative outcome remains even when the effect size is high and power close to one could easily have been attained All the above discussion supports the view that a clinical trial should guarantee power at effect sizes of clinical or commercial interest Smaller effects are not pertinent since as Shih 16 p 517 states trials need to consider sample size to detect a difference that is clinically meaningful not merely to find a statistical significance Limitations occur when the sample size needed to detect a particularly small effect is prohibitive then power must be specified at the smallest value of that resources permit Shun et al 3 p 520 give an example of the dilemma investigators can face It is agreed that the minimum clinically meaningful effect is 5 but the anticipated effect size if 10 Should the trial