STATISTICS IN MEDICINEStatist. Med. 2006; 25:917–932Published online 11 October 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/sim.2251Ecient group sequential designs when there are several eectsizes under considerationChristopher Jennison1and Bruce W. Turnbull2; ∗; †1Department of Mathematical Sciences; University of Bath; Bath; BA2 7AY; U.K.2School of Operations Research and Industrial Engineering; Cornell University; Ithaca; NY 14853; U.S.A.SUMMARYWe consider the construction of ecient group sequential designs where the goal is a low expectedsample size not only at the null hypothesis and the alternative (taken to be the minimal clinicallymeaningful eect size), but also at more optimistic anticipated eect sizes. Pre-speci ed Type I errorrate and power requirements can be achieved both by standard group sequential tests and by morerecently proposed adaptive procedures. We investigate four nested classes of designs: (A) group se-quential tests with equal group sizes and stopping boundaries determined by a monomial error spendingfunction (the ‘-family’); (B) as A but the initial group size is allowed to be dierent from the others;(C) group sequential tests with arbitrary group sizes and arbitrary boundaries, xed in advance;(D) adaptive tests—as C but at each analysis, future group sizes and critical values are updated de-pending on the current value of the test statistic. By examining the performance of optimal procedureswithin each class, we conclude that class B provides simple and ecient designs with eciency closeto that of the more complex designs of classes C and D. We provide tables and gures illustrating theperformances of optimal designs within each class and de ning the optimal procedures of classes Aand B. Copyright? 2005 John Wiley & Sons, Ltd.KEY WORDS: clinical trial; group sequential test; sample size re-estimation; adaptive design; exibledesign; optimal design; error spending function1. INTRODUCTIONAlong with practical considerations, the sample size for a clinical trial is determined by settingup null and alternate hypotheses concerning a primary parameter of interest, , and thenspecifying a Type I error rate and power 1− to be controlled at a given treatment eect size = . Usually, traditional values of and  are used (e.g. =0:025; 0:05,  =0:05; 0:1; 0:2);∗Correspondence to: B. W. Turnbull, School of Operations Research and Industrial Engineering, Cornell University,Ithaca, NY 14853, U.S.A.†E-mail: [email protected]=grant sponsor: National Institutes of Health; contract=grant number: R01 CA66218Received April 2004Copyright?2005 John Wiley & Sons, Ltd. Accepted March 2005918 C. JENNISON AND B. W. TURNBULLhowever, 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’ eectsize—see for example References [1, p. 170], [2, p. 149]. Others such as Shun et al. [3] saythat  can be taken to be the anticipated eect size—a value based on expectations from priorexperimental, observational and theoretical evidence. Pocock [4] suggests that either approachmight be taken: on pp. 125 and 132,  is to be a ‘realistic value’, while in the example onp. 128, it is to be a ‘clinically relevant’ dierence that is ‘important to detect’. In Section 3.5of the ICH Guidance E9 [5], it is also stated that  is to be based on a judgement concerningeither the minimal clinically relevant eect size or the ‘anticipated’ eect.The choice of  is crucial because, for example, a halving in the chosen eect size willlead to a quadrupling in the sample size for a xed sample test (and in the maximum samplesize for a group sequential test). Using the lower sample size appropriate to a high treatmenteect will leave the trial underpowered to detect a smaller but still important eect. Becauseof this, Shun et al. [3] and others have proposed that the trial be designed using the highereect size (and corresponding lower sample size), but that sample size be re-estimated at aninterim analysis based on the emerging observed treatment dierence. This has been termedthe ‘start small then ask for more’ strategy [6]. Liu and Chi [7] present formal two-stagedesigns in which the rst stage sample size is sucient to provide speci ed power at anexpected eect size but additional observations in the second stage increase power at smallereect sizes and guarantee an overall power requirement at a minimal clinically signi canttreatment eect.There have been several accounts in the literature of studies in which sample size has beenadapted in order to increase power at lower eect sizes. Cui et al. [8] report on a placebocontrolled myocardial infarction prevention trial with a sample size of 600 subjects per treat-ment arm, this number being based on a planned eect size of a 50 per cent reduction inincidence and 95 per cent power. However, midway through the trial, only about a 25 per centreduction in incidence was observed, a reduction which was still of clinical and commercialimportance. Because of the low conditional power at this stage, the sponsor of the trial sub-mitted a proposal to expand the sample size. In recent years, classes of procedures termed‘exible’, ‘adaptive’, ‘self-designing’ or ‘variance spending’ have been developed whichenable such sample size re-estimation to be done while preserving the Type I error rate .See References [8–14] among others.Remarks by some authors, e.g. Shen and Fisher [15] and Shun et al. [3], suggest a desireto set a speci c power, 1−, at whatever is the true value of the eect size parameter. Thisaim may lead to adaptive designs with a power curve rising sharply from at  = 0, thenremaining almost at at 1−. In consequence, signi cant risk of a negative outcome remainseven when the eect 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 ateect sizes  of clinical or commercial interest. Smaller eects are not pertinent since, asShih [16, p. 517] states ‘::: trials need to consider sample size to detect a dierence that isclinically meaningful, not merely to nd a statistical signi cance.’ Limitations occur when thesample