STATISTICS IN MEDICINE Statist Med 2006 25 917 932 Published online 11 October 2005 in Wiley InterScience www interscience wiley com DOI 10 1002 sim 2251 E cient group sequential designs when there are several e ect sizes under consideration Christopher Jennison1 and Bruce W Turnbull2 1 Department 2 School of Mathematical Sciences University of Bath Bath BA2 7AY U K of Operations Research and Industrial Engineering Cornell University Ithaca NY 14853 U S A SUMMARY We consider the construction of e cient 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 e ect size but also at more optimistic anticipated e ect sizes Pre speci ed 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 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 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 e cient designs with e ciency close to that of the more complex designs of classes C and D We provide tables and gures illustrating the performances of optimal designs within each class and de ning the optimal procedures of classes A and B Copyright 2005 John Wiley Sons Ltd KEY WORDS clinical trial group sequential test sample size re estimation adaptive design exible 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 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 bwt2 cornell edu E mail Contract grant sponsor National Institutes of Health contract grant number R01 CA66218 Copyright 2005 John Wiley Sons Ltd Received April 2004 Accepted March 2005 918 C JENNISON AND B W TURNBULL 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 e ect size see for example References 1 p 170 2 p 149 Others such as Shun et al 3 say that can be taken to be the anticipated e ect size a value based on expectations from prior experimental observational and theoretical evidence Pocock 4 suggests that either approach might be taken on pp 125 and 132 is to be a realistic value while in the example on p 128 it is to be a clinically relevant di erence 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 e ect size or the anticipated e ect The choice of is crucial because for example a halving in the chosen e ect size will lead to a quadrupling in the sample size for a xed sample test and in the maximum sample size for a group sequential test Using the lower sample size appropriate to a high treatment e ect will leave the trial underpowered to detect a smaller but still important e ect Because of this Shun et al 3 and others have proposed that the trial be designed using the higher e ect size and corresponding lower sample size but that sample size be re estimated at an interim analysis based on the emerging observed treatment di erence 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 rst stage sample size is su cient to provide speci ed power at an expected e ect size but additional observations in the second stage increase power at smaller e ect sizes and guarantee an overall power requirement at a minimal clinically signi cant treatment e ect There have been several accounts in the literature of studies in which sample size has been adapted in order to increase power at lower e ect 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 e ect size of a 50 per cent reduction in incidence and 95 per cent power However midway through the trial only about a 25 per cent 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 exible 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 References 8 14 among others Remarks by some authors e g Shen and Fisher 15 and Shun et al 3 suggest a desire to set a speci c power 1 at whatever is the true value of the e ect size parameter This aim may lead to adaptive designs with a power curve rising sharply from at 0 then remaining almost at at 1 In consequence signi cant risk of a negative outcome remains even 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 at e ect sizes of clinical or commercial interest Smaller e ects are not pertinent since as Shih 16 p 517 states trials need to consider sample size to detect a di erence that is clinically meaningful not merely to nd a statistical signi cance Limitations occur when the sample size needed to detect a particularly small e ect is prohibitive then power must be speci ed at the smallest value of that resources permit Shun et al 3 p 520 gives an example of the dilemma investigators can face It is agreed that the minimum clinically meaningful e ect is 5 but the anticipated e ect size is 10 Should the trial be planned with the large sample size necessary to