Efficient group sequential designs when there are several effect sizes under consideration

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 'rho-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 procedures of classes A and B.