Efficient group sequential tests with unpredictable group sizes

SUMMARY We derive group sequential tests to minimize expected sample size. A parametric family of group sequential tests is proposed containing tests which are nearly optimal, in terms of minimizing expected sample size, amongst all group sequential tests. These parametric tests can be implemented when group sizes are unequal and unpredictable. They are shown to achieve error rates close to their nominal level in a wide range of situations and to be efficient when compared with other tests based on the same group sizes and achieving the same error rates. We consider group sequential procedures for choosing between two hypotheses concerning the mean of a normal distribution with known variance. The group sequential approach (Pocock, 1977) offers a practicable alternative to 'fully' sequential tests, which require accumulating data to be examined as each new observation is recorded. It is also desirable to ensure the maximum sample size needed in a long-term experiment is not much greater than that of the corresponding fixed sample size test, and we shall take the specification of the maximum allowable sample size and the number of groups of observations as our starting point in deriving sequential procedures. In ? 2 we formulate the testing problem and discuss its relevance to clinical trials. In ? 3 we describe the derivation of boundaries which are optimal in the sense of minimizing various expected sample sizes, subject to a fixed number of equally sized groups and a fixed maximum sample size. In ? 4 we introduce a parametric family of group sequential tests, containing tests which are nearly optimal by the criteria of ? 3; these tests are easily adapted to allow unequal and unpredictable group sizes. We show in ? 5 that they remain efficient when actual group sizes are quite different from those anticipated. Programs used for the calculations are available from the author.