Group sequential tests: beyond exponential family models

This manuscript considers group sequential tests powered for multiple ordered alternative hypotheses with a predetermined α-spending function. Theorem 1 shows that if a fixed-sample size likelihood ratio test is monotone with respect to a one-dimensional test statistic, then a group sequential test constructed by interim cumulative likelihood ratio tests is most powerful for a sequence of ordered alternatives, at a given α-spending function. This theorem extends Tarima and Flournoy (Metrika 85: 491-513, 2022) from the exponential family to non-exponential distributions with monotone likelihood ratio. A three-stage design powered for three ordered alternatives shows how the theory applies to uniform data. When the likelihood ratio is not monotone for finite sample sizes, locally most powerful tests can be constructed if a test is locally most powerful for a fixed sample size against a local alternative. A two-stage Cauchy example shows how such tests can be built using either a likelihood ratio test statistic or its MLE. Overall, if a parametric distribution of the data is either known or assumed, MLE-based group sequential tests powered for multiple ordered alternatives are most powerful for this set of hypotheses in either finite or in local asymptotic settings.