Constructing multiple test procedures for partially ordered hypothesis sets

A popular method to control multiplicity in confirmatory clinical trials is to use a so-called hierarchical, or fixed sequence, test procedure. This requires that the null hypotheses are ordered a priori, for example, in order of clinical importance. The procedure tests the hypotheses in this order using alpha-level tests until one is not rejected. It then stops, so that no subsequent hypotheses are eligible for rejection. This procedure strongly controls the familywise error rate (FWE), that is to say, the probability that any true hypotheses are rejected. This paper describes a simple generalization of this approach in which the null hypotheses are partially ordered. It is convenient to display the partial ordering in a directed acyclic graph (DAG). We consider sequentially rejective procedures based on the partial ordering, in which a hypothesis is tested only when all preceding hypotheses have been tested and rejected. In general such procedures do not control the FWE, but it is shown that when certain intersection hypotheses are added, strong control of the FWE is obtained. The purpose of the method is to construct inference strategies for confirmatory clinical trials that better reflect the trial objectives.