Shortcuts for Locally Consonant Closed Test Procedures

The closed testing principle provides a general and simple framework to construct powerful multiple test procedures for k elementary null hypotheses while controlling the familywise error rate in the strong sense. However, the closed testing principle has the disadvantage of leading to the evaluation of O(2(k)) intersection hypotheses. Multiple test procedures based on the closed testing principle may thus require substantial computational efforts. Consonant closed test procedures for unrestricted hypotheses have the advantage of rejecting at least one elementary null hypothesis whenever the global null hypothesis is rejected and thus admit shortcuts of size k. If the elementary null hypotheses are restricted by logical constraints, the closure of common tests, such as max-t or min-p tests, may not be consonant in the usual sense. In this article we introduce a weaker consonance property, denoted as local consonance, and show that many closed test procedures with restricted hypotheses satisfy this condition. We describe a general algorithm to construct related shortcuts and illustrate the results with several applications.