Consonance and the Closure Method in MultipleTesting

Consider the problem of testing s hypotheses simultaneously. In order to deal with themultiplicity problem, the classical approach is to restrict attention to procedures that controlthe familywise error rate (FWE). Typically, it is known how to construct tests of the individualhypotheses, and the problem is how to combine them into a multiple testing procedure thatcontrols the FWE. The closure method of Marcus et al. (1976), in fact, reduces the problemof constructing multiple test procedures which control the FWE to the construction of singletests which control the usual probability of a Type 1 error. The purpose of this paper is toexamine the closure method with emphasis on the concepts of coherence and consonance.[...]