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Exact and approximate stepdown methods for multiple hypothesis testing. (English) Zbl 1117.62416

Summary: Consider the problem of testing k hypotheses simultaneously. In this article we discuss finite- and large-sample theory of stepdown methods that provide control of the familywise error rate (FWE). To improve on the Bonferroni method or on Holm’s stepdown method, Westfall and Young made effective use of resampling to construct stepdown methods that implicitly estimate the dependence structure of the test statistics. However, their methods depend on an assumption known as ”subset pivotality.” Our goal here is to construct general stepdown methods that do not require such an assumption. To accomplish this, we take a close look at what makes stepdown procedures work; a key component is a monotonicity requirement of critical values. By imposing monotonicity on estimated critical values (which is not an assumption on the model but rather is an assumption on the method), we show how to construct stepdown tests that can be applied in a stagewise fashion so that at most k tests need to be computed. Moreover, at each stage, an intersection test that controls the usual probability of a type 1 error is calculated, which allows us to draw on an enormous resampling literature as a general means of test construction. In addition, it is possible to carry out this method using the same set of resamples (or subsamples) for each of the intersection tests.

MSC:

62-XX Statistics
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