# Consonance and the Closure Method in Multiple Testing - Zurich Open Repository and Archive

Romano, Joseph P; Shaikh, Azeem M; Wolf, Michael (2009). Consonance and the Closure Method in Multiple Testing. Working paper series / Institute for Empirical Research in Economics No. 446, University of Zurich.

## Abstract

Consider the problem of testing s hypotheses simultaneously. In order to deal with the multiplicity problem, the classical approach is to restrict attention to procedures that control the familywise error rate (FWE). Typically, it is known how to construct tests of the individual hypotheses, and the problem is how to combine them into a multiple testing procedure that controls the FWE. The closure method of Marcus et al. (1976), in fact, reduces the problem of constructing multiple test procedures which control the FWE to the construction of single tests which control the usual probability of a Type 1 error. The purpose of this paper is to examine the closure method with emphasis on the concepts of coherence and consonance. It was shown by Sonnemann and Finner (1988) that any incoherent procedure can be replaced by a coherent one which is at least as good. The main point of this paper is to show a similar result for dissonant and consonant procedures. We illustrate the idea of how a dissonant procedure can be strictly improved by a consonant procedure in the sense of increasing the probability of detecting a false null hypothesis while maintaining control of the FWE. We then show how consonance can be used in the construction of some optimal maximin procedures.

## Abstract

Consider the problem of testing s hypotheses simultaneously. In order to deal with the multiplicity problem, the classical approach is to restrict attention to procedures that control the familywise error rate (FWE). Typically, it is known how to construct tests of the individual hypotheses, and the problem is how to combine them into a multiple testing procedure that controls the FWE. The closure method of Marcus et al. (1976), in fact, reduces the problem of constructing multiple test procedures which control the FWE to the construction of single tests which control the usual probability of a Type 1 error. The purpose of this paper is to examine the closure method with emphasis on the concepts of coherence and consonance. It was shown by Sonnemann and Finner (1988) that any incoherent procedure can be replaced by a coherent one which is at least as good. The main point of this paper is to show a similar result for dissonant and consonant procedures. We illustrate the idea of how a dissonant procedure can be strictly improved by a consonant procedure in the sense of increasing the probability of detecting a false null hypothesis while maintaining control of the FWE. We then show how consonance can be used in the construction of some optimal maximin procedures.

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Item Type: Working Paper 03 Faculty of Economics > Department of Economics Working Paper Series > Institute for Empirical Research in Economics (former) 330 Economics English September 2009 29 Nov 2011 20:09 05 Apr 2017 23:09 Working paper series / Institute for Empirical Research in Economics 1424-0459 http://www.econ.uzh.ch/wp.html