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Consonance and the Closure Method in Multiple Testing


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.

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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Additional indexing

Item Type:Working Paper
Communities & Collections:03 Faculty of Economics > Department of Economics
Working Paper Series > Institute for Empirical Research in Economics (former)
Dewey Decimal Classification:330 Economics
Language:English
Date:September 2009
Deposited On:29 Nov 2011 20:09
Last Modified:05 Apr 2016 15:09
Series Name:Working paper series / Institute for Empirical Research in Economics
ISSN:1424-0459
Official URL:http://www.econ.uzh.ch/wp.html
Permanent URL: https://doi.org/10.5167/uzh-51898

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