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February 2002 Multiple hypotheses testing and expected number of type I. errors
H. Finner, M. Roters
Ann. Statist. 30(1): 220-238 (February 2002). DOI: 10.1214/aos/1015362191

Abstract

The performance of multiple test procedures with respect to error control is an old issue. Assuming that all hypotheses are true we investigate the behavior of the expected number of type I errors (ENE) as a characteristic of certain multiple tests controlling the familywise error rate (FWER) or the false discovery rate (FDR) at a prespecified level. We derive explicit formulas for the distribution of the number of false rejections as well as for the ENE for single-step, step-down and step-up procedures based on independent $p$-values. Moreover, we determine the corresponding asymptotic distributions of the number of false rejections as well as explicit formulae for the ENE if the number of hypotheses tends to infinity. In case of FWER-control we mostly obtain Poisson distributions and in one case a geometric distribution as limiting distributions; in case of FDR control we obtain limiting distributions which are apparently not named in the literature. Surprisingly, the ENE is bounded by a small number regardless of the number of hypotheses under consideration. Finally, it turns out that in case of dependent test statistics the ENE behaves completely differently compared to the case of independent test statistics.

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H. Finner. M. Roters. "Multiple hypotheses testing and expected number of type I. errors." Ann. Statist. 30 (1) 220 - 238, February 2002. https://doi.org/10.1214/aos/1015362191

Information

Published: February 2002
First available in Project Euclid: 5 March 2002

zbMATH: 1012.62020
MathSciNet: MR1892662
Digital Object Identifier: 10.1214/aos/1015362191

Subjects:
Primary: 62F05 , 62J15
Secondary: 60F99 , 62F03

Keywords: Asymptotic critical value behavior , ballot theorem , Bolshev’s recursion , Bonferroni test procedure , Dempster’s formula, DKW inequality , Empirical distribution function , False discovery rate , familywise error rate , independent $p$ -values , Lagrange–Bürmann theorem , Multiple comparisons , multiple level , multiple test procedure , order statistics , Schur–Jabotinski theorem , step-down test , step-up test

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 1 • February 2002
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