The Annals of Statistics

Generalizing Simes’ test and Hochberg’s stepup procedure

Sanat K. Sarkar

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In a multiple testing problem where one is willing to tolerate a few false rejections, procedure controlling the familywise error rate (FWER) can potentially be improved in terms of its ability to detect false null hypotheses by generalizing it to control the k-FWER, the probability of falsely rejecting at least k null hypotheses, for some fixed k>1. Simes’ test for testing the intersection null hypothesis is generalized to control the k-FWER weakly, that is, under the intersection null hypothesis, and Hochberg’s stepup procedure for simultaneous testing of the individual null hypotheses is generalized to control the k-FWER strongly, that is, under any configuration of the true and false null hypotheses. The proposed generalizations are developed utilizing joint null distributions of the k-dimensional subsets of the p-values, assumed to be identical. The generalized Simes’ test is proved to control the k-FWER weakly under the multivariate totally positive of order two (MTP2) condition [J. Multivariate Analysis 10 (1980) 467–498] of the joint null distribution of the p-values by generalizing the original Simes’ inequality. It is more powerful to detect k or more false null hypotheses than the original Simes’ test when the p-values are independent. A stepdown procedure strongly controlling the k-FWER, a version of generalized Holm’s procedure that is different from and more powerful than [Ann. Statist. 33 (2005) 1138–1154] with independent p-values, is derived before proposing the generalized Hochberg’s procedure. The strong control of the k-FWER for the generalized Hochberg’s procedure is established in situations where the generalized Simes’ test is known to control its k-FWER weakly.

Article information

Ann. Statist. Volume 36, Number 1 (2008), 337-363.

First available in Project Euclid: 1 February 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J15: Paired and multiple comparisons

Global testing multiple testing single-step procedure stepdown procedure stepup procedure generalized Bonferroni procedure generalized Holm’s procedure generalized Hochberg’s procedure


Sarkar, Sanat K. Generalizing Simes’ test and Hochberg’s stepup procedure. Ann. Statist. 36 (2008), no. 1, 337--363. doi:10.1214/009053607000000550.

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