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Inequalities between generalized familywise error rates of a multiple testing procedure

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  • Gordon, Alexander Y.

Abstract

We consider a multiple testing procedure (MTP) that decides which hypotheses to reject based solely on the observed p-values associated with the hypotheses being tested. Let fk be the exact level at which the MTP weakly controls, under a general and unknown dependence structure of the p-values, the kth generalized familywise error rate--the probability of k or more false rejections. The sequence f1,...,fm, where m is the number of hypotheses being tested, is nonincreasing. We show that if the MTP is monotone (reducing p-values can only increase the number of rejections), then the sequence kfk is nondecreasing. This result pertaining to the weak control of generalized FWERs (all hypotheses are true) carries over to the situation where a limited number of hypotheses may be false.

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  • Gordon, Alexander Y., 2009. "Inequalities between generalized familywise error rates of a multiple testing procedure," Statistics & Probability Letters, Elsevier, vol. 79(19), pages 1996-2004, October.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:19:p:1996-2004
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    1. Gordon, Alexander Y. & Salzman, Peter, 2008. "Optimality of the Holm procedure among general step-down multiple testing procedures," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1878-1884, September.
    2. van der Laan Mark J. & Dudoit Sandrine & Pollard Katherine S., 2004. "Augmentation Procedures for Control of the Generalized Family-Wise Error Rate and Tail Probabilities for the Proportion of False Positives," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 3(1), pages 1-27, June.
    3. Mark van der Laan & Sandrine Dudoit & Katherine Pollard, 2004. "Multiple Testing. Part III. Procedures for Control of the Generalized Family-Wise Error Rate and Proportion of False Positives," U.C. Berkeley Division of Biostatistics Working Paper Series 1140, Berkeley Electronic Press.
    4. Ajit C. Tamhane & Lingyun Liu, 2008. "On weighted Hochberg procedures," Biometrika, Biometrika Trust, vol. 95(2), pages 279-294.
    5. Gordon, Alexander Y., 2007. "Unimprovability of the Bonferroni procedure in the class of general step-up multiple testing procedures," Statistics & Probability Letters, Elsevier, vol. 77(2), pages 117-122, January.
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