Open Access
August 2007 Control of generalized error rates in multiple testing
Joseph P. Romano, Michael Wolf
Ann. Statist. 35(4): 1378-1408 (August 2007). DOI: 10.1214/009053606000001622


Consider the problem of testing $s$ hypotheses simultaneously. The usual approach restricts attention to procedures that control the probability of even one false rejection, the familywise error rate (FWER). If $s$ is large, one might be willing to tolerate more than one false rejection, thereby increasing the ability of the procedure to correctly reject false null hypotheses. One possibility is to replace control of the FWER by control of the probability of $k$ or more false rejections, which is called the $k$-FWER. We derive both single-step and step-down procedures that control the $k$-FWER in finite samples or asymptotically, depending on the situation. We also consider the false discovery proportion (FDP) defined as the number of false rejections divided by the total number of rejections (and defined to be 0 if there are no rejections). The false discovery rate proposed by Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289–300] controls $E$(FDP). Here, the goal is to construct methods which satisfy, for a given $γ$ and $α$, $P\{\mathrm{FDP} \gt γ\} ≤ α$, at least asymptotically. In contrast to the proposals of Lehmann and Romano [Ann. Statist. 33 (2005) 1138–1154], we construct methods that implicitly take into account the dependence structure of the individual test statistics in order to further increase the ability to detect false null hypotheses. This feature is also shared by related work of van der Laan, Dudoit and Pollard [Stat. Appl. Genet. Mol. Biol. 3 (2004) article 15], but our methodology is quite different. Like the work of Pollard and van der Laan [Proc. 2003 International Multi-Conference in Computer Science and Engineering, METMBS’03 Conference (2003) 3–9] and Dudoit, van der Laan and Pollard [Stat. Appl. Genet. Mol. Biol. 3 (2004) article 13], we employ resampling methods to achieve our goals. Some simulations compare finite sample performance to currently available methods.


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Joseph P. Romano. Michael Wolf. "Control of generalized error rates in multiple testing." Ann. Statist. 35 (4) 1378 - 1408, August 2007.


Published: August 2007
First available in Project Euclid: 29 August 2007

zbMATH: 1127.62063
MathSciNet: MR2351090
Digital Object Identifier: 10.1214/009053606000001622

Primary: 62J15
Secondary: 62G10

Keywords: bootstrap , false discovery proportion , False discovery rate , generalized familywise error rate , multiple testing , step-down procedure

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 4 • August 2007
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