The Annals of Statistics

Control of generalized error rates in multiple testing

Joseph P. Romano and Michael Wolf

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Abstract

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.

Article information

Source
Ann. Statist., Volume 35, Number 4 (2007), 1378-1408.

Dates
First available in Project Euclid: 29 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1188405615

Digital Object Identifier
doi:10.1214/009053606000001622

Mathematical Reviews number (MathSciNet)
MR2351090

Zentralblatt MATH identifier
1127.62063

Subjects
Primary: 62J15: Paired and multiple comparisons
Secondary: 62G10: Hypothesis testing

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

Citation

Romano, Joseph P.; Wolf, Michael. Control of generalized error rates in multiple testing. Ann. Statist. 35 (2007), no. 4, 1378--1408. doi:10.1214/009053606000001622. https://projecteuclid.org/euclid.aos/1188405615


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