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VOL. 49 | 2006 On stepdown control of the false discovery proportion


Consider the problem of testing multiple null hypotheses. A classical approach to dealing with the multiplicity problem is to restrict attention to procedures that control the familywise error rate ($FWER$), the probability of even one false rejection. However, if $s$ is large, control of the $FWER$ is so stringent that the ability of a procedure which controls the $FWER$ to detect false null hypotheses is limited. Consequently, it is desirable to consider other measures of error control. We will consider methods based on control of the false discovery proportion ($FDP$) defined by the number of false rejections divided by the total number of rejections (defined to be 0 if there are no rejections). The false discovery rate proposed by Benjamini and Hochberg (1995) controls $E (FDP)$. Here, we construct methods such that, for any $\gamma$ and $\alpha$, $P \{ FDP > \gamma \} \le \alpha$. Based on $p$-values of individual tests, we consider stepdown procedures that control the $FDP$, without imposing dependence assumptions on the joint distribution of the $p$-values. A greatly improved version of a method given in Lehmann and Romano is derived and generalized to provide a means by which any sequence of nondecreasing constants can be rescaled to ensure control of the $FDP$. We also provide a stepdown procedure that controls the $FDR$ under a dependence assumption.


Published: 1 January 2006
First available in Project Euclid: 28 November 2007

zbMATH: 1268.62078
MathSciNet: MR2337829

Digital Object Identifier: 10.1214/074921706000000383

Primary: 62J15

Rights: Copyright © 2006, Institute of Mathematical Statistics


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