## The Annals of Statistics

### Generalizations of the familywise error rate

#### Abstract

Consider the problem of simultaneously testing null hypotheses H1,…,Hs. The usual 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. In many applications, particularly if s is large, one might be willing to tolerate more than one false rejection provided the number of such cases is controlled, thereby increasing the ability of the procedure to detect false null hypotheses. This suggests replacing control of the FWER by controlling the probability of k or more false rejections, which we call the k-FWER. We derive both single-step and stepdown procedures that control the k-FWER, without making any assumptions concerning the dependence structure of the p-values of the individual tests. In particular, we derive a stepdown procedure that is quite simple to apply, and prove that it cannot be improved without violation of control of the k-FWER. We also consider 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 [J. Roy. Statist. Soc. Ser. B 57 (1995) 289–300] controls E(FDP). Here, we construct methods such that, for any γ and α, P{FDP>γ}≤α. Two stepdown methods are proposed. The first holds under mild conditions on the dependence structure of p-values, while the second is more conservative but holds without any dependence assumptions.

#### Article information

Source
Ann. Statist., Volume 33, Number 3 (2005), 1138-1154.

Dates
First available in Project Euclid: 1 July 2005

https://projecteuclid.org/euclid.aos/1120224098

Digital Object Identifier
doi:10.1214/009053605000000084

Mathematical Reviews number (MathSciNet)
MR2195631

Zentralblatt MATH identifier
1072.62060

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

#### Citation

Lehmann, E. L.; Romano, Joseph P. Generalizations of the familywise error rate. Ann. Statist. 33 (2005), no. 3, 1138--1154. doi:10.1214/009053605000000084. https://projecteuclid.org/euclid.aos/1120224098

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