# Control of Generalized Error Rates in Multiple Testing

Romano, Joseph P; Wolf, Michael (2005). Control of Generalized Error Rates in Multiple Testing. Working paper series / Institute for Empirical Research in Economics No. 245, University of Zurich.

## Abstract

Consider the problem of testing s hypotheses simultaneously. The usual approach tondealing with the multiplicity problem is to restrict attention to procedures that controlnthe probability of even one false rejection, the familiar familywise error rate (FWER). Innmany applications, particularly if s is large, one might be willing to tolerate more than onenfalse rejection if the number of such cases is controlled, thereby increasing the ability of thenprocedure to reject false null hypotheses One possibility is to replace control of the FWERnby control of the probability of k or more false rejections, which is called the k-FWER.nWe derive both single-step and stepdown procedures that control the k-FWER in finitensamples or asymptotically, depending on the situation. Lehmann and Romano (2005a)nderive some exact methods for this purpose, which apply whenever p-values are availablenfor individual tests; no assumptions are made on the joint dependence of the p-values. Inncontrast, we construct methods that implicitly take into account the dependence structurenof the individual test statistics in order to further increase the ability to detect false nullnhypotheses. We also consider the false discovery proportion (FDP) defined as the numbernof false rejections divided by the total number of rejections (and defined to be 0 if therenare no rejections). The false discovery rate proposed by Benjamini and Hochberg (1995)ncontrols E(FDP).

## Abstract

Consider the problem of testing s hypotheses simultaneously. The usual approach tondealing with the multiplicity problem is to restrict attention to procedures that controlnthe probability of even one false rejection, the familiar familywise error rate (FWER). Innmany applications, particularly if s is large, one might be willing to tolerate more than onenfalse rejection if the number of such cases is controlled, thereby increasing the ability of thenprocedure to reject false null hypotheses One possibility is to replace control of the FWERnby control of the probability of k or more false rejections, which is called the k-FWER.nWe derive both single-step and stepdown procedures that control the k-FWER in finitensamples or asymptotically, depending on the situation. Lehmann and Romano (2005a)nderive some exact methods for this purpose, which apply whenever p-values are availablenfor individual tests; no assumptions are made on the joint dependence of the p-values. Inncontrast, we construct methods that implicitly take into account the dependence structurenof the individual test statistics in order to further increase the ability to detect false nullnhypotheses. We also consider the false discovery proportion (FDP) defined as the numbernof false rejections divided by the total number of rejections (and defined to be 0 if therenare no rejections). The false discovery rate proposed by Benjamini and Hochberg (1995)ncontrols E(FDP).