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Consider the multiple testing problem of testing null hypotheses$H_1,...,H_s$. A classical approach to dealing with the multiplicity problem isto restrict attention to procedures that control the familywise error rate($\mathit{FWER}$), the probability of even one false rejection. But if $s$ islarge, control of the $\mathit{FWER}$ is so stringent that the ability of aprocedure that controls the $\mathit{FWER}$ to detect false null hypotheses islimited. It is therefore desirable to consider other measures of error control.This article considers two generalizations of the $\mathit{FWER}$. The first isthe $k-\mathit{FWER}$, in which one is willing to tolerate $k$ or more falserejections for some fixed $k\geq 1$. The second is based on the false discoveryproportion ($\mathit{FDP}$), defined to be the number of false rejectionsdivided by the total number of rejections (and defined to be 0 if there are norejections). Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995)289--300] proposed control of the false discovery rate ($\mathit{FDR}$), bywhich they meant that, for fixed $\alpha$, $E(\mathit{FDP})\leq\alpha$. Here,we consider control of the $\mathit{FDP}$ in the sense that, for fixed $\gamma$and $\alpha$, $P\{\mathit{FDP}>\gamma\}\leq \alpha$. Beginning with anynondecreasing sequence of constants and $p$-values for the individual tests, wederive stepup procedures that control each of these two measures of errorcontrol without imposing any assumptions on the dependence structure of the$p$-values. We use our results to point out a few interesting connections withsome closely related stepdown procedures. We then compare and contrast two$\mathit{FDP}$-controlling procedures obtained using our results with thestepup procedure for control of the $\mathit{FDR}$ of Benjamini and Yekutieli[Ann. Statist. 29 (2001) 1165--1188].

Stepup procedures for control of generalizations of the familywise error rate