  Consider the multiple testing problem of testing null hypotheses
$H_1,...,H_s$. A classical approach to dealing with the multiplicity problem is
to restrict attention to procedures that control the familywise error rate
($\mathit{FWER}$), the probability of even one false rejection. But if $s$ is
large, control of the $\mathit{FWER}$ is so stringent that the ability of a
procedure that controls the $\mathit{FWER}$ to detect false null hypotheses is
limited. It is therefore desirable to consider other measures of error control.
This article considers two generalizations of the $\mathit{FWER}$. The first is
the $k-\mathit{FWER}$, in which one is willing to tolerate $k$ or more false
rejections for some fixed $k\geq 1$. The second is based on the false discovery
proportion ($\mathit{FDP}$), defined to be the number of false rejections
divided by the total number of rejections (and defined to be 0 if there are no
rejections). Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995)
289--300] proposed control of the false discovery rate ($\mathit{FDR}$), by
which 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 any
nondecreasing sequence of constants and $p$-values for the individual tests, we
derive stepup procedures that control each of these two measures of error
control without imposing any assumptions on the dependence structure of the
$p$-values. We use our results to point out a few interesting connections with
some closely related stepdown procedures. We then compare and contrast two
$\mathit{FDP}$-controlling procedures obtained using our results with the
stepup procedure for control of the $\mathit{FDR}$ of Benjamini and Yekutieli
[Ann. Statist. 29 (2001) 1165--1188].
