  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
\citer10 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.
