then rejecting the null hypotheses corresponding to p(1), . . .,p(k) provides FDR =
m0/m •E ƒ¿ . ƒ¿. If no p-value satisfies this inequality, then no hypothesis test is
called significant. (It is important to keep in mind that for any given set of data
we do not have V/R . ƒ¿. Rather, the long-run behavior of this procedure is such
that FDR . ƒ¿.) The FDR offers less stringent control over Type I errors than the
FWER, and is therefore usually more powerful, as is shown in their simulations.
In this paper, we define the positive false discovery rate (pFDR) to be pFDR =
E[V/R|R > 0]. The term •gpositive•h describes the fact that we have conditioned
on at least one positive finding having occurred. See Section 2 for the motivation
and definition. The aim of this paper is to investigate the statistical properties of
the pFDR. The following are the main results of this paper.