Definition 3.2. (Benjamini and Yekutieli, 2001) The set of p -values P$_{1}$, . . . , P$_{M}$ has property PRDS if for any increasing set D , and for each true null hypothesis i , Pr (( P$_{1}$, . . . , P$_{M}$ ) ∈ D | P$_{i}$ = p ) is nondecreasing in p . 

If the p -values are independent in the primary study, yet have property PRDS in the follow-up study, Theorem S3.1 in the Supplementary Material shows that the result in Theorem 3.2 holds. For arbitrary dependence among the p -values in the primary study, a modification of the cut-off level of Procedure 3.2 will guarantee that the FDR is controlled at the nominal level. The most severe modification, that will guarantee FDR control for any valid selection rule, is to apply Procedure 3.2 with the modification in item 1 of Theorem 3.3 below. However, in item 2 of Theorem 3.3 we show that the modification factor may be smaller than ∑ m i $_{=1}$1 /i ≈ log m if the selected hypotheses for follow-up are a subset of the hypotheses with primary study p -values below a fixed cut-off t . For example, in GWAS it is common to select hypotheses with primary study p -values below 1 /m , where m is the number of hypotheses in the primary study (Lander and Kruglyak, 1995). If t ≥ q$_{1}$ $_{1+}$∑ m$_{−}$ 1 i =1 1 i , then the modification in item 1 of Theorem 3.3 cannot be improved. However, if t < q$_{1}$ $_{1+}$∑ m$_{−}$ 1 i =1 1 i , then the modification in item 2 of Theorem 3.3 is less conservative than the modification in item 1. For typical values of q$_{1}$ (e.g. q$_{1}$ ∈ [0 . 005 , 0 . 045]) and large m , the threshold t will often be below q$_{1}$ $_{1+}$∑ m$_{−}$ 1 i =1 1 i , and therefore item 2 may be useful in applications. Note, moreover, that if t ≤ q$_{1}$/m , then item 2 of Theorem 3.3 states that no modification is required, so for a valid selection rule which selects a subset of the set of hypotheses with primary study p -values below t , where t ≤ q$_{1}$/m , Procedure 3.2 is valid for any form of dependency among the p -values in the primary study. 

Theorem 3.3. Assume that the follow-up study p -values have property PRDS, and are independent of the p -values in the primary study. Then Procedure 3.2 controls 

