Benjamini and Hochberg (2000), Storey et al. (2004), Benjamini et al. (2006), and Blanchard and Roquain (2009), are non-valid selection rules. Theorem 3.2. If all the p -values are jointly independent and the selection rule in step 1 of Procedure 3.2 is a valid selection rule, then Procedure 3.2 controls the FDR at level q for the family of no replicability null hypotheses H$_{NR,}$$_{1}$, . . . , H$_{NR,m}$ . See Appendix A for the proof. The selection rule and the choice of q$_{1}$ affect the power of Procedure 3.2. A natural choice for a selection rule is the set of rejected hypotheses by the BH procedure at level q$_{1}$ on the primary study p -values, since the set of indices of rejected no replicability null hypotheses is a subset of this set. A rule that selects by the BH procedure at level q is not as good as the rule at level q$_{1}$ , since any additional hypotheses selected will not be rejected, but will result in more severe thresholds on the follow-up study p -values. In Figure 4 we showed in a simulated example that the BH procedure at level q$_{1}$ was very close to selecting the optimal number of hypotheses for follow-up. We recommend using it when there are no additional constraints that require choosing only a small number of hypotheses for follow-up. The optimal choice of q$_{1}$ depends on | I$_{00}$ | , | I$_{01}$ | , | I$_{10}$ | , | I$_{11}$ | , and the non-null distribution of the p -values, and therefore guidelines for choosing q$_{1}$ are application specific. In simulated GWAS in Section 5 the choice of q$_{1}$ had little effect on the average number of discoveries. Theorem 3.2 assumes independence of the p -values within each study as well as across the studies. However, the assumption of independence among the p -values within each study may not be realistic in many applications. Particularly, in GWAS there is dependency across the SNPs, therefore the p -values within each study may be dependent. Benjamini and Yekutieli (2001) proved that the BH procedure controls the FDR when the p -values have a special dependency called PRDS. 9