Example 2. To discover associations between SNPs and Crohn’s disease (CD), Barrett et al. (2008) examined 635,547 SNPs on 3230 cases and 4829 controls of European descent, collected in three separate studies: NIDDK4, WTCCC5, and a Belgian-French study. The primary study p -values in this example are the meta- analysis p -values from the combined data from the three studies. Only hypotheses with primary study p -values below 5 × 10 − 5 were considered for follow-up. Although 526 SNPs met the selection criterion, only a subset of 126 SNPs were followed up. These 126 p -values were the smallest two p -values in 63 distinct regions, so the selec- tion rule is a valid selection rule. Procedure 3.2 with ( q$_{1}$, q ) = (0 . 04 , 0 . 01) identified 36 SNPs. In Appendix D, Table 5 shows the p -values from the primary and follow-up studies, as well as the FDR-replicability adjusted p -values for the choice c = 0 . 8, for these 36 replicability discoveries. Since the p -values are not independent within each study, a more conservative analysis approach is to modify the cut-offs as suggested by Theorem 3.3. Assuming PRDS type dependency in the follow-up study, item 1 of Theorem 3.3 suggests using ˜ q$_{1}$ = 0 . 04 / ( ∑ 635 , 547 i =1 1 /i ) = 0 . 0029 for the primary study cut-offs, while the follow-up study cut-offs remain unchanged. The modified proce- dure identified 21 SNPs. Column 7 of Table 5 shows the FDR-replicability adjusted p -values for the choice c = 0 . 8, where the adjustment is made as described in expres- sions (3.1) and (3.2), with p$_{1}$$_{j}$ replaced by ˜ p$_{1}$$_{j}$ = ( ∑ 635 , 547 i =1 1 /i ) p$_{1}$$_{j}$ = 13 . 94 × p$_{1}$$_{j}$, j = 1 , . . . , 635 , 547. Since the SNPs considered for follow-up were only SNPs with pri- mary study p -values below 5 × 10 − $^{5}$, one could use a less conservative procedure suggested in item 2 of Theorem 3.3, with ˜ q$_{1}$ = 0 . 0038, where 0 . 0038 is the solution to 0 . 04 = x ( ∑ ⌈ 635 , 547 × 5 × 10 − $^{5}$/x − 1 ⌉ i =1 1 /i + 1). This procedure resulted in 23 replicability discoveries. The latter procedure is the recommended procedure, if the investigator is not willing to assume that Procedure 3.2 is robust to deviations from independence within the primary study. However, simulations in the next example suggest that for the type of dependencies that occur in GWAS, Procedure 3.2 may actually be con- 15