In Figure 3 we consider the FDR level of Procedure 4.1 with w$_{1}$ ∈ { 0 , 0 . 5 , 1 } , as well as of the naive procedure in the null setting, where all no replicability null hypotheses are true (i.e. f$_{11}$ = 0). The estimated FDR of (BH- i , BH- j ) procedure exceeds 0.05 in the settings where f$_{10}$ = f$_{01}$ = 0 . 5 and f$_{00}$ = 0 . 8 , f$_{10}$ = f$_{01}$ = 0 . 1. In these settings the estimated FDR of both (BH-1, BH-2) and (BH-2, BH-1) procedures are increasing functions of µ = µ$_{1}$ = µ$_{2}$ , reaching one in the setting where f$_{10}$ = f$_{01}$ = 0 . 5 (left), and 0.4 in the setting where f$_{00}$ = 0 . 8 , f$_{10}$ = f$_{01}$ = 0 . 1 (right). Clearly, procedure (BH- i , BH- j ) is not valid since it may be far too liberal in terms of FDR level. 

Finally, we examined how the selection rule affects the power. In Figure 4 we show the power as a function of µ$_{1}$ for Procedure 4.1 with parameters w$_{1}$ = 0 . 5 , q$_{1}$ = 0 . 025 , q = 0 . 05, for the following selection rules: BH at level 0 . 0125; the rule that selects the hypotheses with k smallest primary study p -values, where k ∈ { 25 , 30 , . . . , 100 } . The remaining parameters were: f$_{00}$ = 0 . 9 , f$_{01}$ = f$_{10}$ = 0 . 025 , f$_{11}$ = 0 . 05 , σ$_{1}$ = 0 . 5 , σ$_{2}$ = 1 , µ$_{2}$ = 3. For different values of µ$_{1}$ the optimal k is different, and using the BH procedure for selection is optimal for the entire range of µ$_{1}$ . 

## 7 Discussion 

In many research areas first a primary study is analyzed, then a follow-up study is analyzed with the goal to corroborate the findings, or at least a subset of the findings, of the primary study. We suggested novel testing procedures for corroborating the evidence from a primary study in a follow-up study. We demonstrated their usefulness on a GWAS application. In the setting where there is no division of roles to a primary and a follow-up study, the simulations suggested that our novel Procedure 4.1 with w$_{1}$ = 0 . 5 is more powerful than the BH procedure on maximum p -values. 

We proved that Procedures 3.2 and 4.1 control the FDR when the p -values are inde- 

