c=0.2 c=0.5 c=0.8 Bonf. on max −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 P ower difference c=0.2 c=0.5 c=0.8 Bonf. on max −0.5 −0.4 −0.3 −0.2 −0.1 0.0 P ower difference Figure 6: The difference in power of Procedure 3.1 using Bonferroni with c = α$_{1}$/α ∈ { 0 . 2 , 0 . 5 , 0 . 8 } , as well as the Bonferroni procedure on maximum p -values, from the power of Procedure 3.1 with optimal c . Left panel: for all pairs of configurations where µ$_{11}$ ∈ { 0 , 0 . 5 , 1 . 0 , . . . , 10 } and µ$_{21}$ ∈ { 0 , 0 . 5 , 1 . 0 , . . . , 10 } . Right panel: Subset of configurations of ( µ$_{11}$, µ$_{21}$ ) for which the power with optimal choice c is above 0.90. F Proof of Theorem 3.3 We use the notation given in the first two paragraphs of Appendix A of the main manuscript, including: q$_{2}$ = q − q$_{1}$ ; R$_{j}$ is the indicator of whether H$_{NR,j}$ was rejected for j = 1 , . . . , m , and R = ∑ m j =1 R$_{j}$ . In addition we define: I$_{0}$ = I$_{00}$ ∪ I$_{01}$ ; p$_{1}$ = ( p$_{11}$, . . . , p$_{1}$$_{m}$ ); R $_{1}$( p$_{1}$ ) is the set of hypotheses selected for follow-up based on p$_{1}$, R$_{1}$ ( p$_{1}$ ) = |R $_{1}$( p$_{1}$ ) | . 1. Given p$_{1}$, for j ∈ I$_{10}$ ∩ R $_{1}$( p$_{1}$ ) , R$_{1}$ ( p$_{1}$ ) ∑ r =1 Pr ( C ( j ) r | P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) , P$_{1}$ = p$_{1}$ ) ≤ 1 . 38 Lemma F.1. Assume that the p -values across studies are independent, and the set of p -values within the follow-up study has property PRDS. Then for any valid selection rule, the following results hold: