that FDR ≤ | I$_{0}$ | q$_{1}$/m + q$_{2}$ = | I$_{0}$ | q$_{1}$/m + q − q$_{1}$ ≤ q. Proof of item 2. The first term of the sum in (A.1) equals to: Now it follows from the arguments that lead from (F.9) to (F.18) that the upper bound for the first term of the sum in (A.1) is q$_{1}$. E ( ∑ j ∈ I$_{10}$ R$_{j}$/ max( R, 1) | P$_{1}$ = p$_{1}$ ) = j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) R$_{1}$ ( p$_{1}$ ) ∑ r =1 1 r I [ p$_{1}$$_{j}$ ≤ $^{r}$˜ q$_{1}$ m ] Pr ( P$_{2}$$_{j}$ ≤ rq$_{2}$ R$_{1}$ ( p$_{1}$ ) ∑ R$_{1}$ ( p$_{1}$ ) s =1 1 /s , C ( j ) r | P$_{1}$ = p$_{1}$ ) ≤ ∑ j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) R$_{1}$ ( p$_{1}$ ) ∑ r =1 r ∑ l =1 1 r p$_{jrl}$ ( p$_{1}$ ) , The second term of the sum in (A.1) is E ( ∑ j ∈ $_{I$_{10}$}$R$_{j}$ / max( R, 1) ) . For each p$_{1}$, where p$_{jrl}$ ( p$_{1}$ ) is defined in (G.1). Now it follows from the arguments that lead from (G.4) to (G.6) that the second term of the sum in (A.1) is bounded by q$_{2}$. Therefore, FDR ≤ q$_{1}$ + q$_{2}$ = q$_{1}$ + q − q$_{1}$ = q. Consider now a situation where both studies are available before the analysis, as described in Section 4 of the main manuscript. Without loss of generality, we label the studies as study one and study two. Theorem G.3. Assume the p -values across studies are independent. Procedure 4.1 with parameters ( w$_{1}$, q$_{1}$, q ) controls the FDR at level q for the family of no replicability E ( ∑ j ∈ $_{I$_{0}$}$R$_{j}$ max( R, 1) ) = ∑ j ∈ I$_{0}$ m ∑ r =1 1 r Pr ( j ∈ R $_{1}$, P$_{1}$$_{j}$ ≤ $^{r}$˜ q$_{1}$ m , P$_{2}$$_{j}$ ≤ r ( q − q$_{1}$ ) |R 1 | ∑ |R$_{1}$| s =1 1 /s , C ( j ) r ) ≤ j ∈ I$_{0}$ m r =1 1 r Pr P$_{1}$$_{j}$ ≤ min ( q$_{1}$ m , t ) , C ( j ) r ) . 51