First, we find an upper bound for the first term of the sum in (A.1). E ( ∑ j ∈ $_{I$_{0}$}$R$_{j}$ max( R, 1) ) = ∑ j ∈ I$_{0}$ m ∑ r =1 1 r Pr ( j ∈ R $_{1}$, P$_{1}$$_{j}$ ≤ rq$_{1}$ m , P$_{2}$$_{j}$ ≤ r ( q − q$_{1}$ ) |R 1 | , C ( j ) r ) ≤ ∑ j ∈ I$_{0}$ m ∑ r =1 1 r Pr ( P$_{1}$$_{j}$ ≤ rq$_{1}$ m , C ( j ) r ) = ∑ j ∈ I$_{0}$ m ∑ r =1 1 r Pr ( P$_{1}$$_{j}$ ≤ rq$_{1}$ m ) Pr ( C ( j ) r ) (A.2) ≤ q$_{1}$ m j ∈ I$_{0}$ m r =1 Pr C ( j ) r ) = | I$_{0}$ | m q$_{1}$ (A.3) The equality in (A.2) follows from the independence of the p -values. The inequality in (A.3) follows from the fact that for each j ∈ I$_{0}$ , Pr( P$_{1}$$_{j}$ ≤ x ) ≤ x for all x ∈ [0 , 1] . Finally, the equality in (A.3) follows from the fact that ∪ m r $_{=1}$C ( j ) r is the entire sample space of ( P ( j ) 1 , P ( j ) 2 ) , represented as a union of disjoint events. Next, we find an upper bound for the second term of the sum in (A.1). Let R $_{1}$( p$_{1}$ ) be the set of selected indices using P$_{1}$ = p$_{1}$. Then E ( ∑ j ∈ $_{I$_{10}$}$R$_{j}$ / max( R, 1) | P$_{1}$ = p$_{1}$ equals to: ∑ j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) |R$_{1}$ ( p$_{1}$ ) | ∑ r =1 1 r I [ p$_{1}$$_{j}$ ≤ rq$_{1}$ m ] Pr ( P$_{2}$$_{j}$ ≤ rq$_{2}$ |R $_{1}$( p$_{1}$ ) | , C ( j ) r | P$_{1}$ = p$_{1}$ ) ≤ ∑ j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) |R$_{1}$ ( p$_{1}$ ) | ∑ r =1 1 r Pr ( P$_{2}$$_{j}$ ≤ rq$_{2}$ |R $_{1}$( p$_{1}$ ) | , C ( j ) r | P$_{1}$ = p$_{1}$ ) (A.4) = ∑ j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) |R$_{1}$ ( p$_{1}$ ) | ∑ r =1 1 r Pr ( P$_{2}$$_{j}$ ≤ rq$_{2}$ |R $_{1}$( p$_{1}$ ) | | P$_{1}$ = p$_{1}$ ) Pr ( C ( j ) r | P$_{1}$ = p$_{1}$ ) (A.5) ≤ q$_{2}$ |R $_{1}$( p$_{1}$ ) | ∑ j ∈ I$_{10}$ ∩R$_{1}$ ( p$_{1}$ ) |R$_{1}$ ( p$_{1}$ ) | ∑ r =1 Pr ( C ( j ) r | P$_{1}$ = p$_{1}$ ) = q$_{2}$ |R $_{1}$( p$_{1}$ ) | | I$_{10}$ ∩ R $_{1}$( p$_{1}$ ) | . (A.6) The equality in (A.5) follows from the fact that P$_{2}$$_{j}$, P ( j ) 2 , P$_{1}$ are independent, since then C ( j ) r and the event { P$_{2}$$_{j}$ ≤ rq$_{2}$/ |R $_{1}$( p$_{1}$ ) |} are conditionally independent. The inequality in (A.6) follows from the independence of the p -values across the studies 30