content_image/296765.jpg content_image/296759.jpg content_image/296762.jpg
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: 

<img src='content_image/296765.jpg'>

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$_{1}$. 

The second term of the sum in (A.1) is E ( ∑ j ∈ $_{I$_{10}$}$R$_{j}$ / max( R, 1) ) . For each p$_{1}$, 

<img src='content_image/296759.jpg'>

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, 

<img src='content_image/296762.jpg'>

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 

