The set of indices { 1 , . . . , m } of the elementary null hypotheses may be divided into four (unknown) subsets I$_{00}$ ∪ I$_{10}$ ∪ I$_{01}$ ∪ I$_{11}$ = { 1 , . . . , m } , where each index j is in exactly one of the four subsets, defined as follows: I$_{00}$ = { j : ( h$_{1}$$_{j}$, h$_{2}$$_{j}$ ) = (0 , 0) , j ∈ { 1 , . . . , m }} ; I$_{10}$ = { j : ( h$_{1}$$_{j}$, h$_{2}$$_{j}$ ) = (1 , 0) , j ∈ { 1 , . . . , m }} ; I$_{01}$ = { j : ( h$_{1}$$_{j}$, h$_{2}$$_{j}$ ) = (0 , 1) , j ∈ { 1 , . . . , m }} ; I$_{11}$ = { j : ( h$_{1}$$_{j}$, h$_{2}$$_{j}$ ) = (1 , 1) , j ∈ { 1 , . . . , m }} . Definition 2.1. The no replicability null hypothesis for elementary hypothesis H$_{j}$ is H$_{NR,j}$ : ( h$_{1}$$_{j}$, h$_{2}$$_{j}$ ) ∈ { (0 , 0) , (0 , 1) , (1 , 0) } . By definition, H$_{NR,j}$ is false if and only if the elementary null hypothesis H$_{j}$ is false in both studies considered. In the family of m composite null hypotheses H$_{NR,}$$_{1}$, . . . , H$_{NR,m}$ , the sets of indices of true and false null hypotheses are I$_{00}$ ∪ I$_{01}$ ∪ I$_{10}$ and I$_{11}$ respectively. Our goal is to discover as many indices from I$_{11}$ as possible, i.e. true positives, while controlling for the number of discoveries from I$_{00}$ ∪ I$_{01}$ ∪ I$_{10}$ , i.e. false positives. Let p$_{ij}$ be the p -value for the j th SNP in study i , for i = 1 , 2. Since the studies are in- dependent, the p -values are independent across studies. However, the p -values within each study may be dependent. Inequality x ≥ y for vectors x and y is understood componentwise. Remark 2.1. In a typical meta-analysis (Hedges and Olkin, 1985), the goal is to discover as many indices from I$_{01}$ ∪ I$_{10}$ ∪ I$_{11}$ as possible, while controlling for the number of discoveries from I$_{00}$ . Had we known, and had it been true, that I$_{01}$ =  and I$_{10}$ =  , then the typical methods for meta-analysis could serve to discover replicable findings. However, it is not known in practice whether I$_{01}$ and I$_{10}$ are empty sets, and they need not be empty when the follow-up study is different, in at least one aspect of design, from the primary study. Therefore, typical meta-analysis methods are not 4