The theoretical distribution of the Z statistic, resulting from a meta-analysis of GWAS results under imperfect CGRs, can be found in S1 Derivations. These expressions allow for differences in sample size, hSNP2, and CGR across (pairs of) studies. For intuition, we here present the specific case of a meta-analysis of results from two studies with CGR ρG, with equal SNP-based heritability hSNP2, and equal sample sizes (i.e., N in Study 1 and N in Study 2). Under this scenario, we find that under high polygenicity, the Z statistic of an associated SNP k is normally distributed with mean zero and the following variance: VarZk=EZk2≈1+hSNP2MN1+ρG.(1) We incorporate cross-study genetic heterogeneity by assuming that the data-generating process follows a random-effects model, where cross-study correlations in SNP effects shape the inferred CGRs. When one has random effects, under the null hypothesis a SNP effect follows a degenerate distribution with all probability mass at zero, whereas under the alternative hypothesis a SNP effect follows a distribution with mean zero and a finite non-zero variance. Bearing in mind that we can write a meta-analysis Z
