In order to compute statistical power in a multi-study setting, we first use the generic expression for the variance of the GWAS Z statistic derived in S1 Derivations to characterize the distribution of the Z statistic under the alternative hypothesis. Given a genome-wide significance threshold (denoted by α; usually α = 5 ⋅ 10−8), we use the normal cumulative distribution function under the alternative hypothesis to quantify the probability of attaining genome-wide significance for an associated SNP. This probability we refer to as the ‘power per associated SNP’ (denoted here by β). Given that we use SNPs tagging independent haplotype blocks, we can calculate the probability of rejecting the null for at least one SNP and the expected number of hits, true positives, false positives, false negatives, and positive negatives, as functions of α, β, the number of truly associated SNPs (denoted by M), and the number of non-associated SNPs (denoted by S − M). Letting ‘#’ denote the number of elements in a set, we have that ℙ [# true positives ≥ 1]=1−(1−β)M,ℙ [# hits ≥ 1]=1−[(1−β)M(1−α)S−M],E [# hits]=βM+α(S−M),E [# true positives]=βM,E [# false positives]=α(S−M),E [# false negatives]=(1−β)M, andE [# true negatives]=(1−α)(S−M).
