With the summary statistics from single-SNP analyses and individual-level genotype data of the discovery sample, we can convert the marginal effects to joint effects without using the phenotype data. We know from equation (4) that X′y=Dβ^, and we therefore can rewrite equation (2) with respect to β^ (5)b^=(X′X)−1Dβ^and var(b^)=σJ2(X′X)−1 The proportion of phenotypic variance explained by all the SNPs (coefficient of determination of a multiple regression model) is (6)RJ2=b^′X′yy′y=b^′Dβ^y′y giving the following equation: (7)σ^J2=(1−RJ2)y′yn−N=y′y−b^′Dβ^n−N In an association analysis of a single SNP j, (8)σ^M(j)2=y′y−Djβ^j2n−1 and the squared standard error of the estimate of the effect size is Sj2=σ^M(j)2∕Dj so that y′y=DjSj2(n−1)+Djβ^j2. Although the phenotypes of a quantitative trait are often standardized to z scores, we take the median of DjSj2(n−1)+Djβ^j2 across all the SNPs to calculate y′y instead of relying on the variance being known.
