Under a multivariate extension of the S-LDSC model, the expected value of the product of z statistics for each pairwise combination of phenotypes for SNP j equals: E[z1jz2j]=N1N2∑cτcℓ(j,c)Mc+ρNsN1N2+a where Ni is the sample size for study i, c indexes a genomic annotation, Mc is the number of SNPs in annotation c, ℓ(j,c) is the LD score of SNP j with respect to annotation c (that is, the sum of squared LD this SNP has with all SNPs in the annotation), τc is a vector of free parameters used to compute the conditional contribution to heritability or coheritability (genetic covariance) in annotation c, Ns is the number of individuals included in both GWAS samples, ρ is the phenotypic correlation within the overlapping samples, and a is a term representing unmeasured sources of confounding such as shared population stratification across GWASs53. The inclusion of the term Mc in the above equation produces LD scores (ℓ(j,c)Mc) that are scaled relative to the size of the respective annotations, thereby allowing τc to be interpreted on the same scale as genome-wide estimates of heritability and
