All definitions refer to narrow-sense heritabilities and genetic covariances. Let S denote a set of M SNPs, let X denote a vector of additively (0-1-2) coded genotypes for the SNPs in S, and let y1 and y2 denote phenotypes. Define β:=argmaxα∈RMCor[y1,Xα], where the maximization is performed in the population (i.e., in the infinite data limit). Let γ denote the corresponding vector for y2. This is a projection, so β is unique modulo SNPs in perfect LD. Define hS2, the heritability explained by SNPs in S, as hS2(y1):=∑jβj2 and ρS(y1,y2), the genetic covariance among SNPs in S, as ρS(y1,y2):=∑j∈Sβjγj. The genetic correlation among SNPs in S is rS(y1,y2):=ρS(y1,y2)/hS2(y1)hS2(y2), which lies in [−1,1]. Following [11], we use subscript g (as in hg2, ρg,rg) when the set of SNPs is genotyped and imputed SNPs in GWAS.
