In the framework that follows, all traits and genotypes are standardized to have mean zero and variance one. For SNP j, we denote the vector of marginal (i.e., not controlling for other SNPs), true effects on each of the T traits by βj. We treat these true effects as random effects with E(βj)=0 and Var(βj)=Ω. If the true effects are correlated across traits, then the off-diagonal elements of Ω are non-zero. MTAG’s key assumption is that Ω is homogeneous across SNPs, i.e., it does not depend on j.
