(1)E[z1jz2j]=N1N2ρgMℓj+ρNsN1N2, where Ni is the sample size for study i, ρg is genetic covariance (defined in Methods), ℓj is LD Score [19], Ns is the number of individuals included in both studies, and ρ is the phenotypic correlation among the Ns overlapping samples. We derive this equation in the Supplementary Note. If study 1 and study 2 are the same study, then Equation 1 reduces to the single-trait result from [19], because genetic covariance between a trait and itself is heritability, and χ2=z2. As a consequence of equation 1, we can estimate genetic covariance using the slope from the regression of z1jz2j on LD Score, which is computationally very fast (Methods).
