common factor. Two key results were verified at this stage. First, we confirmed that the standardized factor loadings on the common factor were 1 for the identical phenotypes. Second, we verified that the bivariate ld-score intercepts that are used to account for sample overlap in the sampling covariance matrix were as expected. The equation for the ld-score bivariate intercept is:4 Nsρ/√(N1N2), where Ns = sample overlap, ρ = the phenotypic correlation, N1 = sample size of trait 1, and N2 = sample size of trait 2. In this simulation, we observed bivariate intercepts of .67, which is as expected given sample overlap of 40,000, a phenotypic correlation of 1, and sample sizes of 60,000 (i.e., 40,000*1/√(60,000*60,000) = .67). Finally, estimates from this multivariate GWAS were compared to estimates from the univariate GWAS in PLINK for the full set of 100,000 participants. If sample overlap is not appropriately accounted for in this example, such that data are incorrectly treated as deriving from 180,000 participants (as opposed to 100,000 total participants), we would expect the Z statistics for the SNP effects from Genomic SEM to be upwardly biased relative to those from a univariate GWAS applied directly to the single phenotype in
