and Σ(θ) have been half-vectorized to produce s and σ(θ), respectively, and DS is VS with its off-diagonal elements set to 0. The sampling covariance matrix of the stage 2, Genomic SEM parameter estimates (Vθ) are obtained using a sandwich correction described in Grotzinger et al.13: Vθ=(Δ^′Γ−1Δ^)−1Δ^′Γ−1VSΓ−1Δ^(Δ^′Γ−1Δ^)−1 where Δ^ is the matrix of model derivatives evaluated at the parameter estimates, Γ is the stage 2 weight matrix, DS, and VS is the sampling covariance matrix of S. Validation of Genomic SEM in Grotzinger et al.13 demonstrated that the framework produces unbiased standard errors, appropriately accounts for sample overlap in multivariate GWAS, and produces accurate point estimates for different population generating models. In addition, polygenic scores derived from Genomic SEM summary statistics were found to better predict the individual traits that define the factor than polygenic scores constructed from the summary statistics for the individual traits. As part of the current analyses, we sought to further validate Genomic SEM via a series of simulations based directly on the factor structure identified here and additionally benchmark Genomic SEM against existing multivariate methods.
