It may not always be possible to obtain the full sampling covariance matrix, VS. For example, for highly sensitive data only the matrix S and the SEs of its elements may be available (i.e., the diagonal of VS). However, we note that when there is low sample overlap across the GWASs for each phenotype, off-diagonal elements of the sampling covariance matrix are small and pragmatically ignorable. Moreover, in other contexts with complete sample overlap, SE inflation of the SEM parameters estimated using diagonally-weighted versions of WLS has been estimated to be less than 8%9 without robustness corrections, and nil with robustness corrections.47
