These simulations suggest that in most realistic applications of MTAG, the bias from ignoring sampling variation in Ω^ and ∑^j is negligibly small. A possible exception, not discussed so far, arises if MTAG is used for GWAS meta-analysis across a large number of cohorts (in which case T is large). MTAG may be valuable for that purpose because (i) it accounts for sample overlap and cryptic relatedness across cohorts and (ii) different cohorts often have phenotypic data from different measures, which may be imperfectly genetically correlated and have different heritabilities. For such applications, to reduce bias in the MTAG standard errors, we recommend imposing reasonable parameter restrictions on the Ω^ and ∑^j matrices (e.g., assuming that within groups of cohorts that analyzed identical phenotype measures, the heritability is equal and all pairwise genetic correlations are one).
