Because of the homogeneous-Ω assumption, the above formulas for the MTAG estimator and its standard error effectively use the variance-covariance matrix of true SNP effects across all SNPs, Ω, to calculate the MTAG estimate for each SNP. If in fact there are different types of SNPs characterized by different variance-covariance matrices, then the MTAG estimator remains consistent but could be made more efficient if it took into account the different types of SNPs. In addition, the standard error formula is conservative on average across SNPs, which reduces MTAG’s statistical power to identify truly associated SNPs. Most importantly, the MTAG estimator is in general biased in finite samples, and it is biased away from zero for SNPs that are truly null, which causes the false positive rate to be inflated.
