There are several useful special cases of MTAG (Online Methods). When all estimates are for the same trait (implying ωtωt'ωtt=Ω and ωtωtt=1), equation (1) simplifies to: β^MTAG,j,t=1′∑j−11′∑j−11β^j. When the GWAS estimates are obtained from non-overlapping samples (i.e., ∑j is diagonal), this formula specializes to the well-known formula for inverse-variance-weighted meta-analysis. When the genetic correlations across all traits are zero and there is no sample overlap (i.e., both Ω and ∑j are diagonal), the MTAG estimates are identical to the GWAS estimates. This equivalence is intuitive, since it corresponds exactly to the case of no correlation between the GWAS estimates that can be leveraged.
