The genome-wide MSE of the MTAG estimates is simply equal to their sampling variance (given above): MSE(β^MTAG,j,t)≡E[(β^MTAG,j,t−βj,t)2]=Var(β^MTAG,j,t−βj,t)=1ωt′ωtt(Ω−ωtωt′ωtt+∑j)−1ωtωtt, where the first equality follows because both the true effects βj,t and the MTAG estimates β^MTAG,j,t are mean zero. Illustrative calculations of this formula in a two-trait setting are shown in Supplementary Figure 1. This formula for the MSE holds very generally; in particular, it does not require assuming that Ω is homogeneous across SNPs (because the genome-wide MSE is a property regarding the mean across all the SNPs included in the analysis). In the formula, Ω is (re-)defined as the genome-wide (i.e., across-SNP) variance-covariance matrix of the SNPs’ true effects on the traits. By simulation, we verify that the MSE formula is a good approximation when using estimates of Ω and ∑j (Supplementary Table 1).
