Therefore, MTAG proceeds by running linkage disequilibrium (LD) score regressions14 on the GWAS results and using the estimated intercepts to construct the diagonal elements of ∑^j. Next, bivariate LD score regressions6 are run using each pair of GWAS results, and the estimated intercepts are used to construct the off-diagonal elements of ∑^j. Under the assumptions of LD score regression (including that the LD reference sample and GWAS samples are all drawn from the same population), the resulting matrix ∑^j captures all relevant sources of estimation error, including not only sampling variation but also population stratification, unknown sample overlap, and cryptic relatedness. Because the LD-score-intercept adjustment is already built into the MTAG estimates, MTAG-generated association results do not require further adjustment for these biases.
