To characterize differences in statistical power among the multivariate GWAS, we examine the mean χ2, λGC, and number of genome-wide significant risk loci for each factor and residual SUD (Supplementary Table 1c). We evaluated the novelty of our findings by comparing the genomic risk loci identified in our analyses to 1) loci identified in the original externalizing and addiction risk GWAS and 2) loci previously identified for substance use phenotypes in the GWAS literature. This latter test was performed by comparing the genomic risk loci for our factors and correlated SNPs (r2 > 0.1) to those in the NHGRI-EBI GWAS Catalog29 (version e114_r2025-06-27). Finally, we assessed the relative performance of our two models by comparing the degree of heterogeneity of SNP effects. We did so by calculating QSNP heterogeneity statistics, which can be used to identify SNPs that have an effect on one or more indicator phenotypes that is better explained by pathways independent of the factor. In other words, the QSNP test is designed to ensure that SNPs are not being detected with the shared factor primarily due to
