We developed an approach called mtCOJO (multi-trait-based conditional and joint analysis; URLs) to perform a GWAS analysis for a trait conditioning on other traits using GWAS summary data (Methods; Supplementary Fig. 5). We then re-ran the GSMR analysis using the adjusted GWAS summary data from the mtCOJO analysis (Methods). The mtCOJO analysis requires the estimates of bxy of the covariate risk factors on the target risk factor and disease, rg among the covariate risk factors, SNP-based heritability (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{{\mathrm{SNP}}}^2$$\end{document}hSNP2) for the covariate risk factors, and sampling covariance between SNP effects estimated from potentially overlapping samples, all of which can be computed from summary data (Methods; Supplementary Tables 7–10). Given the similar GSMR results between BMI and WHRadjBMI and between SBP and DBP (Supplementary Fig. 10), we did not include DBP and WHRadjBMI in the conditional analysis to avoid over-correction.
