LDSC analysis between xi and y (ref. 30). Vzx is the sampling variance-covariance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathbf b}}_{zx}$$\end{document}b^zx with the ij-th element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {\hat b_{zx\left( i \right)},\hat b_{zx\left( j \right)}} \right) = \rho _{x_ix_j}r_{p(x_i,x_j)}\sqrt {{\mathrm{var}}\left( {\hat b_{zx_i}} \right){\mathrm{var}}(\hat b_{zx_j})}$$\end{document}covb^zxi,b^zxj=ρxixjrp(xi,xj)varb^zxivar(b^zxj), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{x_ix_j}r_{p(x_i,x_j)}$$\end{document}ρxixjrp(xi,xj) can also be approximated by the intercept of a bivariate LDSC analysis between xi and xj. The multi-trait-based conditional GWAS test can be performed using the test-statistic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{{\mathrm{cond}}} = \left( {\hat b_{zy}{\mathrm{|}}{\hat{\mathbf b}}_{xy}} \right)^2/{\mathrm{var}}\left( {\hat b_{zy}{\mathrm{|}}{\hat{\mathbf b}}_{xy}} \right)$$\end{document}Tcond=b^zy∣b^xy2∕varb^zy∣b^xy. We call this approach mtCOJO (multi-trait-based conditional and joint analysis), and have demonstrated the accuracy of the approximation by simulation (Supplementary Fig. 5). Note that since the estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{x_iy}$$\end{document}βxiy is free of confounding from shared environmental or genetic effects that are not correlated with the valid instruments, our estimate of conditional effect does not suffer from the bias described in Aschard et al.23, as confirmed by simulation (Supplementary Fig. 6). We have implemented mtCOJO in
