Chunk #58 — Methods — Multi-trait conditional GWAS analysis using summary data

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Causal associations between risk factors and common diseases inferred from GWAS summary data.

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The sampling variance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{zy}|{\hat{\mathbf b}}_{xy}$$\end{document}b^zy∣b^xy is approximately\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\hat b_{zy}{\mathrm{|}}{\hat{\mathbf b}}_{xy}} \right) = {\mathrm{var}}\left( {\hat b_{zy}} \right) + {\hat{\mathbf b}}_{xy}^t{\mathbf{V}}_{zx}{\hat{\mathbf b}}_{xy} - 2{\hat{\mathbf b}}_{xy}^t{\mathrm{cov}}\left( {\hat b_{zy},{\hat{\mathbf b}}_{zx}} \right),$$\end{document}varb^zy∣b^xy=varb^zy+b^xytV zxb^xy-2b^xytcovb^zy,b^zx,where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{V}}_{zx} = {\mathrm{var}}({\hat{\mathbf b}}_{zx})$$\end{document}V zx=var(b^zx), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {\hat b_{zy},{\hat{\mathbf b}}_{zx}} \right)$$\end{document}covb^zy,b^zx is a t-length vector with the i-th element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {\hat b_{zy},\hat b_{zx_i}} \right)$$\end{document}covb^zy,b^zxi being the covariance between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{zy}$$\end{document}b^zy and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{zx_i}$$\end{document}b^zxi. We know from our previous study17 that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {\hat b_{zy},\hat b_{zx_i}} \right) = \rho _{x_iy}r_{p(x_i,y)}\sqrt {{\mathrm{var}}\left( {\hat b_{zx_i}} \right){\mathrm{var}}(\hat b_{zy})}$$\end{document}covb^zy,b^zxi=ρxiyrp(xi,y)varb^zxivar(b^zy) where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{x_iy}$$\end{document}ρxiy is the proportion of sample overlap between xi and y and \documentclass[12pt]{minimal} \usepackage{amsmath}