Chunk #51 — Methods — Removal of pleiotropic SNPs by HEIDI-outlier

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

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\begin{document}$$\hat b_{xy}$$\end{document}b^xy as a function of −log10(P-value) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{zx}$$\end{document}b^zx and choose the SNP that shows the strongest association with the exposure in the third quintile of the distribution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat b_{xy}$$\end{document}b^xy to avoid choosing an extreme pleiotropic outlier as the target SNP. If we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i = b_{xy(i)} - b_{xy(top)}$$\end{document}di=bxy(i)-bxy(top), we will have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}} \left( {\hat d_i} \right) {\mathrm{var}}\left( {\hat b_{xy\left( i \right)} - \hat b_{xy\left( {\rm {top}} \right)}} \right) = {\mathrm{var}}\left( {\hat b_{xy\left( i \right)}} \right) + {\mathrm{var}}\left( {\hat b_{xy\left( {\rm {top}} \right)}} \right) - 2{\mathrm{cov}}(\hat b_{xy\left( i \right)},\hat b_{xy\left( {\rm {top}} \right)})$$\end{document}vard^ivarb^xyi-b^xytop=varb^xyi+varb^xytop-2cov(b^xyi,b^xytop), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{ccccc}{\mathrm{cov}}\left( {\hat b_{xy\left( i \right)},\hat b_{xy\left( {\rm {top}} \right)}} \right) = \frac{{r_i\sqrt {{\mathrm{var}}(\hat b_{zy\left( i \right)}){\mathrm{var}}(\hat b_{zy\left( {\rm {top}} \right)})} }}{{b_{zx(i)}b_{zx(\rm {top})}}} +\\ b_{xy(i)}b_{xy(\rm {top})} \left[ \frac{r_i{\sqrt {{\mathrm{var}}\left( {\hat b_{zx\left( i \right)}} \right){\mathrm{var}}{\left( {\hat b_{zx\left( {\rm {top}} \right)}} \right)} }}}{{b_{zx\left( i \right)}b_{zx\left( {\rm {top}} \right)}}} - \frac{{{{\mathrm{var}}\left( {\hat b_{zx\left( i \right)}} \right){\mathrm{var}}{\left( {\hat b_{zx\left( {\rm {top}} \right)}} \right)} }}}{{b^2_{zx\left(