Chunk #82 — Methods — Notation and preliminaries

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Exploring the phenotypic consequences of tissue specific gene expression variation inferred from GWAS summary statistics.

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Thus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {{\mathrm{Var}}}\left( {T_g} \right) = \hat \sigma _g^2$$\end{document}Var^Tg=σ^g2 can be computed as\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{c}} {\hat \sigma _g^2} & = & {\widehat {{\mathrm{Var}}}\left( {\mathop {\sum}\nolimits_{l \in {\mathrm{Model}}_g} {w_{lg}X_l} } \right)} \\ {} & = & {\widehat {{\mathrm{Var}}}\left( {{\bf{W}}_g{\bf{X}}_g} \right)} \\ {} & = & {{\bf{W}}_g^\prime \widehat {{\mathrm{Var}}}\left( {{\bf{X}}_g} \right){\bf{W}}_g} \end{array}$$\end{document}σ^g2=Var^∑l∈ModelgwlgXl=Var^WgXg=Wg′Var^XgWgwhere Wg is the vector of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{lg}$$\end{document}wlg for SNPs in the model of g. By definition, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{\Gamma }}_g$$\end{document}Γg is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {{\mathrm{Var}}}({\bf{X}}_g)$$\end{document}Var^(Xg), the sample covariance of Xg, so that we arrive to4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\mathbf \sigma }}_{\boldsymbol{g}}^2 = {\mathbf{W}}_{\boldsymbol{g}}^\prime {\mathbf{\Gamma W}}_{\boldsymbol{g}}$$\end{document}σ^g2=Wg′ΓWg