Chunk #84 — 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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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat \gamma _g$$\end{document}γ^g can be expressed as\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{c}} {\hat \gamma _g} & = & {\frac{{\widehat {{\mathrm{Cov}}}\left( {T_g,Y} \right)}}{{\hat \sigma _g^2}}} \\ {} & = & {\frac{{\widehat {{\mathrm{Cov}}}\left( {\mathop {\sum }\nolimits_{l \in {\mathrm{Model}}_g} w_{lg}X_l,Y} \right)}}{{\hat \sigma _g^2}}} \\ {} & = & {\mathop {\sum }\limits_{l \in {\mathrm{Model}}_g} \frac{{w_{lg}\widehat {{\mathrm{Cov}}}\left( {X_l,Y} \right)}}{{\hat \sigma _g^2}}} \end{array}$$\end{document}γ^g=Cov ^Tg,Yσ^g2=Cov ^∑l∈ModelgwlgXl,Yσ^g2=∑l∈ModelgwlgCov ^Xl,Yσ^g2where we used the linearity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\mathrm{Cov}}$$\end{document}Cov^ in the last step. Using Eq. (3), we arrive to5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat \gamma _g = \mathop {\sum }\limits_{l \in {\mathrm{Model}}_g} \frac{{w_{lg}\hat \beta _l\,\hat \sigma _l^2}}{{\hat \sigma _g^2}}$$\end{document}γ^g= ∑l∈Modelgwlgβ^lσ^l2σ^g2