Chunk #91 — Methods — Calculation of Z-score

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

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with which we can compute the p-value as p = 2Φ(−|Zg|) where Φ(.) is the normal CDF function. Thus\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {Z_g = \frac{{\hat \gamma _g}}{{{\mathrm{se}}\left( {\hat \gamma _g} \right)}}} \hfill \\ { = \mathop {\sum }\limits_{l \in {\mathrm{Model}}_g} \frac{{w_{lg}\hat \beta _l\hat \sigma _l^2}}{{\hat \sigma _g^2}}\sqrt {\frac{n}{{\hat \sigma _Y^2}}\frac{{\hat \sigma _g^2}}{{(1 - R_g^2)}}} } \hfill \\ { = \mathop {\sum }\limits_{l \in {\mathrm{Model}}_g} \frac{{w_{lg}\hat \beta _l\hat \sigma _l^2}}{{\hat \sigma _g^2}}\sqrt {\frac{{(1 - R_l^2)}}{{{\mathrm{se}}^2\left( {\hat \beta _l} \right)\hat \sigma _l^2}}\frac{{\hat \sigma _g^2}}{{(1 - R_g^2)}}} } \hfill \end{array}$$\end{document}Zg=γ^gseγ^g= ∑l∈Modelgwlgβ^lσ^l2σ^g2nσ^Y2σ^g2(1-Rg2)= ∑l∈Modelgwlgβ^lσ^l2σ^g2(1-Rl2)se2β^lσ^l2σ^g2(1-Rg2)where we used Eqs. (5) and (6) in the second line and Eq. (8) in the last step. So10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_g = \mathop {\sum }\limits_{l \in {\mathrm{Model}}_g} w_{lg}\frac{{\hat \sigma _l}}{{\hat \sigma _g}}\frac{{\hat \beta _l}}{{{\mathrm{se}}\left( {\hat \beta _l} \right)}}\sqrt {\frac{{\left( {1 - R_l^2} \right)}}{{\left( {1 - R_g^2} \right)}}} $$\end{document}Zg= ∑l∈Modelgwlgσ^lσ^gβ^lseβ^l1-Rl21-Rg211\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \approx \mathop {\sum }\limits_{l \in {\mathrm{Model}}_g} w_{lg}\frac{{\hat \sigma _l}}{{\hat \sigma _g}}\frac{{\hat \beta _l}}{{{\mathrm{se}}\left( {\hat \beta _l} \right)}}$$\end{document}≈∑l∈Modelgwlgσ^lσ^gβ^lseβ^l