Chunk #54 — Methods — Bias adjustment

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Adjustment for index event bias in genome-wide association studies of subsequent events.

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yes

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then the asymptotic ordinary least squares estimate is\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\begin{array}{*{20}{c}} {\beta _{GY}^\prime } \\ {\beta _{XY}^\prime } \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {var\left( G \right)} & {cov\left( {G,X} \right)} \\ {cov\left( {G,X} \right)} & {var\left( X \right)} \end{array}} \right]^{ - 1}\left[ {\begin{array}{*{20}{c}} {cov\left( {G,Y} \right)} \\ {cov\left( {X,Y} \right)} \end{array}} \right]$$\end{document}βGY′βXY′=varGcovG,XcovG,XvarX-1covG,YcovX,Y5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$= \frac{1}{{var\left( G \right)var\left( X \right) - cov(G,X)^2}}\left[ {\begin{array}{*{20}{c}} {var\left( X \right)} & { - cov\left( {G,X} \right)} \\ { - cov\left( {G,X} \right)} & {var\left( G \right)} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {cov\left( {G,Y} \right)} \\ {cov\left( {X,Y} \right)} \end{array}} \right]$$\end{document}=1varGvarX-cov(G,X)2varX-covG,X-covG,XvarGcovG,YcovX,YFrom Eq. (1), under the assumptions above,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$var\left( X \right) = \beta _{GX}^2var\left( G \right) + \beta _{UX}^2var\left( U \right) + var\left( {E_X} \right)$$\end{document}varX=βGX2varG+βUX2varU+varEX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$cov\left( {G,X} \right) = \beta _{GX}var\left( G \right)$$\end{document}covG,X=βGXvarGFrom Eq. (2),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$cov\left( {G,Y} \right) = \beta _{GY}var\left( G \right) + \beta _{XY}\beta _{GX}var(G)$$\end{document}covG,Y=βGYvarG+βXYβGXvar(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}