Chunk #55 — 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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\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} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$cov\left( {X,Y} \right) = \beta _{GX}\beta _{GY}var\left( G \right) + \beta _{UX}\beta _{UY}var\left( U \right) + \beta _{XY}var\left( X \right)$$\end{document}covX,Y=βGXβGYvarG+βUXβUYvarU+βXYvarX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$= \left( {\beta _{GX}\beta _{XY} + \beta _{GY}} \right)\beta _{GX}var\left( G \right) + \left( {\beta _{UX}\beta _{XY} + \beta _{UY}} \right)\beta _{UX}var\left( U \right) + \beta _{XY}var\left( {E_X} \right)$$\end{document}=βGXβXY+βGYβGXvarG+βUXβXY+βUYβUXvarU+βXYvarEXSubstituting these covariances into Eq. (5) gives, after some working out, Eq. (3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{GY}^\prime = \beta _{GY} - \frac{{var(U)\beta _{UX}\beta _{UY}}}{{var\left( U \right)\beta _{UX}^2 + var(E_X)}}\beta _{GX}$$\end{document}βGY′=βGY-var(U)βUXβUYvarUβUX2+var(EX)βGXThis derivation is similar to that of Aschard et al.2, except that we allow for the direct effect of X on Y in Eq. (2) and have focussed on the asymptotic estimate of the true \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{GY}^\prime$$\end{document}βGY′.