Now consider Eq. (3) applied to each one of a genome-wide set of SNPs. Assuming it has no interaction with each SNP, the non-genetic component of U is constant (Fig. 3). The genetic component of U equals the entire shared genetic basis of incidence and prognosis, minus any component due to the SNP under consideration. For polygenic traits, the variation explained by individual SNPs is small in relation to the total genetic variance, so we may assume that the genetic component of U is approximately constant across SNPs. Therefore we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b = \frac{{ - var(U)\beta _{UX}\beta _{UY}}}{{var\left( U \right)\beta _{UX}^2 + var(E_X)}}$$\end{document}b=-var(U)βUXβUYvarUβUX2+var(EX) is approximately constant across SNPs, and may be obtained from the linear regression of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta^\prime_{GY}}$$\end{document}βGY′ on βGX, giving the bias-corrected effects4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{GY} = \beta _{GY}^\prime - b\beta _{GX}$$\end{document}βGY=βGY′-bβGXIn practice, we have finite sample estimates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat \beta _{GY}^\prime$$\end{document}β^GY′ and \documentclass[12pt]{minimal} \usepackage{amsmath}
