The effect of interest is the direct SNP effect on prognosis βGY, conditional on incidence X and confounders U. In practice, however, the relevant confounders may not be observed and we can only estimate the SNP effect conditional on incidence, denoted by \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′. In the Methods we show that this estimand is the direct effect βGY plus a bias that is linear in the effect on incidence βGX: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)βGXNotably, the coefficient of βGX is negative if the confounder effects on incidence and prognosis, βUX and βUY, have the same sign and positive if they have opposing signs.
