Recall that we assume incidence X is linear in the coded genotype G, the combined common causes U of incidence and prognosis, and causes EX unique to X (Eq. (1)):\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X = \beta _{GX}G + \beta _{UX}U + E_X$$\end{document}X=βGXG+βUXU+EXSimilarly, assume that prognosis Y is linear in G, U and X (Eq. (2)):\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y = \beta _{GY}G + \beta _{UY}U + \beta _{XY}X + E_Y$$\end{document}Y=βGYG+βUYU+βXYX+EYThese are not necessarily causal models, but reflect a parameterisation of associations between G, U, X and Y that is natural when the conditional independence structure is as in Fig. 1 without conditioning on X. We assume that G, U, EX and EY are pairwise uncorrelated and have no interactions in the models for X and Y. Polygenic effects may contribute to U, EX and EY.
