For a single SNP, we assume that incidence X is linear in the coded genotype G, the combined common causes U of incidence and prognosis, and causes EX unique to X: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 and U with an additional main effect of X: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+EYIf X or Y are binary, we continue to argue from linear models by observing that logistic and probit link functions are approximately linear for small effects (Methods).
