We simulated a binary selection event by treating the incidence trait as a liability with a threshold for disease such that 20% of individuals were affected. We simulated 10,000 cases and 10,000 controls, and obtained estimated effects on incidence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat \beta _{GX}$$\end{document}β^GX from logistic regression of disease on genotype. We then simulated a binary prognosis by thresholding the prognosis trait at its median, so that half the individuals had a good prognosis and half a poor prognosis. We obtained unadjusted estimated effects on prognosis \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′ from logistic regression of prognosis on genotype among cases only.
