\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\mathrm{max}}}^2 = 1 - {\cal{L}}_{{\mathrm{res}}}^{2{\mathrm{/}}N}$$\end{document}Rmax2=1-Lres2∕N, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{L}}_{{\mathrm{res}}}$$\end{document}Lres is the likelihood of a restricted logistic regression model with covariates only (an intercept, current age, sex and top 10 PCs of the genotype data), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{L}}_{{\mathrm{full}}}$$\end{document}Lfull is the likelihood of the full logistic regression model (covariates and the PRS predictor), and N is the sample size. We define the relative increase or decrease in R2 of a polygenic prediction method A compared to method B as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(R_{\mathrm{A}}^2 - R_{\mathrm{B}}^2)/R_{\mathrm{B}}^2$$\end{document}(RA2-RB2)∕RB2. In addition to R2 or Nagelkerke’s R2, we also report area under the ROC curve (known as AUC), area under the precision-call curve, and the odds ratio (OR) comparing top 10% of the participants having high polygenic risk with the remaining 90% of the sample. We adjusted for current age, sex and top 10 PCs of the genotype data in the calculation of all predictive performance metrics.
