Quantitative traits with heritability (h2) of 0.2 and 0.6 were simulated with the UK Biobank genotype data (post–quality control) as input. Briefly, each quantitative trait was simulated on the basis of the following linear model: (3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}\begin{equation*} Y\ = {\rm{\ }}X\beta + \varepsilon, \end{equation*}\end{document}where X is the unstandardized genotype matrix corresponding to 385,794 individuals (rows) and 560,173 SNP genotypes (columns). The β vector corresponds to the effect size associated with each SNP, with 100, 1,000, 10,000, 100,000, and 560,173 (all SNPs) randomly selected to be causal SNPs with effect size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\beta \sim\,N( {0,1} )$\end{document}, β = 0 otherwise, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\varepsilon $\end{document} represents the random error, which follows \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\varepsilon \sim N\left(0,\sqrt{var(X\beta)(1-h^2)/h^2}\right)$\end{document}. To control for batch effects and population structure in the genotype data, a regression of batch and 40 Principal Components (PCs) against the simulated trait were performed as follows: (4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}\begin{equation*} Y\ = \mathrm{Batch} + 40 \ \mathrm{PCs} + \varepsilon. \end{equation*}\end{document}
