We consider a general linear mixed model:4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{y}} = {\bf{Wb}} + {\bf{\epsilon }}$$\end{document}y=Wb+ϵwhere y is the phenotype, W a matrix of SNP genotypes, where values are standardised to give the ijth element as: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{ij} = \left( {x_{ij} - 2p_j} \right){\mathrm{/}}\sqrt {2p_j\left( {1 - p_j} \right)}$$\end{document}wij=xij-2pj∕2pj1-pj, with xij the number of minor alleles (0, 1 or 2) for the ith individual at the jth SNP and pj the minor allele frequency. b are the genetic effects for each SNP, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{\epsilon }}$$\end{document}ϵ the residual error. The dimensions of y, W, b and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{\epsilon }}$$\end{document}ϵ are dependent upon the number of phenotypes, k, the number of SNP markers, M, and the number of individuals, N, and are described in the sections below. We denote the distributional properties var(b) = B, var(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{\epsilon }}$$\end{document}ϵ) = R and var(y) = WBW′ + R.