For human complex diseases and quantitative phenotypes, GWASs have typically estimated the solutions for b of Eq. (1) one SNP at a time using OLS regression29 as:5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}} = {\mathrm{diag}}\left[ {{{{\bf W}{\prime}{\bf W}}}} \right]^{ - 1}{{{\bf W}{\prime}{\bf y}}}$$\end{document}b^OLS=diagW′W-1W′ywhere diag[W′W] has diagonal elements \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_j\prime w_j$$\end{document}wj′wj and off-diagonal elements of zero. However, by analysing one SNP at a time, GWAS effect size estimates do not account for the covariance structure among SNPs and they are not unbiased in the sense that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathit{E}}\left[ {{\bf{b}}|{\hat{\bf b}}} \right] = {\hat{\bf b}}$$\end{document}Eb∣b^=b^12. BLUP of the SNP effects have the property \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\it{E}}\left[ {{\bf{b}}|{\hat{\bf b}}} \right] = {\hat{\bf b}}$$\end{document}Eb∣b^=b^, are used in genomic prediction in animal and plant breeding30 and more recently in human medical genetics, yielding improved prediction accuracy for a number of traits over genetic predictors created from OLS SNP estimates16,17. In a general form, BLUP solutions for b of Eq. (1) can be written using Henderson’s mixed model equations31 as:6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{BLUP}}} = \left[ {{{{\bf W}{\prime}{\bf R}}}^{ - {\mathrm{1}}}{\bf{W}}{{ + }}{\bf{B}}^{ - 1}} \right]^{ - 1}{{{\bf W}{\prime}{\bf R}}}^{ - 1}{\bf{y}}$$\end{document}b^BLUP=W′R-1W+B-1-1W′R-1yand if R is diagonal, then Eq. (6) can be reduced to:7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{BLUP}}} = \left[ {{{{\bf W}{\prime}{\bf W}}} + {\bf{B}}^{ - 1}{\bf{R}}} \right]^{ - 1}{{{\bf W}{\prime}{\bf y}}}$$\end{document}b^BLUP=W′W+B-1R-1W′y