Single-trait approximate BLUP SNP effects can be obtained from GWAS summary statistics (SBLUP: summary statistic approximate BLUP) by replacing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{W}}_k\prime {\bf{W}}_k$$\end{document}Wk′Wk and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{W}}_k\prime {\bf{y}}_k$$\end{document}Wk′yk of Eq. (8) by their expectation, which are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Bbb E}\left[ {{\bf{W}}_k\prime {\bf{W}}_k} \right] = N_k{\bf{L}}$$\end{document}EWk′Wk=NkL and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Bbb E}\left[ {{\bf{W}}_k\prime {\bf{y}}_k} \right] = N_k\widehat {\bf{b}}_{{\mathrm{OLS}}_k}$$\end{document}EWk′yk=Nkb^OLSk, respectively, where L is an M × M scaled SNP LD correlation matrix estimated from a reference SNP data set and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}_k}$$\end{document}b^OLSk are obtained from publically available GWAS summary statistics20. GWAS summary statistics report effect estimates of SNPs on an unstandardised scale and not \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}}$$\end{document}b^OLS as it is defined here. To obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}}$$\end{document}b^OLS from GWAS summary statistics, the effect of each SNP must be multiplied by the standard deviation of each SNP: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}_{\mathrm{j}}}$$\end{document}b^OLSj = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}} - {\mathrm{UNSCALED}}_{\mathrm{j}}} \times \sqrt {2p_j\left( {1 - p_j} \right)}$$\end{document}b^OLS-UNSCALEDj×2pj1-pj. Equation (8) can then be written as:11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \hfill & = \hfill & {\left[ {N_k{\bf{L}} + {\bf{I}}_M\lambda _k} \right]^{ - 1}N_k\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \hfill \\ {} \hfill & = \hfill & {\left[ {{\bf{L}} + {\bf{I}}_M\lambda _k{\mathrm{/}}N_k} \right]^{ - 1}\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \hfill \end{array}$$\end{document}b^SBLUPk=NkL+IMλk-1Nkb^OLSk=L+IMλk∕Nk-1b^OLSk