For multiple phenotypes, the elements of Eq. (11) become: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{{{\rm OLS}{\prime}}}} = \left[ {\widehat {\bf{b}}_{{\mathrm{OLS}}_1{\prime} } \ldots \widehat {\bf{b}}_{{\mathrm{OLS}}_k{\prime} }} \right]$$\end{document}b^OLS′=b^OLS1′…b^OLSk′ and N = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\begin{array}{*{20}{c}} {{\bf{N}}_1} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & {{\bf{N}}_k} \end{array}} \right]$$\end{document}N1000⋱000Nk, meaning that Eq. (11) can be extended as:12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{MT}} - {\mathrm{SBLUP}}} = \left[ {{\bf{I}}_k \otimes {\bf{L}} + {\bf{\Sigma }}_\epsilon {\bf{\Sigma }}_b^{ - 1}{\bf{N}}^{ - 1} \otimes {\bf{I}}_M} \right]^{ - 1}\widehat {\bf{b}}_{{\mathrm{OLS}}}$$\end{document}b^MT-SBLUP=Ik⊗L+ΣϵΣb-1N-1⊗IM-1b^OLSEquation (12) approximates Eq. (9) using only publically available GWAS summary statistic data and an independent genomic reference sample. However, there remains the large computational cost associated with the inversion of the non-diagonal matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {{\bf{I}}_k \otimes {\bf{L}} + {\bf{\Sigma }}_\epsilon {\bf{\Sigma }}_b^{ - 1}{\bf{N}}^{ - 1} \otimes {\bf{I}}_M} \right]$$\end{document}Ik⊗L+ΣϵΣb-1N-1⊗IM.