Just like before, the optimal weights can be derived as:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{w}}_{{\mathrm{OLS}}} = {\bf{V}}_{{\mathrm{OLS}}}^{ - 1}{\bf{C}}_{{\mathrm{OLS}}}$$\end{document}wOLS=V OLS-1COLS, where COLS is now a k × 1 column vector of the covariances of the \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 values of the k traits with the true genetic effects of the SNPs for the focal trait, and VOLS is a k × k variance–covariance matrix of the \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 effects:24\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {{\bf{w}}_{{\mathrm{OLS}}}} \hfill & = \hfill & {{\bf{V}}_{{\mathrm{OLS}}}^{ - 1}{\bf{C}}_{{\mathrm{OLS}}}} \hfill \\ {} \hfill & = \hfill & {\left[ {\begin{array}{*{20}{c}} {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_1}} \right)} & \cdots & {{\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_1},\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \right)} \\ \vdots & \ddots & \vdots \\ {{\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_k},\widehat {\bf{b}}_{{\mathrm{OLS}}_1}} \right)} & \cdots & {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \right)} \end{array}} \right]^{ - 1}} \hfill \\ {} \hfill & {} \hfill & {\left[ {\begin{array}{*{20}{c}} {{\mathrm{cov}}\left( {{\bf{b}}_{\mathit{f}},\widehat {\bf{b}}_{{\mathrm{OLS}}_1}} \right)} \\ \vdots \\ {{\mathrm{cov}}\left( {{\bf{b}}_{\mathit{f}},\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \right)} \end{array}} \right]} \hfill \end{array}$$\end{document}wOLS=V OLS-1COLS=varb^OLS1⋯covb^OLS1,b^OLSk⋮⋱⋮covb ^OLSk,b^OLS1⋯varb^OLSk-1covbf ,b^OLS1⋮covbf ,b^OLSkThe diagonal elements of VOLS are:25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \right) = \frac{{h_k^2}}{M} + \frac{1}{{N_k}}$$\end{document}varb^OLSk=hk2M+1NkThe off-diagonal elements for trait k and l are26\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_k},\widehat {\bf{b}}_{{\mathrm{OLS}}_l}} \right) = \frac{{r_{\mathrm{G}}h_kh_l}}{M}$$\end{document}covb^OLSk,b^OLSl=rGhkhlMCOLS now has elements27\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_{\mathrm{k}},\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \right) = \frac{{r_{\mathrm{G}}h_kh_l}}{M}$$\end{document}covbk,b^OLSk=rGhkhlMIf we again consider a two-trait example, Eqs. (25), (26) and (27) combine as:28\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{w}}_{{\mathrm{OLS}}} = {\bf{V}}_{{\mathrm{OLS}}}^{ - 1}{\bf{C}}_{{\mathrm{OLS}}} = \left[ {\begin{array}{*{20}{c}} {\frac{{h_1^2}}{M} + \frac{1}{{N_1}}} & {\frac{{r_{\mathrm{G}}h_1h_2}}{M}} \\ {\frac{{r_{\mathrm{G}}h_1h_2}}{M}} & {\frac{{h_2^2}}{M} + \frac{1}{{N_2}}} \end{array}} \right]^{ - 1}\left[ {\begin{array}{*{20}{c}} {\frac{{h_1^2}}{M}} \\ {\frac{{r_{\mathrm{G}}h_1h_2}}{M}} \end{array}} \right]$$\end{document}wOLS=V OLS-1COLS=h12M+1N1rGh1h2MrGh1h2Mh22M+1N2-1h12MrGh1h2M