Finally, we can consider the column vector CSBLUP, which is composed of the covariance between the true genetic effects of the SNPs for the focal trait, bf, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}$$\end{document}b^SBLUPk for all of the k traits. The first element of CSBLUP is covariance between the true genetic effects of the SNPs for the focal trait bf and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}_f}$$\end{document}b^SBLUPf for the focal trait \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_f}} \right) = {\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}_f}} \right) = \frac{{R_f^2}}{M}$$\end{document}covbf,b^BLUPf=varb^BLUPf=Rf2M. The remaining elements of CSBLUP are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right)$$\end{document}covbf,b^BLUPk, which can be approximated from theory by considering a regression of bf on bk where the regression coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{f,k} = r_{\mathrm{G}}\sqrt {{\mathrm{var}}\left( {{\bf{b}}_f} \right){\mathrm{/}}{\mathrm{var}}\left( {{\bf{b}}_k} \right)}$$\end{document}βf,k=rGvarbf∕varbk. The covariance of bf and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}$$\end{document}b^BLUPk can then be written as:20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right) = {\mathrm{cov}}\left( {\beta _{f,k}{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right) = r_{\mathrm{G}}\frac{{R_k^2}}{M} \cdot \frac{{h_f}}{{h_k}}$$\end{document}covbf,b^BLUPk=covβf,kbk,b^BLUPk=rGRk2M⋅hfhkIf we consider a two-trait example where the focal trait that we want to predict is matched to the first of the two traits, Eqs. (18), (19) and (20) combine as:21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{w}}_{{\mathrm{SBLUP}}} = {\bf{V}}_{{\mathrm{SBLUP}}}^{ - 1}{\bf{C}}_{{\mathrm{SBLUP}}} = \left[ {\begin{array}{*{20}{c}} {\frac{{R_1^2}}{M}} & {\frac{{r_{\mathrm{G}}R_1^2R_2^2}}{{h_1h_2M}}} \\ {\frac{{r_{\mathrm{G}}R_1^2R_2^2}}{{h_1h_2M}}} & {\frac{{R_2^2}}{M}} \end{array}} \right]^{ - 1}\left[ {\begin{array}{*{20}{c}} {\frac{{R_1^2}}{M}} \\ {r_{\mathrm{G}}\frac{{R_2^2}}{M} \cdot \frac{{h_1}}{{h_2}}} \end{array}} \right]$$\end{document}wSBLUP=V SBLUP-1CSBLUP=R12MrGR12R22h1h2MrGR12R22h1h2MR22M-1R12MrGR22M⋅h1h2giving the index for the focal trait as: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{SBLUP}}_f} = {\bf{w}}_1\widehat {\bf{b}}_{S{\mathrm{BLUP}}_f} + {\bf{w}}_2\widehat {\bf{b}}_{{\mathrm{SBLUP}}_2}$$\end{document}b^wMT-SBLUPf=w1b^SBLUPf+w2b^SBLUP2 with solutions for the index weights of:22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {{{w}}_{\mathrm f}} \hfill & = \hfill & {\left( {1 - \frac{{{{r}}_{\mathrm G}^2R_2^2}}{{h_2^2}}} \right){\mathrm{/}}\left( {1 - \frac{{{{r}}_{\mathrm G}^2R_{\mathrm f}^2R_2^2}}{{h_{\mathrm f}^2h_2^2M}}} \right)} \hfill \\ {} \hfill & = \hfill & {\left( {1 - {{r}}_{\mathrm G}^2R_{{\bf{b}}_2,\widehat {\bf{b}}_{{\mathrm{BLUP}}_2}}^2} \right){\mathrm{/}}\left( {1 - {{r}}_{\mathrm G}^2R_{{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_f}}^2R_{{\bf{b}}_2,\widehat {\bf{b}}_{{\mathrm{BLUP}}_2}}^2} \right),{\mathrm{and}}} \hfill \\ {{{w}}_2} \hfill & = \hfill & {\left( {{{r}}_{\mathrm G}/\left( {h_fh_2} \right)\left( {h_f^2 - R_1^2} \right)} \right){\mathrm{/}}\left( {1 - \frac{{{{r}}_{\mathrm G}^2R_f^2R_2^2}}{{h_f^2h_2^2M}}} \right)} \hfill \\ {} \hfill & = \hfill & {{{r}}_{\mathrm G}\left( {h_f{\mathrm{/}}h_2} \right)\left( {1 - R_{{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_f}}^2} \right){\mathrm{/}}\left( {1 - {{r}}_{\mathrm G}^2R_{{\bf{b}}_f,{\hat{\mathbf b}}_{{\mathrm{BLUP}}_f}}^2R_{{\bf{b}}_2,\widehat {\bf{b}}_{{\mathrm{BLUP}}_2}}^2} \right)} \hfill \end{array}$$\end{document}wf=1-rG2R22h22∕1-rG2Rf2R22hf2h22M=1-rG2Rb2,b^BLUP22∕1-rG2Rbf,b^BLUPf2Rb2,b^BLUP22,andw2=rG∕hfh2hf2-R12∕1-rG2Rf2R22hf2h22M=rGhf∕h21-Rbf,b^BLUPf2∕1-rG2Rbf,b^BLUPf2Rb2,b^BLUP22