The prediction accuracy of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}$$\end{document}b^wMT-BLUP effects obtained from Eq. (15) can be derived by considering the correlation 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}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_k}$$\end{document}b^wMT-BLUPk as:29\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}} = \frac{{{\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}} \right)}}{{\sqrt {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}} \right){\mathrm{var}}\left( {{\bf{b}}_f} \right)} }}$$\end{document}rbf,b^wMT-BLUPf=covbf,b^wMT-BLUPfvarb^wMT-BLUPfvarbfEquation (13) gives \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f} = {{{\bf w}{\prime}}}\widehat {\bf{b}}_{{\mathrm{BLUP}}}$$\end{document}b^wMT-BLUPf=w′b^BLUP and thus 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}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}$$\end{document}b^wMT-BLUPf is:30\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}} \right) = {\mathrm{cov}}\left( {{\bf{b}}_f,{{{\bf w}{\prime}}}\widehat {\bf{b}}_{{\mathrm{BLUP}}}} \right) = {{{\bf w}{\prime}}}{\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}}} \right) = {{{\bf w}{\prime}\bf C}}$$\end{document}covbf,b^wMT-BLUPf=covbf,w′b^BLUP=w′covbf,b^BLUP=w′CThe variance of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}$$\end{document}b^wMT-BLUP effects obtained from Eq. (15) is:31\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right) = {\mathrm{var}}\left( {{{{\bf w}{\prime}}}\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right) = {{{\bf w}{\prime}}}{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right){\bf{w}} = {{{\bf w}{\prime}\bf Vw}}$$\end{document}varb^wMT-BLUP=varw′b^BLUPk=w′varb^BLUPkw=w′V wAdditionally, w = V−1C and Vw = C, and thus w′C = w′Vw or written another way \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}} \right) = {\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right)$$\end{document}covbf,b^wMT-BLUPf=varb^wMT-BLUP following BLUP properties. Substituting into Eq. (19), the correlation 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}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_k}$$\end{document}b^wMT-BLUPk can then be written as:32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {r_{{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}}} \hfill & = \hfill & {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right){\mathrm{/}}\sqrt {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right){\mathrm{var}}\left( {{\bf{b}}_f} \right)} } \hfill \\ {} \hfill & = \hfill & {\sqrt {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right){\mathrm{/}}{\mathrm{var}}\left( {{\bf{b}}_f} \right)} } \hfill \end{array},$$\end{document}rbf,b^wMT-BLUPf=varb^wMT-BLUP∕varb^wMT-BLUPvarbf=varb^wMT-BLUP∕varbf,which gives the squared correlation as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}}^2$$\end{document}Rbf,b^wMT-BLUPf2 = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right){\mathrm{/}}{\mathrm{var}}\left( {{\bf{b}}_f} \right)$$\end{document}varb^wMT-BLUP∕varbf = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{R_f^2}}{M}{\mathrm{/}}\frac{{h_k^2}}{M} = R_f^2{\mathrm{/}}h_k^2$$\end{document}Rf2M∕hk2M=Rf2∕hk2. Therefore, the squared correlation between a phenotype and a multiple trait index weighted BLUP predictor of the phenotype is approximately:33\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{y}}_k,\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_k}}^2 = M{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right) = M{{{\bf w}{\prime}{\bf Vw}}}$$\end{document}Ryk,g^wMT-BLUPk2=Mvarb^wMT-BLUP=Mw′V wIf we consider a two-trait example then prediction accuracy for a focal trait \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_k}}^2$$\end{document}Ryf,g^wMT-BLUPk2 can be written as:34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}}^2 = {\bf{w}}_f^2R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{BLUP}}_f}}^2 + {\bf{w}}_2^2R_{{\bf{y}}_2,\widehat {\bf{g}}_{{\mathrm{BLUP}}_2}}^2 + 2{\bf w}_f{\bf w}_2{\bf V}_{1,2}$$\end{document}Ryf,g^wMT-BLUPf2=wf2Ryf,g^BLUPf2+w22Ry2,g^BLUP22+2wfw2V 1,2where V1,2 is the off-diagonal element of the matrix V of Eqs. (15) and (21). The value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}}^2$$\end{document}Ryf,g^wMT-BLUPf2 can then be compared to the prediction accuracy of the single-trait BLUP predictor of Eq. (16) and to the prediction accuracy of a cross-trait predictor43, where a BLUP predictor of the second trait is used to predict the focal trait phenotype, which is given by: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{BLUP}}_2}}^2 = R_{{\bf{y}}_2,\widehat {\bf{g}}_{{\mathrm{BLUP}}_2}}^2r_{\mathrm{G}}\sqrt {\left( {h_2{\mathrm{/}}h_f} \right)}$$\end{document}Ryf,g^BLUP22=Ry2,g^BLUP22rGh2∕hf. This comparison is of interest, because we expect the multi-trait predictor to be more accurate than any available single-trait predictor, even if the most accurate single-trait predictor is across two different traits. Cross-trait prediction is equivalent to the proxy-phenotype method, which has been used to predict cognitive performance from educational attainment GWAS data44.