In animal and plant breeding, selection indices have been developed, which combine many single-trait BLUP predictors of an individual’s genetic value together in an index weighting to optimise the selection of individuals with the most favourable multi-trait phenotype for breeding programs33–36. Utilising a selection index approach, the solution for wSBLUP of Eq. (13) can be obtained as:14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{w}}_{{\mathrm{SBLUP}}}={\bf{V}}_{{\mathrm{SBLUP}}}^{-1}{\bf{C}}_{{\mathrm{SBLUP}}}$$\end{document}wSBLUP=V SBLUP-1CSBLUPwhere CSBLUP a k × 1 column vector of the covariance of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}$$\end{document}b^SBLUPk values of the k traits, with the true genetic effects of the SNPs for the focal trait, and VSBLUP 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}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP effects:15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {{\bf{w}}_{{\mathrm{SBLUP}}}} \hfill & = \hfill & {{\bf{V}}_{{\mathrm{SBLUP}}}^{ - 1}{\bf{C}}_{{\mathrm{SBLUP}}}} \hfill \\ {} \hfill & = \hfill & {\left[ {\begin{array}{*{20}{c}} {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_1}} \right)} & \cdots & {{\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_1},\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)} \\ \vdots & \ddots & \vdots \\ {{\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k},\widehat {\bf{b}}_{{\mathrm{SBLUP}}_1}} \right)} & \cdots & {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)} \end{array}} \right]^{ - 1}} \hfill \\ {} \hfill & {} \hfill & {\left[ {\begin{array}{*{20}{c}} {{\mathrm{cov}}\left( {{\bf{b}}_{\it{f}},\widehat {\bf{b}}_{{\mathrm{SBLUP}}_1}} \right)} \\ \vdots \\ {cov\left( {{\bf{b}}_{\mathit{f}},\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)} \end{array}} \right]} \hfill \end{array}$$\end{document}wSBLUP=V SBLUP-1CSBLUP=varb^SBLUP1⋯covb^SBLUP1,b^SBLUPk⋮⋱⋮covb ^SBLUPk,b^SBLUP1⋯varb^SBLUPk-1covbf,b^SBLUP1⋮covbf ,b^SBLUPk