Consider two genetically correlated traits for which we have individual-level genetic predictors with BLUP properties. For each individual, i, and focal trait of interest, f, we have a genetic prediction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{BLUP}}_{i,k}}} \right)$$\end{document}g^BLUPi,k for each trait, k, that we can combine together using the index weights, wi,k, for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{g}}_{{\mathrm{BLUP}}_{i,k}}$$\end{document}g^BLUPi,k effect to produce a weighted multi-trait BLUP genetic predictor:1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_{i,f}} = \mathop {\sum }\limits_k w_{i,k}{\hat{\bf g}}_{{\mathrm{BLUP}}_{i,k}} = {\bf{w}}_i\prime \widehat {\bf{g}}_{{\mathrm{SBLUP}}_i}$$\end{document}g^wMT-BLUPi,f= ∑kwi,kg^BLUPi,k=wi′g^SBLUPiIn the Methods section, we show that the optimal index weights can be calculated as:2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${w} = \left[ {\begin{array}{*{20}{c}} {w_1} \\ {w_2} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {R_1^2} & {\frac{{r_{\mathrm{G}}R_1^2R_2^2}}{{\sqrt {h_1^2h_2^2} }}} \\ {\frac{{r_{\mathrm{G}}R_1^2R_2^2}}{{\sqrt {h_1^2h_2^2} }}} & {R_2^2} \end{array}} \right]^{ - 1}\left[ {\begin{array}{*{20}{c}} {R_1^2} \\ {r_{\mathrm{G}}\sqrt {\frac{{h_1^2}}{{h_2^2}}} R_2^2} \end{array}} \right]$$\end{document}w=w1w2=R12rGR12R22h12h22rGR12R22h12h22R22-1R12rGh12h22R22where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_k^2$$\end{document}hk2 is the SNP heritability of trait k (proportion of phenotypic variance explained by genome-wide SNPs), rG is the genetic correlation between trait k and the focal trait and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_k^2$$\end{document}Rk2 is the expected squared correlation between a phenotype and a BLUP predictor, calculated as:3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_k^2 = \frac{{h_k^2}}{{1 + M_{{\mathrm{eff}}}\frac{{1 - R_k^2}}{{N_kh_k^2}}}}$$\end{document}Rk2=hk21+Meff1-Rk2Nkhk2where Meff is the effective number of chromosome segments and Nk is the sample size of trait k. These weights will ensure that the contribution of each added trait is approximately proportional to the square root of its sample size, its SNP heritability and its genetic correlation with the focal trait (trait 1), while accounting for different variances of single-trait BLUP predictors.