Equations (16), (17), (18), (19), (20), (21), (22), (23), (24), (25), (26), (27), (28), (29), (30), (31), (32), (33) and (34) assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)$$\end{document}covbk,b^SBLUPk = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right) = {\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right)$$\end{document}varb^SBLUPk=varb^BLUPk, or in other words that SBLUP SNP solutions have BLUP properties. The use of an independent LD reference sample to create an approximate single-trait BLUP predictor in Eq. (11) does not affect the covariance between the true SNP effect sizes and the approximate BLUP SNP solution, meaning that the approximate single-trait BLUP predictors have BLUP properties. However, the variance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP is likely affected, which may potentially result in a loss of prediction accuracy of a weighted multi-trait BLUP predictor. The variance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP is:35\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {\sigma _{\widehat {\bf{b}}_{{\mathrm{SBLUP}}}}^2} \hfill & = \hfill & {\left[ {\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}{{{\bf W}{\prime}}}} \right]\left[ {{{{\bf W}{\prime}{\bf W}}}\sigma _{\mathit{b}}^2 + {\bf{I}}\sigma _{\mathit{e}}^2} \right]} \hfill \\ {} \hfill & {} \hfill & {\left[ {{\bf{W}}\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}} \right]} \hfill \\ {} \hfill & = \hfill & {\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}\left[ {\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\sigma _{\mathit{b}}^2} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { + {{{\bf W}{\prime}{\bf W}}}\sigma _{\mathit{e}}^2} \right]\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}} \hfill \\ {} \hfill & = \hfill & {\left[ {\left( {\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}} \right.} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. {\left. { + \left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}{{{\bf W}{\prime}{\bf W}}}\lambda _k} \right)\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}} \right]\sigma _{\mathit{b}}^2} \hfill \end{array}$$\end{document}σb^SBLUP2=NL+IMλ-1W′W′Wσb2+Iσe2WNL+IMλ-1=NL+IMλ-1W′WW′Wσb2+W′Wσe2NL+IMλ-1=NL+IMλ-1W′WW′WNL+IMλ-1+NL+IMλ-1W′WλkNL+IMλ-1σb2The loss of information from using an independent data set as an LD reference to obtain L, rather than directly using the individual-level data to calculate W′W, can be approximated by considering the scenario where SNP makers are unlinked, resulting in diag[L]. The diagonal elements of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\widehat {\bf{b}}_{{\mathrm{SBLUP}}_{jj}}}^2$$\end{document}σb^SBLUPjj2 for SNP j are then:36\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\widehat {\bf{b}}_{{\mathrm{SBLUP}}_{jj}}}^2 = \left( {\left[ {N + \lambda } \right]^{ - 2}{\mathrm{diag}}\left[ {\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\left( {{{{\bf W}{\prime}{\bf W}}}} \right)} \right] + N\lambda \left[ {N + \lambda } \right]^{ - 2}} \right)\sigma _{\mathit{b}}^2$$\end{document}σb^SBLUPjj2=N+λ-2diagW′WW′W+NλN+λ-2σb2The diagonal elements of diag[(W′W)(W′W)] can be approximated as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{diag}}\left[ {\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\left( {{{{\bf W}{\prime}{\bf W}}}} \right)} \right]$$\end{document}diagW′WW′W ≈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^2\left( {1 + {\Bbb E}\left[ {r^2} \right]M} \right)$$\end{document}N21+Er2M + \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^2\left( {1 + M{\mathrm{/}}N} \right)$$\end{document}N21+M∕N, where the expectation of the LD correlation of the SNPs, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Bbb E}\left[ {r^2} \right]$$\end{document}Er2, is 1/N as the SNP markers are unlinked. Equation (36) can then be written as:37\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {\sigma _{\widehat {\bf{b}}_{{\mathrm{SBLUP}}_{jj}}}^2} \hfill & = \hfill & {\left( {\left( {N^2 + NM + N\lambda } \right){\mathrm{/}}\left( {N + \lambda } \right)^2} \right)\sigma _{\mathit{b}}^2} \hfill \\ {} \hfill & = \hfill & {\sigma _{\mathit{b}}^2N{\mathrm{/}}(N + \lambda ) + \sigma _{\mathit{b}}^2NM{\mathrm{/}}(N + \lambda )^2} \hfill \end{array}$$\end{document}σb^SBLUPjj2=N2+NM+Nλ∕N+λ2σb2=σb2N∕(N+λ)+σb2NM∕(N+λ)2From Eq. (37), the squared correlation between true SNP effects and SBLUP SNP effects can be written as:38\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{b}},\widehat {\bf{b}}_{{\mathrm{SBLUP}}}}^2 = N{\mathrm{/}}\left( {N + M + \lambda } \right) = N{\mathrm{/}}\left( {N + M{\mathrm{/}}h^2} \right)$$\end{document}Rb,b^SBLUP2=N∕N+M+λ=N∕N+M∕h2This can be contrasted to Eq. (17), which gives the squared correlation between the true genetic effects of the SNPs, bk, 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 as:39\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2} \hfill & = \hfill & {\frac{{R_k^2}}{M}{\mathrm{/}}\frac{{h_k^2}}{M} = R_k^2{\mathrm{/}}h_k^2} \hfill \\ {} \hfill & = \hfill & {1{\mathrm{/}}\left( {1 + M\left( {1 - R_k^2} \right){\mathrm{/}}\left( {N_kh_k^2} \right)} \right)} \hfill \\ {} \hfill & = \hfill & {N_k{\mathrm{/}}\left( {N_k + M\left( {1 - R_k^2} \right){\mathrm{/}}h_k^2} \right)} \hfill \end{array}$$\end{document}Rbk,b^BLUPk2=Rk2M∕hk2M=Rk2∕hk2=1∕1+M1-Rk2∕Nkhk2=Nk∕Nk+M1-Rk2∕hk2Equation (39) is similar to Eq. (38) apart from the factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 - R_k^2$$\end{document}1-Rk2. Therefore, the relative loss of prediction accuracy from using an SBLUP predictor is given as a ratio of Eqs. (39) and (38) as:40\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{R_{{\bf{b}},\widehat {\bf{b}}_{{\mathrm{SBLUP}}}}^2}}{{R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2}} = \frac{{Nh^2 + M}}{{Nh^2 + M\left( {1 - R_k^2} \right)}}$$\end{document}Rb,b^SBLUP2Rbk,b^BLUPk2=Nh2+MNh2+M1-Rk2