# Multiple Trait Evaluation

Multiple-trait genetic evaluation differs from a single-trait evaluation in that several phenotypic traits are evaluated at the same time. Reasons for doing a multiple-trait genetic evaluation include greater prediction accuracy and reduced prediction bias.

As with single-trait genetic evaluation most traits use BLUP to obtain EPD by solving the mixed-model equations. For single traits with breeding values as the only random effect the mixed-model equations take the form $\begin{pmatrix} \mathbf{X}'\mathbf{X}&\mathbf{X}'\mathbf{Z}\\ \mathbf{Z}'\mathbf{X}&\mathbf{Z}'\mathbf{Z}+\mathbf{K}^{-1}\lambda \end{pmatrix} \begin{pmatrix} \hat{\boldsymbol{\beta}}\\ \hat{\mathbf{u}} \end{pmatrix}= \begin{pmatrix} \mathbf{X}'\mathbf{y}\\ \mathbf{Z}'\mathbf{y} \end{pmatrix}$

where $\mathbf{X}$ and $\mathbf{Z}$ are the incidence matrices for the fixed and random effects, $\mathbf{K}$ is a relationship matrix, $\hat\boldsymbol{\beta}$ are estimated fixed effects, $\hat\mathbf{u}$ the predicted breeding values, $\mathbf{y}$ the vector of observed phenotypes, and $\lambda=\sigma^2_e/\sigma^2_g$ is the ratio of the environmental variance $\sigma^2_e$ and the additive genetic variance $\sigma^2_g$.

In a multiple-trait genetic evaluation we have covariances in addition to addition to variances associated with the random effects. In addition, different animals could have observed phenotypes for different subsets of traits. In the case where we have two traits and with breeding values as the only random effects, the mixed-model equations take the form $\begin{pmatrix} \mathbf{X}'\mathbf{R}^{-1}\mathbf{X}&\mathbf{X}'\mathbf{R}^{-1}\mathbf{Z}\\ \mathbf{Z}'\mathbf{R}^{-1}\mathbf{X}&\mathbf{Z}'\mathbf{R}^{-1}\mathbf{Z}+\mathbf{G}^{-1} \end{pmatrix} \begin{pmatrix} \hat{\boldsymbol{\beta}}\\ \hat{\mathbf{u}} \end{pmatrix}= \begin{pmatrix} \mathbf{X}'\mathbf{R}^{-1}\mathbf{y}\\ \mathbf{Z}'\mathbf{R}^{-1}\mathbf{y} \end{pmatrix}$

where $\mathbf{G}$ is the genetic covariance matrix and is a function of a relationship matrix $\mathbf{K}$ and genetic variances of the two traits and the genetic covariance between the two traits, and $\mathbf{R}$ is the environmental covariance matrix and is a function of the environmental variances of the two traits and the environmental covariance between the two traits.