# 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[1][2]. 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.

## References

1. Henderson, C. R., and R. L. Quaas. 1976. Multiple trait evaluation using relatives' records, J. Anim. Sci. 43:1188–1197.
2. Mrode, R. A. 2005. Best linear unbiased prediction of breeding value: multivariate models. In: Linear models for the prediction of animal breeding values 2nd ed. CAB Int., Wallingford, Oxfordshire, UK. p. 83-119.