# 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

${\displaystyle {\begin{pmatrix}\mathbf {X} '\mathbf {X} &\mathbf {X} '\mathbf {Z} \\\mathbf {Z} '\mathbf {X} &\mathbf {Z} '\mathbf {Z} +\mathbf {K} ^{-1}\lambda \end{pmatrix}}{\begin{pmatrix}{\hat {\boldsymbol {\beta }}}\\{\hat {\mathbf {u} }}\end{pmatrix}}={\begin{pmatrix}\mathbf {X} '\mathbf {y} \\\mathbf {Z} '\mathbf {y} \end{pmatrix}}}$

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

In a multiple-trait genetic evaluation we have covariances in 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

${\displaystyle {\begin{pmatrix}\mathbf {X} '\mathbf {R} ^{-1}\mathbf {X} &\mathbf {X} '\mathbf {R} ^{-1}\mathbf {Z} \\\mathbf {Z} '\mathbf {R} ^{-1}\mathbf {X} &\mathbf {Z} '\mathbf {R} ^{-1}\mathbf {Z} +\mathbf {G} ^{-1}\end{pmatrix}}{\begin{pmatrix}{\hat {\boldsymbol {\beta }}}\\{\hat {\mathbf {u} }}\end{pmatrix}}={\begin{pmatrix}\mathbf {X} '\mathbf {R} ^{-1}\mathbf {y} \\\mathbf {Z} '\mathbf {R} ^{-1}\mathbf {y} \end{pmatrix}}}$

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