# 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

where and are the incidence matrices for the fixed and random effects, is a relationship matrix, are estimated fixed effects, the predicted breeding values, the vector of observed phenotypes, and is the ratio of the environmental variance and the additive genetic variance .

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

where is the genetic covariance matrix and is a function of a relationship matrix and genetic variances of the two traits and the genetic covariance between the two traits, and 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

- ↑ Henderson, C. R., and R. L. Quaas. 1976. Multiple trait evaluation using relatives' records, J. Anim. Sci. 43:1188–1197.
- ↑ 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.