# Heterogeneous variance

Heterogenous variance[1] between groups of animals within a trait in a single genetic evaluation can exist. Often the heterogeneous variance between groups results from differences in trait expression by, for example, breed and/or sex. This difference in variance between groups of animals within a genetic evaluation may simply be the result of scaling effects. For example, bulls may express more variation in yearling weight than heifers simply because they are, on average, larger. As animals become larger the variation tends to increase.

Another situation where variance may be heterogenous is when different procedures are used to measure or score a trait between groups of cattle. For example, it is apparent that using a hoof tape measure is highly correlated to birth weight and is frequently used, especially by larger cattle breeding enterprises. However, it has been observed that the variance in the birth weight values predicted by the hoof tape measures is typically about 75% of that of real birth weights. Additionally, reporting weights to the nearest 5 or 10 pounds can result in heterogeneous variance.

In a model where no heterogeneous variance is considered the observation on individual i, $y_{i}$, for some trait may be modelled as,

$y_{ij}=b_{j}+u_{i}+e_{i}$

where $b_{j}$ is some jth fixed effect (e.g., contemporary group) on the observation, $u_{i}$ is the breeding value of the ith animal for the trait, and $e_{i}$ is the random residual (error) on the observation with a distribution of $N(0,I\sigma_{e}^{2})$. Notice that for all observations, the residual variance is assumed to be the same. When considering heterogeneous variance[2], an additive model may be expressed as,

$y_{ijk}=b_{j}+\epsilon_{k}u_{i}+e_{i}^k$

where $\epsilon_{k}$ is a coefficient that scales the true breeding value's expression, and $e_{i}^k$ is the residual effect with a residual variance from the kth residual variance group. This group may be a sex class, a breed class or other category (e.g., hoof tape birth weight observation). The residual variance does not need to be a class and could be scaled according to some continuous criteria.

The $\epsilon_{k}$ and $\sigma_{e_{k}}^{2}$ do not necessarily have to scale together to keep the heritability of the trait the same between heterogeneous variance groups. In some situations, this may be desirable and in others, there may be a difference in the observed (effective) heritability of the trait between groups.

An alternative to an additive model is a multiplicative approach. However, multiplicative modeling can have an order of magnitude more computing effort in performing a genetic evaluation.[3][4] An alternative is to adjust the observations to a standardized value.[5]

## References

1. Garrick, D. J., E. J. Pollak, R. L. Quaas and L. D. Van Vleck. 1989 Variance Heterogeneity in Direct and Maternal Weight Traits by Sex and Percent Purebred for Simmental-Sired Calves. v67:10 2515-2528.
2. Quaas, R. L., D. J. Garrick, and W. H. McElhenny. 1989. Multiple Trait Prediction for a Type of Model with Heterogeneous Genetic and Residual Covariance Structures. J. Anim. Sci. v67:10 2529-2535.
3. Lidauer, M. H., R. Emmerling and E.A. Mäntysaari. 2008. Multiplicative random regression model for heterogeneous variance adjustment in genetic evaluation for milk yield in Simmental. J. Anim. Breeding and Genetics. 125:147-159.
4. Márkus, S., Mäntysaari, E., Strandén, I., Eriksson, J. and Lidauer, M. 2014. Comparison of multiplicative heterogeneous variance adjustment models for genetic evaluations. Journal of animal breeding and genetics = Zeitschrift fur Tierzuchtung und Zuchtungsbiologie. 131. 10.1111/jbg.12082.
5. Macneil, M., Lopez-Villalobos, N. and Northcutt, S. 2011. A prototype national cattle evaluation for feed intake and efficiency of Angus cattle. Journal of animal science. 89. 3917-23. 10.2527/jas.2011-4124.