Heterogeneous variance

Heterogenous variance 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 very highly correlated to birth weight (very near 1) and is frequently used, especially by larger cattle breeding enterprises. However, it has been observed that the variance in the birth weight values forecasted 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, $y_{i}$, for some trait may be modelled as,

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

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 the model becomes,

$y_{i}=b_{j}+\epsilon _{k}u_{i}+e_{ik}$

where $\epsilon _{k}$ is a coefficient that scales the true breeding value's expression and $e_{ik}$ 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 $\epsilon _{k}$ 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.