Use of linear mixed models for genetic evaluation of gestation length and birth weight allowing for heavy-tailed residual effects

Abstract

Background: The distribution of residual effects in linear mixed models in animal breeding applications is typically assumed normal, which makes inferences vulnerable to outlier observations. In order to mute the impact of outliers, one option is to fit models with residuals having a heavy-tailed distribution. Here, a Student's-t model was considered for the distribution of the residuals with the degrees of freedom treated as unknown. Bayesian inference was used to investigate a bivariate Student's-t (BSt) model using Markov chain Monte Carlo methods in a simulation study and analysing field data for gestation length and birth weight permitted to study the practical implications of fitting heavy-tailed distributions for residuals in linear mixed models.

Methods: In the simulation study, bivariate residuals were generated using Student's-t distribution with 4 or 12 degrees of freedom, or a normal distribution. Sire models with bivariate Student's-t or normal residuals were fitted to each simulated dataset using a hierarchical Bayesian approach. For the field data, consisting of gestation length and birth weight records on 7,883 Italian Piemontese cattle, a sire-maternal grandsire model including fixed effects of sex-age of dam and uncorrelated random herd-year-season effects were fitted using a hierarchical Bayesian approach. Residuals were defined to follow bivariate normal or Student's-t distributions with unknown degrees of freedom.

Results: Posterior mean estimates of degrees of freedom parameters seemed to be accurate and unbiased in the simulation study. Estimates of sire and herd variances were similar, if not identical, across fitted models. In the field data, there was strong support based on predictive log-likelihood values for the Student's-t error model. Most of the posterior density for degrees of freedom was below 4. Posterior means of direct and maternal heritabilities for birth weight were smaller in the Student's-t model than those in the normal model. Re-rankings of sires were observed between heavy-tailed and normal models.

Conclusions: Reliable estimates of degrees of freedom were obtained in all simulated heavy-tailed and normal datasets. The predictive log-likelihood was able to distinguish the correct model among the models fitted to heavy-tailed datasets. There was no disadvantage of fitting a heavy-tailed model when the true model was normal. Predictive log-likelihood values indicated that heavy-tailed models with low degrees of freedom values fitted gestation length and birth weight data better than a model with normally distributed residuals.Heavy-tailed and normal models resulted in different estimates of direct and maternal heritabilities, and different sire rankings. Heavy-tailed models may be more appropriate for reliable estimation of genetic parameters from field data.

Figure 1

Posterior densities of degrees of freedom obtained from bivariate Student's-t (BSt) model fitted to gestation length (GL) and birthweight (BW). M represents posterior mean, L represents the 2.5^th percentiles of the posterior density, U represent 97.5^th percentiles of the posterior density.

Figure 2

Posterior densities of direct (D) and maternal (M) heritabilities of gestation length (GL) and birth weight (BW) obtained from bivariate Student's-t (BSt) or normal (BN) models. h²D and h²M represent direct and maternal heritabilities.

Figure 3

Posterior densities of genetic correlations between direct (D) and maternal (M) effects for gestation length (GL) and birth weight (BW) obtained from bivariate Student's-t (BSt) or normal (BN) models.

Figure 4

Distribution of outlier posterior mean values of scale λ_i (for each animal) from a Student's-t model of residuals plotted against the corresponding estimated residuals for gestation length (GL) and birth weight (BW). Distribution of posterior mean values of λ_i less than 0.3 on the left. Distribution of posterior mean values of λ_i less than 0.2 on the right.

Figure 5

Scatter plots of posterior means of all and top 100 sire effects for gestation length (GL) and birth weight (BW) in Italian Piemontese cattle, obtained by bivariate Student's-t (BSt) or normal (BN) models.