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Use of Linear Models with Normal, Student-t or Slash Distributed Error for the Analysis of Quantitative Traits
Some symmetric and heavy-tailed distributions, such as Student’s-t and Slash, have been suggested for robust inference in linear mixed models. These robust models are characterized by the degrees of freedom of these distributions, and include the normal distribution when the degrees of freedom approach infinity. The objective of this this studywas to investigate joint estimation of degrees of freedom for the residual and all other genetic and non-genetic parameters. In a simulation study, five different populations with five replicates were simulated using multivariate linear mixed effects animal models withNormal (NOR), three (ST3) or ten (ST10) degrees of freedom Student-t, and one and half (ST1.5) or three (SL3) degrees of freedom Slash distributions. Multivariate data within each replicate were generated for 18000 progeny from 10 sires and 20 dams mating which is selectedthrough three generations.Models with multivariate Student’s-t, Slash and Normal residuals were fitted to each dataset using a hierarchical Bayesian approach. Predictive log-likelihood(PLL) values strongly favoured the multivariate Student’s-tand Slash models over the Normal models for simulated heavy-tailed datasets. Posterior mean estimates of degrees of freedom parameters seemed to be accurate and unbiased. Estimates of sire and herd variances were similar, if not identical, across fitted models. Posterior mean and 95% posterior probability interval (PPI) estimates of error variances in simulated datasets were found to be (downwardly or upwardly) biased when the fitted model was not the true model. Reliable estimates of degrees of freedom were obtained in all simulated heavy-tailed and normal datasets. The predictive log-likelihood was able to identify the correct model among the models fitted to heavy-tailed datasets. The results obtained indicated that there was no disadvantage of fitting a heavy-tailed model when the true model was normal.
Keywords: Multivariateheavy-tailed distributions, robust linear mixed model, MCMC