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Random Regression Test-Day Models with Residuals Following a Student’s-t Distribution

¹ Centre for Genetic Improvement of Livestock, Department of Animal and Poultry Science, University of Guelph, Guelph, ON, N1G 2W1, Canada
² MTT Agrifood Research Finland, Animal Production Research FIN-31600 Jokioinen, Finland

Corresponding author: J. Jamrozik; e-mail: jamrozik{at}sherlock.aps.uoguelph.ca.

First-lactation milk yield test-day records of Canadian Holsteins were analyzed by single-trait random regression test-day models that assumed normal or Student’s-t distribution for residuals. Objectives were to test the performance of the robust statistical models that use heavy-tailed distributions for the residual effect. Models fitted were: Gaussian, Student’s-t, and Student’s-t with fixed number of degrees of freedom (equal to 5, 15, 30, 100 or 1000) for the t distribution. Bayesian methods with Gibbs sampling were used to make inferences about overall model plausibility through Bayes factors, posterior means for covariance components, estimated breeding values for regression coefficients, solutions for permanent environmental regressions, and residuals of the models. Bayes factors favored Student’s-t model with the posterior mean of degrees of freedom equal to 2.4 over all other models, indicating very strong departure from normality. Number of outliers in Student’s-t model was reduced by 35% in comparison with the Gaussian model. Differences in covariance components for regression coefficients between models were small, and rankings of animals based on additive genetic merit for the first two regression coefficients (total yield and persistency) were similar. Results from the Gaussian and Student’s-t models with fixed degrees of freedom become more alike (smaller departures from normality for Student’s-t models) with increasing number of degrees of freedom for the t-distributions. For any pair of Student’s-t models, the one with the smaller number of degrees of freedom for the t-distribution was shown to be superior. Similarly, number of outliers increased with increasing degrees of freedom for the t distribution.

Key Words: random regression model • test-day data • Student’s-t distribution

Abbreviation key: ASD = age at calving by season of calving by DIM interval, BF = Bayes factor, GS = Gibbs sampling, PE = permanent environment, RR = random regression, TD = test day