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Comparison of Random Regression Models with Legendre Polynomials and Linear Splines for Production Traits and Somatic Cell Score of Canadian Holstein Cows

J. Bohmanova^*,1, F. Miglior^,, J. Jamrozik^*, I. Misztal and P. G. Sullivan

¹ *Centre for Genetic Improvement of Livestock, Department of Animal and Poultry Science, University of Guelph, Guelph, Ontario, N1G 2W1, Canada
Agriculture and Agri-Food Canada, Dairy and Swine Research and Development Centre, Sherbrooke, Quebec, J1M 1Z3, Canada
Canadian Dairy Network, Guelph, Ontario, N1K 1E5, Canada
Department of Animal and Dairy Science, University of Georgia, Athens 30602

¹ Corresponding author: jbohmano{at}uoguelph.ca

A random regression model with both random and fixed regressions fitted by Legendre polynomials of order 4 was compared with 3 alternative models fitting linear splines with 4, 5, or 6 knots. The effects common for all models were a herd-test-date effect, fixed regressions on days in milk (DIM) nested within region-age-season of calving class, and random regressions for additive genetic and permanent environmental effects. Data were test-day milk, fat and protein yields, and SCS recorded from 5 to 365 DIM during the first 3 lactations of Canadian Holstein cows. A random sample of 50 herds consisting of 96,756 test-day records was generated to estimate variance components within a Bayesian framework via Gibbs sampling. Two sets of genetic evaluations were subsequently carried out to investigate performance of the 4 models. Models were compared by graphical inspection of variance functions, goodness of fit, error of prediction of breeding values, and stability of estimated breeding values. Models with splines gave lower estimates of variances at extremes of lactations than the model with Legendre polynomials. Differences among models in goodness of fit measured by percentages of squared bias, correlations between predicted and observed records, and residual variances were small. The deviance information criterion favored the spline model with 6 knots. Smaller error of prediction and higher stability of estimated breeding values were achieved by using spline models with 5 and 6 knots compared with the model with Legendre polynomials. In general, the spline model with 6 knots had the best overall performance based upon the considered model comparison criteria.

Key Words: random regression model • Legendre polynomial • linear spline

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