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

Abstract

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.

Modeling lactation curves of Polish Holstein-Friesian cows. Part II: Prediction of 305-d lactation milk, fat and protein yields

“…This parameter is still useful for producers making their management and breeding decisions (Druet et al, 2003). In the last two decades, genetic evaluations of dairy bulls and cows based on test-day (TD) yields employing TD models have been widely studied (Strabel et al, 2005;Bohmanova et al, 2008). Many different mathematical functions have been applied to model lactation curves.…”

“…With time, these functions have been replaced by orthogonal polynomials that model the lactation curve properly and tend to converge faster because of a lack of correlation among the coefficients. Recently, splines have been proposed as a good alternative to Legendre polynomials (Bohmanova et al, 2008). A variety of different mathematical models have also been examined for modeling lactation curves of the Polish population of cows Frącz, 2002, 2003;Frącz and Ptak, 2003;Ptak et al, 2004;Strabel et al, 2005;Grzesiak et al, 2006;Otwinowska-Mindur et al, 2013).…”

Genotype by production environment interaction for birth and weaning weights in a population of composite beef cattle

“…To evaluate the predictive ability of these models, two additional data sets were considered for each trait as suggested by Bohmanova et al (2008). The first data set included all weight records available until 2004 (DATA04), and the second included all records available until 2008 (DATA08).…”

“…The differences in average of EBV between the two genetic evaluations were accounted by subtracting the average change in EBV from GE04 to GE08 for a set of bulls whose average EBV was not expected to change. As suggested by Bohmanova et al (2008), the bulls chosen for the adjustment were required to have at least 25 progeny records in GE04, no new progeny record, and no more than 10 new grandsons between DATA04 and DATA08. The ERP was also computed for maternal breeding value.…”

“…Percentage of squared bias (PSB) was computed similarly to Ali and Schaeffer (1987) and Bohmanova et al (2008):…”