Estimation of (co)variance function coefficients for test day yield with a expectation-maximization restricted maximum likelihood algorithm

Coefficients for (co)variance functions were obtained via random regression models using the expectation-maximization REML algorithm. Data included milk, fat, and protein yields from 176,495 test days of 22,943 first lactation Holstein cows that calved in Pennsylvania and Wisconsin from 1990 through 1996. Three approximately equal-sized data sets were created: one for Pennsylvania and two for Wisconsin. Random regressions were on third order Legendre polynomials. Genetic and permanent environmental (co)variances each were described by three coefficients. The model contained a fixed effect for age, season, and lactation stage rather than a fixed regression on days in milk. Fixed contemporary groups were based on herd, test day, and milking frequency. The coefficient matrices were dense and included around 70,000 equations. Estimated (co)variance function coefficients, as well as the heritabilities and correlations computed from them, were quite variable across data sets. Heritabilities were at a minimum (0.14 for milk and fat and 0.13 for protein) around peak yield, increased to a maximum (0.24 for milk and protein and 0.21 for fat) around the eighth month in milk, and declined slightly afterwards. Genetic correlations between early and late lactation were low (values of <0.10), especially for protein. Phenotypic correlations among test day yields were between 0.21 and 0.99.