The use of covariance functions and random regressions for genetic evaluation of milk production based on test day records

In the analysis of test day records for dairy cattle, covariance functions allow a continuous change of variances and covariances of test day yields o n different lactation days. The equivalence between covariance functions as an infinite dimensional extension of multivariate models and random regres sion models is shown in this paper. A canonical transformation procedure is proposed for random regression models in large-scale genetic evaluations. Two methods were used to estimate covariance function coefficients for firs t parity test day yields of Holsteins: 1) a two-step procedure fitting cova riance functions to matrices with estimated genetic and residual covariance s between predetermined periods of lactation and 2) REML directly from data with a random regression model. The first method gave more reliable estima tes, particularly for the periphery of the trajectory. The goodness of fit of a random regression model based on covariables describing the shape of t he lactation curve was nearly the same as random regression on Legendre pol ynomials. In the latter model, two and three regression coefficients were s ufficient to fit the covariance structure for additive genetic and permanen t environment, respectively. The eigenfunction pattern revealed the possibi lity of selection for persistency. Covariance functions can be usefully imp lemented in large-scale test day models by means of random regressions.