Estimating the covariance structure of traits during growth and ageing, illustrated with lactation in dairy cattle.

Quantitative variation in traits that change with age is important to both evolutionary biologists and breeders. We present three new methods for estimating the phenotypic and additive genetic covariance functions of a trait that changes with age, and illustrate them using data on daily lactation records from British Holstein-Friesian dairy cattle. First, a new technique is developed to fit a continuous covariance function to a covariance matrix. Secondly, this technique is used to estimate and correct for a bias that inflates estimates of phenotypic variances. Thirdly, we offer a numerical method for estimating the eigenvalues and eigenfunctions of covariance functions. Although the algorithms are moderately complex, they have been implemented in a software package that is made freely available. Analysis of lactation shows the advantages of the new methods over earlier ones. Results suggest that phenotypic variances are inflated by as much as 39% above the underlying covariance structure by measurement error and short term environmental effects. Analysis of additive genetic variation indicates that about 90% of the additive genetic variation for lactation during the first 10 months is associated with an eigen-function that corresponds to increased (or decreased) production at all ages. Genetic tradeoffs between early and late milk yield are seen in the second eigen-function, but it accounts for less than 8% of the additive variance. This illustrates that selection is expected to increase production throughout lactation.