Page 147 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

covariance functions (CFs) to handle the analysis of longitudinal data, illustrating
their methodology with growth data. The application of RR models in animal breed-
ing for the analysis of various types of data has been comprehensively reviewed by
Schaeffer (2004). Prior to the development of the RR model for genetic evaluation,
milk yield test day records were analysed by Ptak and Schaefer (1993) using a fixed
regression model. The details of this model are discussed and illustrated in the next
section, followed by its extension to an RR model. This is then followed by a brief
presentation of CF, and the equivalence of the RR model and CF is demonstrated.

9.2 Fixed Regression Model

The theoretical framework for the fixed regression model and its application for the
analysis of longitudinal data such as test day milk production traits were presented by
Ptak and Schaefer in 1993. On a national scale, a fixed regression model was imple-
mented for the genetic evaluation of test day records of milk production traits and
somatic cell counts in Germany from 1995 until 2002. The model involved the use of
individual test day records, thereby avoiding the problem of explicitly extending test
day yields into 305-day yield, and accounted for the effects peculiar to all cows on the
same test day within herds (herd–test–day (HTD) effect). Therefore, corrections for
temporary environmental effects on the day of test are more precise compared to evalu-
ations based on 305-day yields. The model also accounted for the general shape of the
lactation curve of groups of similar age, and calving in the same season and region. The
latter was accomplished by regressing lactation curve parameters on days in milk (hence
the name of the model) within the groupings for cows. Inclusion of the curve therefore
allows for correction of the means of test day yields at different stages of lactation.
Fitting residual variances relevant to the appropriate stage of lactation could also
account for the variation of test day yields with days in milk. The only major disadvan-
tage is that the volume of data to be analysed is much larger, especially in the dairy situ-
ation, as ten or more test day observations are stored relative to a single 305-day yield.
Similar to the repeatability model, at the genetic level, the fixed regression model
assumes that test day records within a lactation are repeated measurements of the
same trait, i.e. a genetic correlation of unity among test day observations. Usually, the
permanent environmental effect is included in the model to account for environmen-
tal factors with permanent effects on all test day yields within lactation.
The fixed regression model is of the form:
nf
i å
y = htd + + u + pe + e
tij fb j j tij
tjk k
k=0
where y is the test day record of cow j made on day t within HTD subclass i; b are
tij k
fixed regression coefficients; u and pe are vectors of animal additive genetic and
j j
permanent environmental effects, respectively, for animal j; f is the vector of the kth
tjk
Legendre polynomials or any other curve parameter, for the test day record of cow j
made on day t; nf is the order of fit for Legendre polynomials used to model the fixed
regressions (fixed lactation curves) and e is the random residual. In matrix notation,
tij
the model may be written as:
y = Xb + Qu + Zpe + e (9.1)

Analysis of Longitudinal Data 131