Page 153 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

associated with different stages of lactation could be fitted with the fixed regression
model, the model did not account for the covariance structure at the genetic level.
Schaeffer and Dekkers (1994) extended the fixed regression model for genetic
evaluation by considering the regression coefficients on the same covariables as
random, therefore allowing for between-animal variation in the shape of the curve.
Thus the genetic differences among animals could be modelled as deviations from
the fixed lactation curves by means of random parametric curves (see Guo and
Swalve, 1997) or orthogonal polynomials such as Legendre polynomials
(Brotherstone et al., 2000), or even non-parametric curves such as natural cubic
splines (White et al., 1999). Most studies have used Legendre polynomials as they
make no assumption about the shape of the curve and are easy to apply. The RR
model has also been employed for the analysis of growth data in pigs (Andersen and
Pedersen, 1996) and beef cattle (Meyer, 1999). An additional benefit of the RR
model in dairy cattle is that it provides the possibility of genetic evaluation for
persistence of the lactation. A typical random regression model (RRM) especially
for the analysis of dairy cattle test day records is of the form:

nf nr nr
i ∑
+
+
y = htd + f b k ∑ f u jk ∑ f pe + e
tijk jtk jtk jtk jk tijk
k=0 k=0 k=0
where y is the test day record of cow j made on day t within HTD subclass i; b are
tijk k
fixed regression coefficients; u and pe are vectors of the kth random regression for
jk jk
animal and permanent environmental effects, respectively, for animal j; f is the vec-
jtk
tor of the kth Legendre polynomials for the test day record of cow j made on day t;
nf is the order of polynomials fitted as fixed regressions; nr is the order of polynomi-
als for animal and pe effects; and e is the random residual. The model in matrix
tljk
notation is:
y = Xb + Qu + Zpe + e
The vectors y, b and the matrix X are as described in Example 9.1. However, u
and pe are now vectors of random regressions for animal additive genetic and pe
effects. The matrices Q and Z are covariable matrices and, if only animals with
records are considered, the ith row of these matrices contains the orthogonal
polynomials (covariables) corresponding to the DIM of the ith TD yield. If the
order of fit is the same for animal and pe effects, Q = Z, considering only animals
with records. This would not be the case if the order of fit is different for animal
and pe effects. In general, considering animals with records, the order of either
Q or Z is n (number of TD records) by nk, where nk equals nr times the number
td
∗
∗
of animals with records. It is assumed that var(u) = A G, var(pe) = I P and var(e) =
∗
2
Is = R, where A is the numerator relationship matrix, is the Kronecker product
e
and G and P are of the order of polynomial fitted for animal and pe effects. The
MME are:
XR y⎞
⎛ XR X X R Q XR Z⎞ ⎛ ˆ b ⎞ ⎛ ′ − 1
′
′
′
−1
−1
−1
⎜ −1 −1 −1 −1 ⎟ ⎜ ⎟ ⎜ − 1 ⎟ ⎟
′
′
′
′
⎜ QR X Q R Q + A ⊗ G QR Z ⎟ ⎜ ˆ u ⎟ = ⎜ QR y ⎟
⎜ −1 −1 −1 1 ⎟ ⎜ ˆ p ⎟ e ⎜ − 1 ⎟
′
′
′
⎝ ZR X Z R Q ZR ZP+ ⎠ ⎝ ⎠ ⎝ ′ ⎠
ZR y
Analysis of Longitudinal Data 137