Page 164 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

DIM
0 1 2 3
F F F F
4 1.000 0.000 0.000 0.000
38 0.333 0.667 0.000 0.000
72 0.667 0.333 0.000 0.000
106 0.000 1.000 0.000 0.000
140 0.000 0.333 0.667 0.000
174 0.000 0.667 0.333 0.000
208 0.000 0.000 1.000 0.000
242 0.000 0.000 0.333 0.667
276 0.000 0.000 0.667 0.333
310 0.000 0.000 0.000 1.000
As an illustration, the covariates for DIM 38 can be computed as:
F (38) = (38 − 4)/106 − 4) = 0.333 and F (38) = 1 − 0.333 = 0.667
1 2
Thus F(38) = [0.333, 0.667, 0 0]
A random regression model can therefore be fitted as:
nf nr nr
i ∑
+
+
y = htd + f b k ∑ f u jk ∑ f pe + e
tijk jtk jtk jtk jk tij
k=0 k=0 k=0
where all terms are as defined in Section 9.3 but the f is the vector of the kth spline
jtk
function for the test day record of cow j made on day t. The same procedure described
in Section 9.3.1 can be used in the application of the model for the analysis of data
and interpretation of results.
9.3.6 Random regression model for maternal traits

Maternal genetic effects are important in growth traits in beef cattle, and models that
account for these effects have been discussed in Chapter 7. However, the RR model
could also be augmented to include random regressions for maternal genetic and
maternal permanent environmental effects. Albuquerque and Meyer (2001) examined
different orders of fit for the random regressions for both effects. One of the favoured
models was the one in which the order of Legendre polynomials for direct genetic,
maternal genetic, animal pe and maternal pe effects were 5, 5, 5 and 3, respectively.
Such a model, excluding all fixed effects, could be written as:

3 −
4 −
1−
2 −
k 1 k 1 k 1 k 1
+
=
y ijktd ∑ f u ji + ∑ f m ji ∑ f pe ji + ∑ f pp + e ijktd
dtti
jti
jti
di
jti
i=0 i=0 i=0 i=0
where y is the body weight of cow j taken at age t that has a dam d; u , m and
ijktd ji ji
pe are the random regressions for direct, maternal genetic and animal pe effects for
ji
animal j, respectively; pp is the random regression for dam pe effects and e is
di ijktd
random error; f and f are the vector of the ith Legendre polynomial for body
jti dti
weight at age t for cow j and dam d, respectively. They assumed a zero covariance
between direct and maternal genetic effects to simplify the computation. The vari-
ance for direct effects increased from birth to 365 days while maternal genetic
variance increased from birth to about 115 days and decreased thereafter.
148 Chapter 9