Page 154 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

9.3.1 Numerical application

Example 9.2
Analysis of the data in Table 9.1 is undertaken fitting an RR model with Legendre
polynomials of order 4 fitted for the fixed lactation curve and Legendre polynomials
of order 2 fitted for both random animal and pe effects. The covariance matrices for
the random regression coefficients for animal effect and pe effects are:

.
⎡ 3 297. 0 594 − 1 381. ⎤ ⎡ 6 .872 − .254 − .101⎤
0
1
⎢ ⎢
G = ⎢ ⎢ 0 594 0 921 − 0 289. ⎥ ⎥ ⎥ ; P = − .0 254 . 3 171 . 0 167 ⎥ ⎥
.
.
⎢
⎣ − ⎢ 1 381. − 0 289. 1 005⎥ ⎦ ⎢ ⎣ − .101 . 0 167 . 2 457 ⎥ ⎦
1
.
and the residual variance equals 3.710 for all stages of lactation.
As indicated earlier, the above G or P matrix models the genetic or permanent
environment covariance structure of fat yields over the whole lactation length. Thus
the genetic covariance between DIM i and j along the trajectory can be calculated
from G. For instance, the genetic variance for DIM i, (v ) can be calculated as:
ii
v = t Gt′
ii i i
where t = f , the ith row vector of F, for day i, and k is the order of fit. The genetic
i ik
covariance between DIM i and j (v ) therefore is:
ij
v = t Gt′
ij i j
Using the G matrix in Example 9.1, the genetic variance for DIM 106 equals
2
2.6433 kg , with t = [0.7071 −0.4082 −0.5271], and the genetic covariance between
106
DIM 106 and 140 equals 3.0219 kg, with t = [0.7071 −0.1361 −0.7613]. The plots
140
of daily genetic and permanent environmental variances against DIM are shown in
Fig. 9.1, indicating how these variances change through the lactation length.
SETTING UP THE MATRICES FOR THE MME
−1
The setting of the matrix X has been described in Example 9.1. The matrix X′R X
can easily be obtained by matrix multiplication. Considering only animals with
records, Q′ can be represented as:
⎡ Q′ 4 0 0 0 0⎤
⎢ 0Q′ ⎥
⎢ 5 0 0 0 ⎥
Q′ = ⎢ 0 0 Q′ 0 0⎥
⎢ 6 ⎥
⎢ 0 0 0 Q′ 7 0 ⎥
⎢ ⎣ 0 0 0 0 Q′ 8 ⎦ ⎥

where Q′ is the matrix of order nr by k (number of TD records for animal i). Thus
i
for animal 6, Q′ is:
6
é 0 7071 0 7071 0 7071 0 7071 0 7071ù
.
.
.
.
.
ê
Q′ = - 1 2247 - 0 9525 - 0 6804 - 0.4082 - . 0 1361 ú
.
.
.
.
6 ê ú
ê ë . 1 5811 . 0 6441 - . 0 0586 - . 0 5271 - . 0 7613ú û
138 Chapter 9