Page 127 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Table 7.1. Birth weight for group of beef calves.
Calf Sire Dam Herds Pen Birth weight (kg)

5 1 2 1 1 35.0
6 3 2 1 2 20.0
7 4 6 1 2 25.0
8 3 5 1 1 40.0
9 1 6 2 1 42.0
10 3 2 2 2 22.0
11 3 7 2 2 35.0
12 8 7 3 2 34.0
13 9 2 3 1 20.0
14 3 6 3 2 40.0

progeny with records. Suppose that the genetic parameters are g = 150, g = −40,
11 12
g = 90, s = 40 and s = 350. Then:
2
2
22 pe e
⎡0.00756 0.00336 ⎤ 2 ⎤ 2 ⎡ .647 1.176⎤
−1
G = ⎢ ⎥ and ⎡a 1 a ⎥ = ⎢ ⎥
⎢
⎣ 0.00336 0.0126 ⎦ ⎣a 2 a 3 ⎦ ⎣ 1.176 4.412 ⎦
and a = 350/40 = 8.75.
4
The model for the analysis is as presented in Eqn 7.1.

SETTING UP THE DESIGN MATRICES

Considering only animals with records, the first three rows of matrix X relate records
to herd effects and the last two rows to pen effects. The transpose of X is:

⎡1111 000000⎤
⎢ ⎥
⎢ ⎢ 0000 111 000 ⎥
⎢
X ′ = 0000000 111⎥
⎢ ⎥
⎢ 10 0 1 10 0 0 10 ⎥
⎢ ⎣ 01 1 0 01 1 1 01 ⎥ ⎦

Excluding ancestors, each animal has one record; therefore Z is an identity
matrix. However, Z is augmented with columns of zeros equal to the number of
ancestors to take account of ancestors in the pedigree. The matrices W and S relate
records through the dam to their effects, i.e. maternal genetic effect and permanent
environmental effect, respectively. However, since maternal effect is genetic and is
passed from parent to offspring, estimates of maternal effect are for all animals
in the analysis while estimates of permanent environmental effects are only for
dams of progeny with records. Thus, in setting up W, all animals are considered,
while only four dams with progeny having records are taken into account for S.

Maternal Trait Models: Animal and Reduced Animal Models 111