Page 95 - Linear Models for the Prediction of Animal Breeding Values

SETTING UP THE DESIGN MATRICES
The X ′ and X ′ matrices, which relate sex effects for WWG and PWG, respectively, are:
1 2
é 10 0 1 10ù é 000 11 0ù
X¢ = ê ú and X¢ = ê ú
1
2
ë 01 1 0 01 û ë 0 1 1 000 û
é 30ù é 20ù
X¢ 1 X = ê ú and X¢ 2 X = ú
2 ê
1 1
ë 03 û ë 02 û
11
In setting up X ′ R X , it is necessary to account for the fact that animals (one
1 1
male and one female) have missing records for PWD. Thus:
⎡10 ⎤ ⎡20 ⎤ ⎡0.081 0.000 ⎤
11
11
11
′
XR X = r m W ′W + r o ′ B B = 0.025 ⎢ ⎥ + 0.028 ⎢ ⎥ ⎥ = ⎢ ⎥
1
1
⎣ 01 ⎦ ⎣ 02 ⎦ ⎣ 0.000 0.081 ⎦
where the matrix W relates WWG records for animals 4 and 9 with missing records
for PWD to sex effects and B relates WWG records for calves 5, 6, 7 and 8 to sex
effects. The matrices W ¢ and B ¢ are:
⎡10 ⎤ ⎡00 11 ⎤
W ′ = ⎢ ⎥ and ′ B = ⎢ ⎥
⎣ 01 ⎦ ⎣ 11 00 ⎦
However, all animals recorded for PWG also had records for WWG, therefore:
é 20ù é0.074 0 ù
¢
22
XR X 2 = r 22 X ¢ X 2 = 0.037 ê ú = ê ú ú
o
2
2
ë 02 û ë 0 0.074 û
and:
⎡ 20⎤ ⎡ −0.02 0.00⎤
′
′
12
XR X = r o 12 X X = − 0.010 ⎢ ⎥ = ⎢ ⎥
2
1
2
1
⎣ 02 ⎦ ⎣ 0.00 −0.02 ⎦
2
Excluding ancestors, the matrix Z is an identity matrix because every animal has a
1
record for WWG. Therefore, Z′ Z = I and:
1 1
11
Z′R Z = diag(0.025, 0.028, 0.028, 0.028, 0.028, 0.025)
1 1
However:
Z = diag(0, 1, 1, 1, 1, 0)
2
indicating that calves 4 and 9 have no records for PWG, and:
22
Z′R Z = diag(0.0, 0.037, 0.037, 0.037, 0.037, 0.0)
2 2
To account for ancestors (animals 1 to 3), Z′R Z and Z′R Z given above
22
11
1 1 2 2
augmented with three rows and columns of zeros.
The other matrices in the MME can be calculated through matrix multiplica-
−1
−1
tion. The matrix A can be set up and A −1* G (where * means the Kronecker
product) added to the appropriate matrices, as described in Section 5.2.2, to obtain
the MME. The MME are too large to be presented but solutions from solving the
equations are shown below, together with solutions from the univariate analyses of
WWG and PWG.
Multivariate Animal Models 79