Page 128 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

For the example data set, W (with rows and columns numbered by the relevant
animal they relate to) is:
12 3 4 5 6 7 8 9 10 11 12 13 14
⎡
5 0 1 00000000000 0 ⎤
⎢ ⎥
6 0 1 000000000000 0 ⎥
⎢
⎢
7 00000 1 0000000 0 ⎥
⎢ ⎥
8 0000 1 00000000 0 ⎥
⎢
⎢
9 00000 1 0000000 0 ⎥
W = ⎢ ⎥
⎢
10 0 1 000000000000 0 ⎥
⎢
11 000000 1 000000 0 ⎥
⎢ ⎥
⎢
12 000000 1 000000 0 ⎥
⎢
13 0 1 00000000000 0 ⎥ ⎥
⎢
⎢
⎣
14 00000 1 00000000 0 ⎥ ⎦
and:
56 7 8 9 10 11 12 13 14
2 1 1 000 1 0 0 1 0⎤
⎡
⎢
5 0 0 0 1 000000 ⎥
S′ = ⎢ ⎥
6 0 0 1 0 1 0000 1⎥
⎢
⎢ ⎥
7 0 000000 11 00 ⎦
⎣
The matrix S above implies, for instance, that animals 5, 6, 10 and 13 have the same
dam (animal 2), while animals 11 and 12 are from another dam (animal 7).
The transpose of the vector of observations is:
y′ = [35 20 25 40 42 22 35 34 20 40]
The other matrices in the MME can be calculated through matrix multiplication.
The inverse of the relationship matrix is calculated applying the rules in Section 2.4.1.
−1
−1
The matrix A a is added to animal equations, A a to the equations for maternal
1 2
−1
genetic effects, A a to the animal by maternal genetic equations and a to the diagonals
3 4
of the equations for permanent environmental effects to obtain the MME. The MME are
not presented because they are too large. There is dependency between the equations for
herds and pen; thus the row for the first herd was set to zero in solving the MME by direct
inversion. Solutions to the MME are:
Effects Solutions
Herd–year–season
1 0.000
2 3.386
3 1.434
Pen
1 34.540
2 27.691
Continued

112 Chapter 7