Page 222 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

12.3.1 Solving for animal and dominance genetic effects separately

Example 12.1
Suppose the data below are the weaning weights for some piglets in a herd.

Pig Sire Dam Sex Weaning weight (kg)

5 1 2 Female 17.0
6 3 4 Female 20.0
7 6 5 Female 18.0
8 0 5 Female 13.5
9 3 8 Male 20.0
10 3 8 Male 15.0
11 6 8 Male 25.0
12 6 8 Male 19.5

The aim is to estimate sex effects and predict solutions for animal and dominance
2
2
2
genetic effects, assuming that s = 120, s = 90 and s = 80. This has been illustrated
e a d
below, solving for animal and dominance effects separately (Eqn 12.3). From the
above parameters, a = 1.333 and a = 1.5.
1 2
SETTING UP THE MME
The matrix X relates records to sex effects. Its transpose, considering only animals
with records, is:

é 1111 0000ù
X′ = ê ú
ë 0000 1111 û
The matrices Z and W are both identity matrices since each animal has
one record. The transpose of the vector of observations y′ = [17 20 18 13.5 20
15 25 19.5].
−1
−1
The other matrices in the MME, apart from A and D , can be obtained
through matrix multiplication from the matrices already calculated. The inverse of
the additive relationship matrix is set up using rules outlined in Section 2.4.1. Using
Eqn 12.1, the dominance relationship matrix is:
é 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000ù
0
ê ú
ê 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0..000 0.000 ú
ê 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0000 0.000 0.000ú
ê ú
ê 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 00.000 0.000 0.000 ú
ê 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.0000 0.000 0.000 0.000 ú
ê ú
ê 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000ú
0
D = ê ú
ê 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0..000 0.062 0.062 0.125 0.125 ú
ê 0.000 0.000 0.000 0.000 0.000 0.000 0.0000 1.000 0.000 0.000 0.000 0.000ú
ê ú
ê 0.000 0.000 0.000 0.000 0.000 0.000 00.062 0.000 1.000 0.250 0.125 0.125 ú
ê 0.000 0.000 0.000 0.000 0.000 0.0000 0.062 0.000 0.250 1.000 0.125 0.125 ú
ê ú
ê 0.000 0.000 0.000 0.000 0.000 0.000 0.125 0.000 0.125 0.125 1.000 0.250ú
0
ê ê ú
ë 0.000 0.000 0.000 0.000 0..000 0.000 0.125 0.000 0.125 0.125 0.250 1.000 û
206 Chapter 12