Page 268 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

15.3 Numerical Example of Sire Model

Consider the data in Table 15.1 for the pre-weaning gain (WWG) of beef calves. The
objective is to illustrate the estimation of variance components on a very small exam-
ple so that the calculations can be expressed concisely.
The model to describe the observations is:
y = o + s + e
ij j i
where y = the WWG of the ith calf, o = the overall effect, s = random effect of the
ij j
jth sire (j = 1, 2, 3) and e = random error effect (i = 1, 2, 3, 4).
i
In matrix notation, the model is the same as described in Eqn 3.1, with n = 4,
p = 1 and q = 3.
The matrix X in the MME relates records to the overall effects. For the example
data set, its transpose is:
X ′ = [1 1 1 1 1]
The matrix Z then relates records to sires. In this case it is:

é01 0 ù
ê ú
Z = ê 10 0 ú
ê00 1 ú
ê ú
ë 01 0 û

An analysis of variance can be constructed as:

2
Source Degrees of freedom Sums of squares (kg )
Overall 1 F = 48.3085
Sire 2 S = 0.6075
Residual 1 R = 0.1800

with:
−1
2
y ′ X(X ′ X) X ′ y = F = (2.9 + 4 + 3.5 + 3.5) /4 = 48.3025
2
2
−1
2
−1
y ′ Z(Z ′ Z) Z ′ y − y ′ X(X ′ X) X ′ y = S = (4) /1 + (2.9 + 3.5) /2 + (3.5) /1 − 48.3085 = 0.6075
2
y ′ y − y ′ Z(Z ′ Z) Z ′ y = R = (2.9) + (4) + (3.5) + (3.5) − F − S = 0.18
2
2
2
−1
Table 15.1. Pre-weaning gain (kg) for four beef calves.
Calf Sire WWG (kg)
4 2 2.9
5 1 4.0
6 3 3.5
7 2 3.5

252 Chapter 15