Page 271 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The contrasts Q ′ Z ′ Sy are now:

é 0 000 - 0 5000 - 0 5000 0 5000ù
.
.
.5
.
′′
QZ S = ê ú y
ë ë 0 0000 0 7071 - 0.7071 0 0000 û
.
.
.
So the first contrast, (y - y - y + y )/2 = (2.9 - 4.0 - 3.5 + 3.5)/2 = −1.1/2 =-0.55
1 2 3 4
is a scaled contrast comparing sire 2 with sire 1 and sire 3 and the second con-
trast (y - y )/ 2 = (-4.0 = 3.5)/ 2 = (-0.5)/ 2 is a scaled contrast between sire
2 3
1 and sire 3.
An analysis of variance can be constructed:
Expected mean
2
2
Source Degrees of freedom Sums of squares (kg ) squares (kg )
Overall 1 F = 48.3085
2
Sire 2 compared 1 (−0.55) = 0.3025 2 2
e s
s + 1.5s
with sires 1 and 3
2
Sire 1 compared 1 (−0.5) /2 = 0.1250 2 2
e s
s + s
with sire 3
Residual 1 R = 0.1800 2
e
s
2
Fitting a linear model in s and s to the three sums of squares 0.3025, 0.1250
2
e s
2
and 0.1800, gives estimates of s = 0.143 (kg ) and s = 0.079 (kg ). If a generalized
2
2
2
e s
linear model is fitted iteratively to the sum of squares with weights proportional to
2
the variance of the sum of squares when the procedure converges, the estimate of s e
2
2
2
is 0.163 (kg ) and of s is 0.047 (kg ). The estimated variances of these estimates
s
2
(from the inverse of the generalized least squares coefficient matrix) are 0.216 (kg )
2
and 0.234 (kg ).
15.6 Animal Model
It has been shown that estimates can be obtained from analysis of variance for some
models. Now consider a more general model – the animal model introduced in
Chapter 3. This linear model (Eqn 3.1) is:
y = Xb + Za + e
and the variance structure is defined, with:
2
2
var(e) = Is = R; var(a) = As = G and cov(a, e) = cov(e, a) = 0
e a
2
a
where A is the numerator relationship matrix, and there is interest in estimating s
2
and s . A popular method of estimation is by restricted (or residual) maximum like-
e
lihood (REML) (Patterson and Thompson, 1971). This is based on a log-likelihood
of the form:
1
−1
L a( 2 ){−(y − Xb) ′ V (y − Xb) − logdet(V) − logdet(X ′ V X)}
−1
where b is the generalized least squares (GLS) solution and satisfies:
−1
X ′ V Xb = X ′ V y
−1
Estimation of Genetic Parameters 255