Page 126 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

7.2 Animal Model for a Maternal Trait

The model for maternally influenced traits in matrix notation is:
y = Xb + Zu + Wm + Spe + e (7.1)

where y = vector of observations, b = vector of fixed effects, u = vector of random
animal effects, m = vector of random maternal (indirect) genetic effects, pe = vector
of permanent environmental effects as explained in item 3 in Section 7.1, e = vector
of random residual effects, and X, Z, W and S are incidence matrices relating records
to fixed, animal, maternal genetic and permanent environmental effects, respectively.
It is assumed that:

⎡ u⎤ g ⎡ 11 A g 12 A 0 0 ⎤
⎢ m ⎥ ⎢ g A g A 0 0 ⎥
var ⎢ ⎥ = ⎢ ⎢ 21 22 2 ⎥ ⎥
⎢ pe⎥ 0 0 Is pe 0
⎢ ⎥ ⎢ ⎥
⎣ e ⎦ ⎢ ⎣ 0 0 0 Is ⎥
2
e e ⎦
where g = additive genetic variance for direct effects, g = additive genetic variance for
11 22
maternal effects, g = additive genetic covariance between direct and maternal effects,
12
2
2
s = variance due to permanent environmental effects and s = residual error
pe e
variance.
The variance of y, using the same arguments as in Section 3.2, is:
var( ) = [ Z W] ⎢ ⎡ g 11 A g 12 A⎤ ⎡ Z ⎤ ′ ⎥ + 2 pe ′ 2 e
⎥ ⎢
y
⎣ g 21 A g 22 A W′ ⎦ SIs S + Is
⎦ ⎣
The BLUE of estimable functions of b and the BLUP of u, m and pe in Eqn 7.1 are
obtained by solving the following MME:
′
′
′
⎡ ˆ b ⎤ ⎡ XX X Z X W X S ′ ⎤ −1 ⎡ X′y⎤
y
⎢ ⎥ ⎢ − 1 − 1 ⎥ ⎢ ⎥
′
′+
′
⎢ u ˆ ⎥ = ⎢ ZX ZZ A a 1 ZW + A a 2 Z S ′ ⎥ ⎢ Zy ′ ⎥
⎢ m ˆ ⎥ ⎢ W′ ′+ −1 WW + −1 WS ′ ⎥ ⎢ W y ′ ⎥ (7.2)
′
W X W Z
⎢ ⎥ ⎢ A a 2 A a 3 ⎥ ⎢ ⎥
′
′ +
′
′
⎣ pe ˆ ⎦ ⎣ SX S Z S W S S Ia 4 ⎦ ⎣ Sy ′ ⎦
12
⎡ g g ⎤ ⎡ g 11 g ⎤ ⎡ 1 a ⎤ ⎡ g 11 g 12 ⎤
−
2
2
2
1
withG = ⎢ 11 12 ⎥ ; G = ⎢ ⎥ and ⎢ a ⎥ = s ⎢ ⎥ and a = s /s 2 pe
e
4
e
⎣ g 21 g 22⎦ ⎣ g ⎢ 21 g ⎥ ⎦ ⎣ a 2 2 a 3⎦ ⎣ ⎢g 21 g 22 ⎦ ⎥
22
7.2.1 An illustration
Example 7.1
Assume the data in Table 7.1 to be the birth weight for a group of beef calves. The
aim is to estimate solutions for herd and pen effects and predict solutions for direct
and maternal effects for all animals and permanent environmental effects for dams of
110 Chapter 7