Page 131 - Linear Models for the Prediction of Animal Breeding Values

7.3 Reduced Animal Model with Maternal Effects

In Section 3.5, the use of the reduced animal model (RAM), with only one random
effect apart from residual error in the model, was considered. The records of non-
parents in the MME were expressed as the average of parental breeding values plus
Mendelian sampling. This has the advantage of reducing the number of random
animal equations in the MME. The application of RAM with multiple random effects
in the model is illustrated in this section using the example data used for the full
animal model in Section 7.2. The model for the analysis is the same but design matri-
ces and the variance of non-parental animals are different. From the arguments in
Section 3.5, the model for the RAM can be expressed as:

y ⎡ ⎤ ⎡ X p⎤ ⎡ Z p⎤ p e ⎡ ⎤
⎢ p ⎥ = ⎢ ⎥ b + ⎢ ⎥ u p + Z m Z pe + ⎢ ⎥ (7.6)
+
3
2
⎣ y ⎢ n⎦ ⎥ ⎣ X n⎦ ⎣ Z n⎦ n e ⎣ ⎦
where y , y = vector of observations for parent and non-parents, respectively,
p n
b = vector of fixed effects, u = vector of random animal effect for parents, m = vector
p
of maternal genetic effects for parents, pe = vector of permanent environmen-
tal effects and e , e = vector of residual error for parents and non-parents,
p n
respectively.
The incidence matrices Z and Z relate records to maternal genetic and perma-
2 3
nent environmental effect, respectively. The matrices Z and X relate records of
p p
parents to animal and fixed effects, respectively, while Z and X relate records of
n n
non-parents to parents (animal effect) and fixed effects, respectively.
It is assumed that:
u ⎡ p⎤ ⎡ g 11 A g 12 A 0 0 0⎤
⎢ ⎥ ⎢ g A g A 0 0 0 ⎥
⎢ m ⎥ ⎢ 21 22 ⎥
⎥
var p ⎢ e = ⎢ 0 0 Is 2 p pe 0 0⎥
⎢ ⎥ ⎢ ⎥
⎢ e p⎥ ⎢ 0 0 0 I s 2 ep 0⎥
⎢ ⎣ e n⎦ ⎥ ⎢ ⎣ ⎢ 0 0 0 0 I s en⎦ ⎥ ⎥
2

2
2
where s is the residual variance for parents, which is equal to s in Section 7.2,
ep e
2
s is the residual variance for non-parents and is equal to I + Dg , with D being a
en 11
3
diagonal matrix containing elements d , which are equal to or depending on
1
jj 4 2
2
whether one or both parents are known. The matrix G and s are defined as in
pe
Section 7.2. Let:
⎡ X p⎤ ⎡ Z p⎤ 2 ep 0 ⎤ ⎡ R n n 0⎤ ⎡ R − p 1 0⎤
X = ⎢ ⎥ , Z = ⎢ ⎥ , R = ⎢ I ⎡ s 2 ⎥ ⎢ ⎥ and R − 1 = ⎢ − 1⎥
=
1
⎣ X n⎦ ⎣ Zn⎦ ⎣ 0 Is en ⎦ ⎣ 0 R p ⎦ ⎣ 0 R n ⎦
Again, the MME provide the basis of the BLUE of estimable functions of b and BLUP
of a, m and pe in Eqn 7.6. The relevant MME are:
Maternal Trait Models: Animal and Reduced Animal Models 115