Page 221 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Thus D can be generated from the additive genetic relationship. However, the
prediction of dominance effects requires the inverse of D. This could be obtained by
calculating D by Eqn 12.1 and inverting it: this is not computationally feasible with
large data sets. Hoeschele and VanRaden (1991) developed a methodology for
obtaining a rapid inversion of D and this is presented in Section 12.4. Initially, the
−1
principles involved in using D from Eqn 12.1 for the prediction of dominance
effects are discussed.

12.3 Animal Model with Dominance Effect

The model with dominance included is:
y = Xb + Za + Wd + e (12.2)

where y = vector of observations, b = vector of fixed effects, a = vector for random
animal additive genetic effects, d = vector of random dominance effects and e = random
residual error.
It is assumed that:

var(a) = As , var(d) = Ds 2 d and var(e) = s 2 e
2
a
2
e
var(y) = ZAZ′ + WDW′ + Is
The MME to be solved for the BLUP of a and d and the BLUE of b are:

ˆ
¢
¢
é XX X Z ¢ X Wù é bù é X y ¢ ù
ê ¢ -1 ¢ ú ê ú ê Z y ¢ ú
a ˆ =
¢ +
ê
ê ZX ZZ A a 1 ZW ú ê ú ê ú (12.3)
ˆ
ê ë WX W Z WW A a ú d ê ú êW y ¢ ú û
¢
¢
-1
¢
+
2 û ë û
ë
2
2
2
2
with a = s /s and a = s /s . However, we are interested in the total genetic merit
1 e a 2 e d
(g) of the animal, which is g = a + d. The MME could be modified such that the total
genetic merit is solved for directly. Since g = a + d, then:
2 2
a
var(g) = G = As + Ds d
The MME become:
′
′
⎡ XX X Z⎤ ⎡ ⎤ ˆ b ⎡ Xy ′ ⎤
⎢ 2 ⎥ ⎢ ⎥ = ⎢ ⎥ (12.4)
′
′+
g ˆ
⎣ ZX ZZ Gs e⎦ ⎣ ⎦ ⎣ Z y ′ ⎦
The individual components of g can be obtained as:
−1
2
aˆ = s AG gˆ and
a
ˆ
−1
2
d = s DG gˆ
d
Non-additive Animal Models 205