Page 140 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

⎡1 r r ⎤
⎢ ⎥ 2
⎢
⎥
R = r 1 r s e
⎣ ⎢r r ⎦ ⎥ 1
All elements between the various block diagonals are zero. However, Bergsma et al. (2008)
indicated that the residual covariance within groups (cov ) equals the variance among
penmates
2
group means (s ). Thus when cov penmates or r is > 0, instead of fitting the correlated residual
g
structure described above, a random group effect can be fitted as an equivalent model, with:
s = 2 + ( n − 2) 2
2
g s E DS s E S
andresidualvariance now defined as:
2 = 2 2
∗
e
s = s e − s g
Therefore, the equivalent model to Eqn 8.2 is:
y = Xb + Z u + Z u + Vg + Wc + e (8.5)
D D S S
2
where g is the vector of random group effects with g ∼ N(0, I s ). The MME to be
g g
solved then are:
⎛ X′X X′Z D X′Z S X′V X′W ⎞
⎜ −1 −1 ⎟
⎜ Z′ X Z′ Z + A a 1 Z′ Z + A a 2 Z′ V Z′ W ⎟
D
D
D
D
S
D
D
⎜ Z′ X Z′Z + −1 Z Z + −1 ′ Z V ′ Z W ⎟
′
Z
⎜ S S D A a 2 S S A a 3 S S ⎟
⎜ V ′ X V ′ Z D V ′ Z S V ′ V + Ia 4 V ′ W ⎟
⎜ ′ ′ W′ W′ + ⎟
⎝ W X W Z D W ′ZZ S V W Ia ⎠ 5
(8.6)
⎛ ˆ b ⎞ ⎛ X X′y ⎞
⎜ ⎟ ⎜ ⎟
′
⎜ u ˆ D ⎟ ⎜ Zy ⎟
D
⎜
⎜ ˆ u ⎟ = Zy ′ ⎟
⎜ S ⎟ ⎜ S ⎟
⎜ g ˆ ⎟ ⎜ Vy ′ ⎟
⎜ ⎟ ⎜ ⎟
⎝ c ˆ ⎠ ⎝ Wy ′ ⎠
⎡ g 11 g ⎤ 2⎤ ⎡ g 11 g ⎤
2
12
12
2
2
If G −1 = ⎢ 21 22 ⎥ then ⎢ ⎡a 1 a ⎥ = s 2 ∗ e ⎢ 21 22 ⎥, a = s /s 2 g and a = s /s 2 c
e*
4
5
e*
⎣ g ⎢ g ⎥ ⎦ ⎣a 2 a 3⎦ ⎣ g ⎢ g ⎥ ⎦
However, when cov is ≤ 0, then the MME to be solved are:
penmates
⎡ XR XX ′R Z X ′R Z X ′R W ⎤
′
−1
−1
−1
−1
⎢ D S ⎥
−
′
′
′
−1 12
−1
1
−1
−1 11
−1
⎢ ZR X Z ′ R Z + A g ZR Z + A g ZR W ⎥
1
D
S
D
D
D
D
⎢ − 1 − 1 − 121 − 1 − 122 − 1 ⎥
′
′
′
′
⎢ ZR X Z R Z + A g ZR Z + A g ZR W ⎥
S
S
S
S
S
D
⎢ −1 −1 −1 −1 2 ⎥
′
′
′
′
⎣ W WR X W R Z D WR Z S W R W + Is c ⎦ (8.7)
′
−1
⎡ ˆ ⎤ ⎡XR y ⎤
b
⎢ ⎥ ⎢ −1 ⎥
′
⎢ ⎢ u ˆ D⎥ ⎢ZR y ⎥
D
⎢ ⎥ = ⎢ ′ −1 ⎥
⎢ u ˆ S ⎥ ⎢ ZR y ⎥
S
′
⎣ c ˆ ⎢ ⎥ ⎦ ⎢ ⎣ WR y ⎥ ⎦
−1
124 Chapter 8