Page 284 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

ii
-1
G
b ik | b i k ,b j , ,R e , , ~ N (b ik ,(x′ ik r x ik ) ); j = 1 n , and j ¹ i (16.21)
y
u
-
,
,
,
,
,
with:
ˆ
b = (x′ r x ) x′ (r y + r y ) r− ii (x′ b + z u )
−1
ij
ii
ii
−
, k
, ik , ik , ik , ik i j , ik i − i i i
− rx b + z u ); j = 1 , n and j i
≠
ij
(
j
j
j
j
Similarly, for the random animal effect, the conditional distribution for animal k
of the ith trait is:
−1
ii
ii
, ,
n
,
,
u | u , u b R G y , ~ N(uˆ ,( r z′ z + g A −1 ) ), j = 1, and j ≠ i (16.22)
,
,
ik i,− k j e ik ik ik kk
,
,
,
with:
, ik ( ii -1 , k k ) { ( ii ij r x b - rrx b )
-1
ii
ij
ii
′
ˆ u = z r z , ik + A g z′ , ik r y + r y - i i j j
j
, ik
i
-1
ij
ii
ij
- z ( ′ r ij z + A -1 g u jk) - A ( g u + g u )}
u
,
,
,
ik jk , kk , k s , is js
,
where s represents the known parents of the kth animal.
However, instead of sampling for each level of fixed or random effects for one
trait at a time, it is more efficient to implement block sampling for each level of fixed
or random effect across all traits at once. The conditional distribution for level k of
a fixed effect required for block sampling, assuming n = 2, is:
b ⎡b ˆ ⎤
X R X ) ⎥
1,k 1,k − 1 − 1
b ,, , , ⎢ , ( ′ (16.23)
u R G y ~ N
b − k b ˆ k k ⎥
2,k ⎣ ⎢ 2,k ⎦
where:
ˆ ⎛ ⎞
b , 1 k
−
−
′
′
⎜ ⎟ = (XR X k ) (X R − 1 (y − X b − k − Zu ˆ ))
1
1
−
ˆ ⎜
k
k
k
⎝ b , 2 k ⎠ ⎟ k
which is equivalent to Eqn 5.4.
For the random animal effect, block sampling for animal k, assuming n = 2, the
conditional distribution is:
u ⎡u ˆ ⎤
1,k 1,k − 1 − 1 − 1 −1
Z R Z +
R G y ,
bu , , j − k , , ~ N ⎢ ,( ′ k k A , k k ⊗ G ) ) ⎥ (16.24)
u 2,k ⎣ ⎢ u ˆ 2,k ⎦
where:
æ ˆ u , 1 k ö
-
′
1
-1
ç ç ÷ = (ZR Z + A - 1 Ä G - 1 - 1 ′ k - 1 (y - Xb - - 1 Ä Ä G (ˆ u s + u ˆ )}
) A
) {(Z R
÷
k
d
k
k
è ˆ u , 2 k ø
where s and d are the sire and dam of the kth animal.
From Eqn 16.20, the full conditional distribution of the residual variance is:
P(R|b, u, y) ∝ P(R)P(y|b, u, R)
268 Chapter 16