Page 133 - Linear Models for the Prediction of Animal Breeding Values

2
The residual variance for parents s = 350, and because both parents of non-parents
ep
in the data are known:
2
1
1
2
s = s + (g ) = 350 + (150) = 425
en e 2 11 2
with:
⎡ 2 0⎤
R = ⎢ Is ep 2 ⎥
⎣ 0 Is ⎦
en
Then:
R = diag(350, 350, 350, 350, 350, 425, 425, 425, 425, 425)
and:
−1
R = diag(0.00286, 0.00286, 0.00286, 0.00286, 0.00286, 0.00235, 0.00235,
0.00235, 0.00235, 0.00235)
1
2
1/s = = 0.025
pe 40
SETTING UP THE DESIGN MATRICES
The matrix X, which relates records to fixed effects, is the same as in Section 7.2.1,
considering only animals with records. The matrix X¢R X in the MME can be cal-
−1
culated through matrix multiplication from X and R already set up. For illustrative
−1
purposes, the matrix X¢R X, when expressed as the sum of the contributions from
−1
parents’ and non-parents’ records, is:
′
−1
′
11
11
X ′R X = r X X p + r X X n
p
p
n
n
⎡ 0.0114 0.0 0.0 0.0057 0.0057⎤ ⎡ 0.0 0.0 0.0 0.0 0..0 ⎤
7
⎢ ⎥ ⎢ ⎥
⎢ 0.0 0.0029 0.0 0.0029 0.0 ⎥ ⎢ 0.0 0.0047 0.0 0.0 0.0047 ⎥
⎢
⎢
= 0.0 0.0 0.0 0.0 0.0 ⎥ + 0.0 0.0 0.0071 0.0024 0.0047⎥
⎢ ⎥ ⎢ ⎥
⎢ 0.0057 0.0029 0.0 0.008660.0 ⎥ ⎢ 0.0 0.0 0.00224 0.0024 0.0 ⎥
⎢ ⎣ 0.0057 0.0 0.0 0.0 0.0057 ⎦ ⎥ ⎢ ⎣ 0.0 0.0047 0.0047 0.0 0.0094 ⎦ ⎥
⎡ 0.0114 0.0 0.0 0.0057 0.0057⎤
⎢ ⎥
⎢ 0.0 0.0076 0.0 0.0029 0.0047 ⎥
⎢
= 0.0 0.0 0 0.0071 0.0024 0.0047⎥
⎢ ⎥
⎢ 0.0057 0.0029 0.0024 0.0109 0.0 ⎥
⎢ ⎥
⎣ 0.0057 0.0047 00.0047 0.0 0.0151 ⎦
where X and X are matrices relating parents and non-parents to fixed effects,
p n
respectively, and are:
⎡1111 0 ⎤ ⎡ 0000 00⎤
⎢ ⎥ ⎢ ⎥
⎢ 0000 1 ⎥ ⎢ 11 000 ⎥
⎢
X′ p = ⎢00000 ⎥ and X′ n = 00 111⎥
⎢ ⎥ ⎢ ⎥
⎢ 10 0 1 1 ⎥ ⎢ 000 1 0 ⎥
⎢ ⎣ 01 1 0 0 ⎥ ⎦ ⎢ ⎣ 111 0 1 ⎥ ⎦
Maternal Trait Models: Animal and Reduced Animal Models 117