Page 116 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The transformed additive genetic covariance matrix (M) is:

− 1 − 1 ⎡ 0.5000 0.380539⎤ − 1 ⎡ 2.654723 − 0.862556⎤
(
M = T G T )′ = ⎢ ⎥ and M = ⎢ ⎥
⎣ 0.380539 1.171972 ⎦ ⎣ − 0.862556 1.133334 ⎦
−1
The transformed variables are calculated using the transforming matrix T . For the
first two animals the transformation is as follows:
Animal 1:
11
*
y = t y = 0.1581139(4.5) = 0.712
11 11
Animal 2:
*
11
y = t y = 0.1581139(2.9) = 0.459
11 11
y = t y + t y = −0.052948(2.9) + 0.1925393(5.0) = 0.809
22
21
*
22 11 22
where y and y * are the original and transformed observations, respectively, for
ij ij
the ith trait and jth animal. The transformed variables for all calves are shown in the
table below.
Original traits Transformed traits
Calves Sex Sire Dam WWG PWG y * y *
1 2
4 Male 1 – 4.5 – 0.712 –
5 Female 3 2 2.9 5.0 0.459 0.809
6 Female 1 2 3.9 6.8 0.617 1.103
7 Male 4 5 3.5 6.0 0.553 0.970
8 Male 3 6 5.0 7.5 0.791 1.179
9 Female 7 – 4.0 – 0.632 –

The model for analysis is the same as in Section 5.4.1 except that the variance of
*
y now is:
−1
−1
*
var(y ) = T G(T )′ + I = M + I
The MME for the transformed variables are:
ˆ ⎡ * ⎤ ⎡ 0 0⎤ −1 ⎡ X ′ 1 * y ⎤
1 ′
1 b
⎢ * ⎥ ⎥ ⎢ XX 1 XZ1 ⎥ ⎢ * 1 ⎥ ⎥
′ 1
0
′ 2
′ 2
⎢
⎢ b ˆ ⎢ 2 = ⎢ ⎢ 0 X′ 2 X2 − 1 11 XZ2 ⎥ ⎥ ⎢ X y 2 ⎥
⎥
−
1
12
1 ′
ˆ ⎢ * 1 a ⎥ ⎢ Z 1 ′ X 1 1 0 ZZ1 + A m A m ⎥ ⎢ 1 ′ Z y * 1 ⎥
⎢ ⎥ 0 2 ′ − 1 21 Z Z2 + − 1 22 ⎢ ⎥
2 a ⎣
ˆ ⎦ ⎣ ZX2 A m 2 ′ A m ⎦ ⎣ ⎢ ′ Z y * 2 ⎦ ⎥
*
2
The design matrices X , X , Z and Z and the inverse of the relationship matrix
1 2 1 2
are exactly as in Section 5.4.1. The vector observations y* now contain the trans-
formed variables shown in the above table. All other matrices in the MME above can
be derived from the design matrices and vector of observations through matrix mul-
−1
22
−1
11
tiplication and the addition of the A m and A m to the animal equations for trait
−1
12
one and two, respectively, and A m to animal equations for trait one by trait two
−1
21
and A m to equations for trait one by trait two that pertains to animals. The MME
have not been shown because they are too large. However, solving the MME gives
100 Chapter 6