Page 113 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

6.2.2 An illustration

Example 6.1
The multivariate analysis for WWG and PWG in Section 5.2.2 is repeated below,
carrying out a canonical transformation assuming the same genetic parameters.
The calculation of the transformation Q and the diagonal matrix W are given in
Appendix E, Section E.1. Presented in Table 6.1 are the data for all calves in the original
scale and as transformed variables (VAR1 and VAR2). The observations are trans-
formed into new uncorrelated variables using the matrix Q. Thus for animal 4, the
record would be transformed as:
é 0.1659 - 0.0792ù é 4.5ù é 0.208ù
Qy = ú ê ú ê ú
=
4 ê
ë 0.0168 0.1755 û ë 6.8 û ë 1..269 û
The residual variance for each of the transformed variables is 1, thus heritability for
the ith transformed variable = w /(1 + w ) and a = 1/w .
ii ii i ii
2
2
Therefore h = 0.247, h = 0.573, a = 1/0.3283 = 3.046 and a = 1/1.3436 =
1 2 1 2
0.744. A single trait analysis is carried out on the transformed variates for WWG and
PWG using the model and the MME in Section 5.3.1 and solutions are transformed
back to the original scale.
SETTING UP THE DESIGN MATRICES
The matrix X, which relates records for either VAR1 or VAR2 to sex effects,
is exactly as the matrix X in Section 5.2.2. Similarly, Z is the same as Z in
1 1
*
*
Section 5.2.2. For animals with records, the vector of observations y and y are
1 2
equal to the column of transformed variates for WWG and PWG gains, respec-
tively, in Table 6.1. The matrices in the MME are easily obtained through matrix
−1
multiplication and the addition to the animal equations of A a for VAR1 and
1
−1
−1
A a for VAR2. A has been given earlier in Section 5.2.2. For instance, the
2
MME for VAR1 only are:
-1
é ˆ ù é 3.0 0.0 0.000 0.000 0.000 1.000 0.000 0.000 1.000 1.0000ù é0.549ù
*
1 b
ê ˆ * ú ê 0.0 2.0 0.000 0.000 0.000 0.000 1.000 1.000 0.000 0.000 ú ê 0.194 ú
ê b 2 ú ê ú ê ê ú
ê * 1 ˆ a ú ê 0.0 0.0 5.5884 1.523 0.000 - 2.031 0.000 - 3.046 0.000 0.000 ú ê 0.000 ú
ê 2 ˆ a ú ú ê 0.0 0.0 1.523 6.092 1.5523 0.000 - 3.046 - 3.046 0.000 0.000 ú ê 0.000 ú
*
ê * ú ê 0.0 0.0 0.000 1.523 6.092 0.000 -33.046 1.523 0.000 - 3.046ú ê 0.000ú
3 ˆ a
ê * ú = ê 1.0 0.0 - 2.031 0.000 0.000 6.584 1.523 0.000 - 3.046 0.000 ú ê 0.208 ú
0
ê 4 ˆ a ú ê ú ê ú
ê 5 ˆ a * ú ê 0.0 1.0 0.000 - 3.046 - 3.046 1.523 8.615 0.000 - 3.046 0..000 ú ú ê 0.085 ú
ê 6 ˆ a ú ê 0.0 1.0 - 3.046 - 3.046 1.523 0.000 0.000 8.615 0.000 - 3.046 ú ê 0.108 ú
*
ê * ú ê 1.0 0..0 0.000 0.000 0.000 - 3.046 - 3.046 0.000 7.092 0.000ú ê 0.105ú
ê 7 ˆ a * ú ê ú ê ú
ë 8 ˆ a û ë 1.0 0.0 0.000 0.0000 - 3.046 0.000 0.000 - 3.046 0.000 7.092 û ë 0.235 û
Solving the MME for each transformed trait by direct inversion of the coefficient
matrix gives the following solutions on the canonical scales. Given also are solutions
for WWG and PWG after transforming the solutions for the transformed variates to
the original scale.
Methods to Reduce the Dimension of Multivariate Models 97