Page 114 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Table 6.1. Weaning gain and post-weaning gain for beef calves on the original and
transformed scales.

Original scale Transformed scale
Calves Sex Sire Dam WWG PWG VAR1 VAR2
4 Male 1 – 4.5 6.8 0.208 1.269
5 Female 3 2 2.9 5.0 0.085 0.926
6 Female 1 2 3.9 6.8 0.109 1.259
7 Male 4 5 3.5 6.0 0.106 1.112
8 Male 3 6 5.0 7.5 0.236 1.400

Canonical scale Original scale

Effects VAR1 VAR2 WWG PWG
Sex
Male 0.185 1.266 4.361 6.800
Female 0.098 1.089 3.397 5.880
Animals
1 0.003 0.052 0.151 0.280
2 −0.002 −0.002 −0.015 −0.008
3 0.000 −0.031 −0.078 −0.170
4 −0.001 −0.002 −0.010 −0.013
5 −0.007 −0.088 −0.270 −0.478
6 0.005 0.095 0.276 0.517
7 −0.015 −0.089 −0.316 −0.479
8 0.009 0.073 0.244 0.392

The solutions are exactly the same as those obtained from the multivariate analysis
in Section 5.2. The solutions are transformed to the original scale using Eqns 6.3 and
6.4. For instance, the solutions for animal 1 for both traits on the original scale are:

ˆ a é ù é 5.7651 2.6006 0.0029ù é 0.151ù
ù é
=
=
ê 11 ú ê ú ê ú ê ú
ˆ a ë 12 û ë - 0.5503 5.4495 0.05116 û ë 0.280 û
û ë
6.3 Cholesky Transformation
When all records are measured in all animals, MBLUP may be simplified by a canoni-
cal transformation as described in Section 6.2. However, if animals have some records
missing and the loss of records is sequential then a Cholesky transformation can be
applied (Quaas, 1984). Such situations can arise, for example, in dairy cattle due to
sequential culling and different lactations being regarded as different traits.

6.3.1 Calculating the transformation matrix and defining the model

Cholesky transformation involves forming transformed variables (traits) that are
environmentally independent of each other; that is, there is no residual covariance
among them, therefore the residual covariance matrix for the transformed traits is an

98 Chapter 6