Page 117 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

the following solutions on the transformed scale. The solutions transformed to the
original scale are also shown.
Transformed scale Original scale
Effects WWG PWG WWG PWG

Sex
Male 0.691 1.085 4.367 6.834
Female 0.578 0.963 3.657 6.007
Animals
1 0.021 0.044 0.130 0.266
2 −0.013 −0.010 −0.084 −0.075
3 −0.015 −0.032 −0.098 −0.194
4 0.001 0.003 0.007 0.016
5 −0.054 −0.089 −0.343 −0.555
6 0.030 0.075 0.192 0.440
7 −0.049 −0.077 −0.308 −0.483
8 0.032 0.056 0.201 0.349
9 −0.003 −0.022 −0.018 −0.119

These are exactly the same solutions as those obtained in Section 5.3 without any
transformation. The number of non-zero elements was 188 in the analysis on the
transformed variables, compared with 208 when no transformation is carried out.
This difference could be substantial with large data sets and reduces storage require-
ments when data is transformed. The solutions were transformed to the original scale
using Eqns 6.6 and 6.7. Thus the solutions for male calves on the original scale are:
ˆ ⎡ ⎤ ⎡6.324555 0.000 ⎤ 0.690633⎤ ⎡ 4.367⎤
⎡
b 11
=
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
=
b ⎣ ˆ 12⎦ ⎣ 1.739253 5.193746 ⎦ ⎣ 1.0846 ⎦ ⎣ 6.834 ⎦
6.4 Factor and Principal Component Analysis
In Sections 6.2 and 6.3, the simplification of multivariate analysis using canonical trans-
formation and Cholesky decomposition were discussed. Both approaches involved the
transformation of the vector of observations as well as the residual and genetic covari-
ance matrices. However, for multivariate analysis with a large number of traits and with
high genetic correlations among the traits, a factorial or principal component analysis
might be more appropriate in reducing the dimension of such analysis. Neither of these
methods involve the transformation of the vector of observations. The principal compo-
nent and factor analysis (FA) methods provide efficient means for reducing the rank of
the genetic covariance matrix in multivariate analysis, resulting in the substantial spar-
sity of the MME for genetic evaluation and estimation of genetic parameters (Meyer,
2009). Therefore, both methodologies have attracted considerable attention in multi-
variate analysis involving many traits for parameter estimation and genetic evaluation
(Kirkpatrick and Meyer, 2004; Meyer, 2005, 2007; Tyriseva et al., 2011a, 2011b).
FA is mainly concerned with identifying the common factors that give rise to
correlations between variables. It assumes that the traits studied are linear combina-
tions of few latent variables, referred to as common factors. Then any variance not
explained by these common factors is modelled separately as trait specific, by fitting

Methods to Reduce the Dimension of Multivariate Models 101