Page 111 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

6 Methods to Reduce the

Dimension of Multivariate

Models

6.1 Introduction

One of the limitations of multivariate analysis is the large computational requirements
of such high-dimensional analyses. The number of effects in multivariate analyses
tend to increase linearly with the number of traits considered. In some cases, the
available or small number of records for some of the traits can hamper the reli-
able estimation of a large number of covariance components simultaneously.
However, developments in methodologies to model higher-dimensional data more
parsimoniously (Kirkpatrick and Meyer, 2004) implies that such multivariate
analysis is more feasible in terms of parameter estimation and therefore genetic
evaluation.
Reducing the dimension of multivariate analysis includes methods such as canon-
ical transformation and Cholesky decomposition, which involve the transformation
of the vector of observations in addition to residual and genetic covariance matrices.
Other approaches, such as principal component analysis and factor analysis, only
involve reducing the rank of the genetic covariance matrix. Initially, methods that
include the transformation of the vector of observations are discussed.

6.2 Canonical Transformation

In the example discussed in Section 5.2.2 both traits were affected by the same fixed
effect and all animals were measured for both traits. Thus the design matrices
X and Z were the same for both traits or, in other words, the traits are said to have
equal design matrices. In addition there was only one random effect (animal effect)
for each trait apart from the residual effect. Under these circumstances, the multi-
variate analysis can be simplified into n (number of traits) single trait analyses
through what is called a canonical transformation (Thompson, 1977b). Canonical
transformation involves using special matrices to transform the observations on
several correlated traits into new variables that are uncorrelated with each other.
These new variables are analysed by the usual methods for single trait evaluation,
but the results (predictions) are transformed back to the original scale of the obser-
vations. Ducrocq and Besbes (1993) have presented a methodology for applying
canonical transformation when design matrices are equal for all traits but with
some animals having missing traits; details of the methodology, together with an
illustration, are given in Appendix E, Section E.2.

© R.A. Mrode 2014. Linear Models for the Prediction of Animal Breeding Values, 95
3rd Edition (R.A. Mrode)