Page 100 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

5.5 Multivariate Models with No Environmental Covariance

In some cases, a multivariate analysis may be necessary when individual animals have
records for one trait (or subset of traits) but relatives have records on a different trait (or
subset of traits). For instance, in beef cattle, if selection is for dual-purpose sires, male
and female calves might be reared in different environments (different feedlots) and body
weight recorded in male calves and milk yield in female calves. The evaluation of the
sires will be based on multivariate analysis of these two traits. A special feature of such
a multivariate analysis is that there is no environmental covariance between the traits as
the two traits are not observed in the same individual. In Section 5.5.1, the details of
such a model are discussed and its application to example data is illustrated.
Also, when the same trait is measured on relatives in different environments
such that the genetic correlation between performances in the two environments is
not one, a multivariate analysis might be the optimum means to evaluate sires. For
example, milk yield may be recorded on the daughters of a bull in two different
environments, say, in a tropical environment and a temperate environment. Such
a multivariate analysis will treat milk yield in the various environments as different
traits. However, as the number of environments increases, the data might be associ-
ated with a heterogeneous fixed effects structure that might be difficult to model
correctly in multivariate analysis, such that it might be useful, for practical purposes
of implementation, to analyse not the original data but summaries of the data. A very
good illustration of such a multivariate analysis is the multi-trait sire model used by
the international bull evaluation service Interbull (Uppsala, Sweden), for the across-
country evaluation of dairy sires. This multi-trait sire model, commonly referred to
as MACE (multi-trait across-country evaluations), analyses deregressed breeding
values (DRB) of sires in different countries as different traits. The use of DRB could
be regarded as utilizing a variable that summarizes the daughter performances of
bulls in different countries. This avoids the need to model at the Interbull centre the
heterogeneous fixed effects structure, such as different herd management systems
and complex national climatic conditions associated with the daughters’ milk per-
formance records in the different countries. MACE plays a very important role in
the international trade of dairy cattle and in Section 5.5.2 the model for MACE is
discussed and illustrated.

5.5.1 Different traits recorded on relatives

Defining the model

In this situation, with different traits recorded on relatives in different environments,
the different traits are not observed on the same individual, and so there is not
environmental covariance between the traits. Therefore, the residual covariance
matrix R is diagonal. Thus for n traits:
2
2
2
R = diag(s , s ,...,s ) = diag(r , r ,...,r )
e1 e2 en 11 22 nn
and:
nn
22
−1
11
R = diag(r , r ,...,r )
84 Chapter 5