Page 35 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

very popular in the last decade. In addition to the recognition that more than one
trait contributes to profitability, the broadening of selection goals has also been due
to the need to incorporate health and welfare traits to accommodate public con-
cerns. Examples of indices constructed with PTAs or BVs of several traits and used
in genetic improvement of dairy cattle include production index (PIN), combining
PTAs for milk, fat and protein in the UK, production life index (PLI), which is PIN
plus PTAs for longevity and somatic cell count in the UK; and in the Netherlands,
index net (INET), which combines BVs for milk, fat and protein and durable per-
formance sum (DPS), which is INET plus durability (Interbull, 2000). The principles
for calculating these indices are similar to those outlined in previous sections. Given
that the PTAs or BVs are from a complete multivariate analysis, the optimal index
weights (b) are the sum of the partial regression coefficients of each goal trait on
each index trait, weighted by the economic value of the goal trait (Veerkamp et al.,
1995). Thus given m traits in the selection goal and n traits in the index, then the
partial regressions can be calculated as:
−1
R = G G
ig
and:
b = Rw
where R is a matrix of partial genetic regression, G is the matrix of genetic covari-
ig
ance between m goal and n index traits, G is the genetic covariance matrix between
the index traits, and w is the vector of economic weights. It is obvious that when
goal and index traits are the same, G = G and b = w. In the case where the index
ig
and goal traits are not the same, R can be estimated directly from a regression of
phenotype on the EBVs for the index traits (Brotherstone and Hill, 1991). However,
if PTAs or BVs are from a univariate analysis, rather than from a multivariate analy-
sis, the use of b above results only in minimal loss of efficiency in the index
(Veerkamp et al., 1995).
Selection based on breeding values from BLUP is usually associated with an
increased rate of inbreeding as it favours the selection of closely related individu-
als. Quadratic indices can be used to optimize the rate of genetic gain and
inbreeding. This does not fall within the main subject area of this text and inter-
ested readers should see the work by Meuwissen (1997) and Grundy et al. (1998).

1.7.5 Restricted selection index

Restricted selection index is used when the aim is to maximize selection for a given
aggregate genotype, subject to the restriction that no genetic change is desired in
one or more of the traits in the index for H. This is achieved by the usual index
procedure and setting the covariance between the index and the breeding value
(cov(I, a ) for the ith trait specified not to change to zero. It was Kempthorne and
i
Nordskog (1959) who introduced the idea of imposing restrictions on the general
index procedure.
For instance consider the aggregate genotype composed of two traits:
H = w a + w a
1 1 2 2

Genetic Evaluation with Different Sources of Records 19