Page 62 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

on calves 4 and 5 is very similar; each has a record, a common sire and parents of the
same progeny, hence they have the same reliability. Calf 8 has the highest reliability
and this is due to the information from the parents (its sire has another progeny and
the dam has both parents known) and its record. The standard errors are large due
to the small size of the data set but follow the same pattern as the reliabilities.
In practice, obtaining the inverse of the MME for large populations is not feasible
and various methods have been used to approximate the diagonal element of the
inverse. A methodology published by Meyer (1989) is presented in Appendix D and
was used in the national dairy evaluation programme in Canada (Wiggans et al.,
1992) in the 1990s.

3.4 A Sire Model

The application of a sire model implies that only sires are being evaluated using prog-
eny records. Most early applications of BLUP for the prediction of breeding values,
especially in dairy cattle, were based on a sire model. The main advantage with a sire
model is that the number of equations is reduced compared with an animal model
since only sires are evaluated. However, with a sire model, the genetic merit of the
mate (dam of progeny) is not accounted for. It is assumed that all mates are of similar
genetic merit and this can result in bias in the predicted breeding values if there is
preferential mating.
The sire model in matrix notation is:
y = Xb + Zs + e (3.15)
All terms in Eqn 3.15 are as defined for Eqn 3.1 and s is the vector of random sire
effects, Z now relates records to sires and:
2
var(s) = As s
2
var(y) = ZAZ′s + R
s
2
2
2
where A is the numerator relationship matrix for sires, s = 0.25s and R = Is . The
s a e
2
2
2
2
MME are exactly the same as in Eqn 3.4 except that a = s /s = (4 − h )/h .
e s
3.4.1 An illustration
Example 3.2
An application of a sire model is illustrated below using the same data as for the
animal model evaluation in Table 3.1. Assigning records to sires, and including the
pedigree for sires, the data can be presented as:
Sex of progeny Sire Sire of sire Dam of sire WWG (kg)
Male 1 – – 4.5
Female 3 – – 2.9
Female 1 – – 3.9
Male 4 1 – 3.5
Male 3 – – 5.0

46 Chapter 3