Page 71 - Linear Models for the Prediction of Animal Breeding Values

2
zero and common variance s . The breeding values of animals in subsequent generations
a
are usually expressed relative to those of the base animals. However, if it is known that base
animals were actually from populations that differ in genetic means, for instance, sires from
different countries, this must then be accounted for in the model. In the dairy cattle situation,
due to differences in selection intensity, the genetic means for sires of bulls, sires of cows,
dams of bulls and dams of cows may all be different. These various sub-population structures
should be accounted for in the model to avoid bias in the prediction of breeding values. This
can be achieved through a proper grouping of base animals using available information.
Westell and Van Vleck (1987) presented a procedure for grouping, which has gen-
erally been adopted. For instance, if sires have been imported from several countries
over a period of time and their ancestors are unknown, these sires could be assigned
to groups on the basis of the expected year of birth of the ancestors and the country
of origin. The sires born within a similar time period in a particular foreign country
are assumed to come from ancestors of similar genetic merit. Thus each sire with one
or both parents unknown is initially assigned phantom parents. Phantom parents are
assumed to have had only one progeny each. Within each of the foreign countries, the
phantom parents are grouped by the year of birth of their progeny and any other fac-
tor, such as sex of progeny. In addition, for the dairy cattle situation, the four selection
paths – sire of sires, sire of dams, dam of sires and dam of dams – are usually assumed
to be of different genetic merit and this is accounted for in the grouping strategy.
With groups, the model (Thompson, 1979) is:
n
i ∑
ij y = h + a + t ik k g + ij e (3.27)
j
k=1
where h = effect of the jth herd, a = random effect of animal i, g = fixed group effect
j
k
i
containing the kth ancestor, t = the additive genetic relationship between the kth and
ik
ith animals and the summation is over all n ancestors of animal i, and e = random
ij
environmental effect. From the model, it can be seen that the contribution of the
group to the observation is weighted by the proportion of genes the ancestors in the
group passed on to the animal with a record.
In matrix notation, the model can be written as:
y = Xb + ZQg + Za + e (3.28)
where:
Q = TQ *
Q * assigns unidentified ancestors to groups and T, a lower triangular matrix, is obtained
from A = TDT′ (see Section 2.3). With this model the breeding value of an animal k is:

a = Qgˆ + a ˆ
k* k
The MME are:
′
′
′
⎡ XX XZ XZQ ⎤ ⎡ ˆ ⎤ ⎡ Xy
′ ⎤
b
⎢ − 1 ′ ⎥ ⎢ ⎥ ⎢ ′ ⎥
a ˆ =
′ +
⎢ Z ′X ZZ A a ZZQ ⎥ ⎢ ⎥ ⎢ Zy ⎥
′′ ⎥
′′
′′
′′
⎣ ⎢QZX Q ZZ QZZQ ⎥ g ˆ ⎢ ⎥ ⎢QZy ⎦
⎣
⎣
⎦ ⎣ ⎦
Solving the MME above will yield vectors of solutions for a and g but the ranking
criterion (breeding value) is a = Qˆg + aˆ for animal k. Modification of the MME
ˆ
k* k
Univariate Models with One Random Effect 55