Page 185 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

limited to populations of small size as the tabular method of calculating A and its
v
inverse may not be computationally feasible in large populations.

10.8 Predicting Total Additive Genetic Merit

Van Arendonk (1994) showed that total additive genetic merit (a) for animals that
includes marker information could be predicted directly. This implies that only a
single equation is needed for an animal in the MME to predict breeding values with
marker information included. Let Eqn 10.17 be written as:
y = Xb + Za + e (10.21)
where a = u + Kv with u and v as defined in Eqn 10.17. The matrix K, which relates
allelic effects to animals, is identical to W in Eqn 10.17 when all animals have obser-
vations. The variance–covariance matrix of a (V ) is:
a
V = var(u + Kv)
a
= var(u) + Kvar(v)K′
2 2
u
v
u
= A s + KG K′s v
2 2
u
u
v
= A s + 2A s v
2 2
v
u
u
= A s + A s q
The combined numerator relationship matrix among animals with marker informa-
tion included (A ) is:
a
2 2 2 2 (10.22)
u
a
v
q
u
a
A = A s /s + A s /s a
with:
2 2 2
u
a
s = s + s q
The MME for Eqn 10.21 are:
′
′
⎡XX XZ X ′W ⎤ ⎡ ⎤ ˆ b ⎡Xy
′ ⎤
⎢ −1 ⎥ ⎢ ⎥ = ⎢ ⎥ (10.23)
′
′
⎣ ZX ZZ a 1 Z ′W ⎦ ⎣ ⎢ ⎦ ⎥ a ˆ ⎣ Zy ⎦
′ + A a
where:
2 2
e a
a = s /s
The use of Eqn 10.23 would require the inverse of A to be calculated. Initially,
a
−1
A is computed using Eqn 10.22, then A can be calculated using Eqn 10.11, with
a a
the vector s containing the contributions from ancestors computed using Eqn 10.10.
i
The calculation of both matrices is illustrated in the following example.
10.8.1 Numerical application
Example 10.4
Using the same data set as in Example 10.1 and the same genetic parameters, the total
additive genetic effects of animals, which included marker information, are directly
Use of Genetic Markers in Breeding Value Prediction 169