Page 192 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

p
The marker haplotype inherited by calf 4 from its sire is v ; therefore, r = q in
s21 o
Eqn 10.7. Thus:
g( ) = (1 − q)g( ) + qg( ) = p(1) + q(0) = p = 0.8
4p4p,1p1p 1p1p,1p1p 1m1m,1p1p
g( ) = (1 − q)g( ) + qg( ) = p(0) + q(1) = q = 0.2
4p4p,1m1m 1p1p,1m1m 1m1m,1m1m
g( ) = (1 − q)g( ) + qg( ) = p(0) + q(0) = 0
4p4p,2p2p 1p1p,2p2p 1m1m,2p2p
g( ) = (1 − q)g( ) + qg( ) = p(0) + q(0) = 0
4p4p,2m2m 1p1p,2m2m 1m1m,2m2m
g( ) = (1 − q)g( ) + qg( ) = p(q) + q(p) = 2pq = 0.32
4p4p,3m3m 1p1p,3m3m 1m1m,3m3m
The inverse of G is:
v
é 3.125 1.000 0.000 0.000 -0.625 -2.500 0.000ù
ê 0.000 - 2.500 - ú
0
ê 1.000 3.125 0.000 0.625 0.000 ú
ê 0.000 0.000 1.000 0.000 0.000 0.000 0.000ú
0
ê
-1
G = 0.000 0.000 0.000 3.000 0.500 0.000 - 2.500 ú
v ê ú
ê - 0.625 - 2.500 0.000 0.5000 3.250 0.000 - 0.625 ú
ê ú
- ê 2.500 - 0.625 0.000 0.000 0.000 3.125 0.000 ú
ê ú
ë 0.0000 0.000 0.000 - 2.500 - 0.625 0.000 3.125 û
The matrix G was computed using the rules outlined earlier. Thus for the first two
−1
animals (first four alleles), add 1 to the diagonal elements since parents of both calves
are unknown. For paternal allele of calf 3, add 1/2pq to the diagonal element
(3p3p,3p3p) of G , q/2p to element (1p1p,1p1p), p/2q to element (1m1m,1m1m),
−1
−1/2p to elements (1p1p,3p3p) and (3p3p,1p1p), −1/2q to elements (1m1m,3p3p)
and (3p3p,1m1m) and 0.5 to the elements (1p1p,1m1m) and (1m1m,1p1p).
−1
The matrix A for the example data can be calculated using the usual rules; therefore,
the MME can easily be set up from the design matrices and inverse of the covariance
matrices given. Solving the MME by direct inversion gave the following results:
Effects Solutions
Sex of calf
Male 7.475
Female 5.091
Breeding values for animals
1 0.034
2 –0.034
3 0.246
4 0.280
Additive effects for animals at the MQTL
1p –0.008
1m 0.005
2p –0.047
2m 0.049
3p 0.024
4p 0.010
4m 0.059

A similar model to that in Example 10.5 has been used by Boichard et al. (2002)
for incorporating MQTL information into genetic evaluation for milk production
traits in young bulls.

176 Chapter 10