Page 230 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

−1
The contribution of f to F therefore is:
6,8
1 2 3 6
⎡ 1.00 − 0.50 − 0.50 0.25 ⎤
⎢
11 = − 0.25 − ⎥
cc′ r 0.50 0.25 0.125 1.778
11 ⎢ ⎥
− ⎢ 0.50 0..25 0.25 − 0.125 ⎥
⎢ ⎥
⎣ 0.25 − 0.125 0.125 0.0625 ⎦

11
where r = 1/(1 − (b′ F b )) = 1/(1 − 0.4375) = 1.778 (see Eqn 12.8).
1 1 1
−1
Processing of all subclasses gives F as:
1 2 3 4 5 6
⎡ 1.778 −0.889 −0.889 0.000 0.000 0.445 ⎤
⎢ 0.000 0.000 − ⎥
5
⎢ −0.889 1.778 0.445 0.889 ⎥
⎢
−1 = − 0.000 0.000 − 0.889 ⎥
F 0.889 0.445 1.778
⎢ ⎥
⎢ 0.000 0.0000 0.000 1.000 0.000 0.000 ⎥
⎢ 0.000 0.000 0.000 0.000 1.000 0.000 ⎥
⎢ ⎥
⎣ ⎢ 0.445 −0.889 −0.889 0.000 0.000 1.778 ⎥ ⎦
−
The methodology can be verified by calculating the dominance relationship
matrix among animals as D = (0.25)SFS′ + I(0.75), which should give the same D as
that calculated using Eqn 12.1. S, as defined earlier, relates dominance effects to sub-
class effects. For the example pedigree:

5 6 7 9 10 11 12
⎡
1 0000 0 1 1 ⎤
⎢ ⎥
2 0010 0 0 0 ⎥
⎢
⎢
′ S = 3 000 1 1 0 0 ⎥ ⎥
⎢
4 01 0 0 0 0 0 ⎥
⎢
⎢
5 1 0 0 0 0 0 0 ⎥ ⎥
⎢
6 ⎢ ⎣ 000 0 0 0 0 ⎥ ⎦
0
and:
D = (0.25)SFS′ + I(0.75)
5 6 7 9 10 11 12
⎡ 10 0 0 0 0 0 ⎤
⎢ ⎥
⎢ 01 0 0 0 0 0 ⎥
⎢ 0 0 1 0.0625 0.0625 0.125 0.125 ⎥
= ⎢ ⎥
1
⎢ 0 0 0.0625 1 0.25 0.125 0.125 ⎥
⎢ 0 0 0.0625 0.25 1 0.125 0.125 ⎥
⎢ ⎥
⎢ 0 0 0.125 0.125 0.125 1 0..25 ⎥
⎢ ⎥
⎣ 0 0 0.125 0.125 0.125 0.25 1 ⎦

214 Chapter 12