Page 43 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

As an illustration, the inverse of the relationship matrix in Section 2.2 can be
calculated as below. Initially list all animals in the pedigree:

Calf Sire Dam
1 Unknown Unknown
2 Unknown Unknown
3 1 2
4 1 Unknown
5 4 3
6 5 2
Then set up a 6 × 6 table for the animals. For animals 1 and 2, both parents are
unknown, therefore a = a = 1. Add 1 to their diagonal elements (1,1 and 2,2). For
1 2
animal 3, both parents are known therefore a = 2. Add 2 to the 3,3 element, −1 to the
3
1
(3,1), (1,3), (3,2) and (2,3) elements and to the (1,1), (1,2), (2,1) and (2,2) elements.
2
4
4
For animal 4, only one parent is known, therefore a = . Add to the (4,4) element,
3
3
4
2
1
− to the (4,1) and (1,4) elements and to the (1,1) element. After the first four
3
3
animals, the table is:
1 2 3 4 5 6
1
1 1 + + 1 1 −1 − 2
2 3 2 3
2 1 1 + 1 −1
2 2
3 −1 −1 2
4 − 2 3 4 3
5
6
After applying the relevant rules to animals 5 and 6, the inverse of A then is:
1 2 3 4 5 6
1 1.83 0.5 −1.0 −0.67 0.0 0.0
2 0.5 2.0 −1.0 0.0 0.5 −1.0
3 −1.0 −1.0 2.5 0.5 −1.0 0.0
4 −0.67 0.0 0.5 1.83 −1.0 0.0
5 0.0 0.5 −1.0 −1.0 2.5 −1.0
6 0.0 −1.0 0.0 0.0 −1.0 2.0

Using Eqn 2.3, the inverse of A can be calculated directly. If inbreeding is ignored,
D for the pedigree is:
D = diag(1.0, 1.0, 0.5, 0.75, 0.5, 0.5)

and:
−1
D = diag(1, 1, 2, 1.33, 2,2)

Genetic Covariance Between Relatives 27