Page 233 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

4. If both parents and subclass are unknown, add 1 to the individual diagonal.
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5. For sire × dam subclasses, the contribution of the ith subclass to U is the same
as for the inverse of the dominance matrix (see rule 3 in Section 12.4.1) except that
the coefficients are multiplied by 8.
6. Sort coefficients by row and by columns within a row, and sum coefficients with
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identical row and columns to obtain U .
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The method can be verified by inverting U to form U. The animal by animal sub-
matrix of U should be equal to the epistatic relationship matrix calculated as A#A.
12.5.2 Calculating the inverse relationship matrix for epistasis
and the subclass matrix for an example pedigree

Example 12.4
The calculation of U is illustrated below using the pedigree information in
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Example 12.1.
The identification of sire and dam subclasses and their ancestors subclasses
treated as known has been discussed in Section 12.4.3. Thus the list of animals and
known subclasses is:

Animal Sire Dam
1 0 0
2 0 0
3 0 0
4 0 0
5 1 2
6 3 4
7 6 5
8 0 5
9 3 8
10 3 8
11 6 8
12 6 8

Subclasses Parent subclasses
6,8 6,5, 3,8, 3,5
6,5 3,8, 3,5
3,8 3,5
3,4
1,2
3,5

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In setting up U , animals 1 to 12 have been regarded as rows 1 to 12 while sub-
classes have been assigned rows 13 (subclass (6,8)) to 18 (subclass (3,5)). The first
four animals have both parents and sire–dam subclasses unknown and therefore each
contributes 1 to their respective diagonals. The parents of animals 5, 6, 7, 10, 11 and
12 and their sire × dam subclass effects are known; therefore, the contributions of

Non-additive Animal Models 217