Page 227 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The diagonal elements R can be obtained from Eqn 12.8. The off-diagonals are
zeros if all ancestor subclasses providing relationship ties are included in f. To ensure
a diagonal R, Hoeschele and VanRaden specified two conditions to be used in decid-
ing which subclasses should be included in f as known. These are:

1. A subclass should remain in f if any of its parent subclasses remain in f.
2. A subclass should remain in f if f contains two or more of its immediate progeny
subclasses.
−1
Equation 12.14 implies that F can be calculated from a list of subclasses and
ii
their parent subclass effects by computing for the ith subclass, r (the diagonal ele-
−1
ment i of R ) and c (the ith row of (I − Q)). Then the contribution of the ith subclass
i
−1
ii
to F is calculated as c c ′r . In summary, the following procedure could therefore be
i i
used to calculate F :
−1
1. List animals and their sires and dams. Parents not in the list of animals with more
than one progeny should be added to the list while those with one progeny may be
treated as unknown.
2. Form a list of all filled (S and D known) subclasses and add ancestor subclasses that
provide ties. Ancestors are identified by listing subclasses for the sire with parents of the
dam and for the dam with parents of the sire for each filled subclass and then repeating
this process for the subclasses just added until no further ancestors are known. The same
sex subclasses of animal i with animal j and of animal j with animal i should be treated
as identical when listing ancestor subclasses. The list of subclasses is sorted such that
progeny subclass precedes its parent subclasses. Commencing with the oldest ancestor
subclass, subclasses could be regarded as unknown if they are not filled, have no known
parents and provide no ties for at least two filled descendant subclasses.
The number of connections provided by an ancestor subclass may be approxi-
mately determined from counts formed when ancestor subclasses are being identi-
fied originally. Progeny subclass (f ) would contribute 1 to parent subclasses of
SD
type f and f but −1 to parent subclasses of type f . The substraction of 1
S,SD SS,D SS,SD
is due to the fact that f and f are regarded as progeny subclasses of f and
S,SD SS,D SS,SD
both may have come from one f . It should be noted, however, that some sub-
SD
classes which should be deleted for having a count of less than 2 may be needed in
order to achieve a diagonal R. Thus if both f and f are known, for instance,
S,SD SS,D
it may be necessary to add back subclasses of type f if they have been deleted
SS,SD
for a count of less than 2.
3. Go through the list of all subclasses and calculate contributions (coefficients) of
11
−1
each subclass i to F as r c c ′. The vector c contains non-zero coefficients, which is
i i i
equal to 1 in subclass i and equal to −b for parent subclasses, with b computed as in
Eqn 12.7.
4. Sort the coefficients by columns within rows and sum those with identical columns
−1
and rows to obtain F .
12.4.2 Prediction of dominance effects

So far, the discussion has been on the inverse of the relationship matrix for subclass
effects but the major interest is the prediction of dominance effects.

Non-additive Animal Models 211