Page 104 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

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Then A can be obtained by the rules outlined in Section 2.4, which can be briefly
summarized in the table below, taking into account the contribution to the groups for
MGDs. Given a list of pedigrees with the ith line consisting of a bull, its sire or group,
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its MGS or group and a group for its MGD, then contributions to A are as follows:

Bull Sire MGS MGD
Bull d −0.5d −0.25d −0.25d
Sire −0.5d 0.25d 0.125d 0.125d
MGS −0.25d 0.125d 0.0625d 0.0625d
MGD −0.25d 0.125d 0.0626d 0.0625d

where d = 16/(11 + m) and m = 0 if both sire and MGS are known, m = 1 if the sire
is known but MGS is unknown, m = 4 if the sire is unknown and the MGS is known,
and m = 5 if both sire and MGS are unknown.
Usually there are dependencies among group effect equations and 1 is added to the
diagonals of the phantom group effects in the inverse of the relationship matrix to over-
come these dependencies. Then the group solutions sum to zero, and so the solutions for
bulls are relative to the same genetic base within each country. The addition of 1 to the
diagonals of the phantom groups implies that group effects are random, with expected
values of zero. Since group effects represent differences in the effects of previous selec-
tion, which should not have expected values of zero, Schaeffer (1994) indicated that this
approach could also be regarded as a biased estimation of the fixed effects of phantom
groups. That is, a small amount of bias in the estimates of the phantom groups is
accepted in exchange for the hope of getting estimates with smaller mean square errors.

Computing effective daughter contribution

The use of EDC instead of the number of daughters as a weighting factor was proposed
by Fiske and Banos (2001) from a simulation study in which they demonstrated that using
the numbers of daughters resulted in biased estimates of sire variances used in MACE and
international reliabilities. The computation of EDC for a bull accounts for such factors as
contemporary group (CG) structure for the bull’s daughters, the correlation between
observations on the same daughter and the reliability of the performance of the daughters’
dams. Thus the EDC provides a measure of the precision of the daughter information used
to compute the DRP of the bull. The formula for the computation of EDC (Fiske and
Banos, 2001), which included the performance of the dam of the daughter k of bull i is:

i ∑
EDC = l rel ko ()
k 4 − rel ko () · (1 + rel damo () )
2
2
where the summation is over all the k daughters of the bull, l = (4 − h )/h , rel
dam(o)
is the reliability of the dam’s own performance, rel is the reliability of the animal
k(o)
k’s own performance computed as:
nh 2
rel = k
ko ()
1 + n ( k − r ) 1
with r being the reliability of the animal’s records, n the number of lactations of the
k
daughter k of the sire adjusted for the CG size computed as:
88 Chapter 5