Page 304 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

In processing a type 2 record in the pedigree file for the kth animal, contributions
to DIAG and ARHS according to whether the mate of k is known are:

Mate is unknown Mate is known
2
ARHS = ARHS + g (uˆ ) ARHS = ARHS + (u − 0.5u )g 11
11
ˆ
ˆ
k k 3 o k k o ma
1
1
DIAG = DIAG + g 11 DIAG = DIAG + g 11
k k 3 k k 2
where u and u are current solutions for direct effects for the progeny and mate of
ˆ
ˆ
o ma
the animal k.
Accounting for contributions from the maternal effect to ARHS:
Mate is unknown Mate is known
2
12
ARHS = ARHS + g (mˆ ) ARHS = ARHS + (m − 0.5m ˆ )g 12
ˆ
k k 3 o k k o ma
ˆ
ˆ
ARHS = ARHS − (m )1/3g 12 ARHS = ARHS − (m )½g 12
k k k k k k
ˆ
where m and m are current maternal solutions for the progeny and mate of the
ˆ
o ma
animal k.
If the animal has a yield record:
−1
DIAG = DIAG + r if it is a parent
k k n
or:
−1
DIAG = DIAG + (r )0.5 if it is a non-parent
k k n
The diagonals of non-parents are multiplied by 0.5 instead of 0.25 because records
of non-parents have been written twice (see Section 17.4).
Contributions to the RHS are accumulated as:
ˆ
ˆ
−1
ˆ
ARHS = ARHS + r (y − hd − b − m − p ˆ e )
k k pa ijklt i j l t
for parent records and:
ˆ
ˆ
−1
ˆ
ˆ
ARHS = ARHS + r (y − hd − b − 0.5(u ) − m − p ˆ e
k k np ijklt i j ma l t
ˆ
ˆ
ˆ
ˆ
for non-parent records. In the equations above, hd , b, m , p ˆ e and u are current
i j l t ma
solutions for herd i, jth level for sex effect, lth maternal effect level, tth level of per-
manent environment effect and animal solution for the other parent (mate), respec-
tively. The solution for animal k is computed as usual when all records for the animal
in the pedigree and data file have been read as:
ˆ
u = ARHS /DIAG
k k k
The solution for animal 2 in the example data in the first round of iteration is as
follows.
The contribution to the diagonal from pedigree is:
DIAG = (1 + + )0.00756 = 0.01512
1
1
2 2 2
288 Chapter 17