Page 297 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

ˆ
where b are current solutions for level i of sex effect.
i
When all pedigree and yield records for the kth animal have been processed, the
solution for the animal is computed as:
ˆ
u = ARHS /DIAG
k k k
For the example data, the solutions for animal 5 in the first round of iteration is
computed as follows.
Contribution to diagonal from pedigree is:
DIAG = (2 + 0.5)a = 5.00
5
Accounting for yield record, diagonal becomes:
DIAG = 5.00 + 1 = 6.00
5
Contribution to RHS from yield is:

ARHS = 2.9 − 3.40 = −0.5
5
Contribution to RHS from parents and progeny (pedigree) is:
ARHS = ARHS + a(uˆ + uˆ ) + a(uˆ − 0.5(uˆ ))
5 5 2 3 7 4
= −0.5 + 2(−0.083 + (−0.021)) + 2(−0.833 − 0.5(−0.119))
= −2.255
and:

ˆ
u = −2.255/6.00 = −0.376
5
When all animals have been processed, the current round of iteration is com-
pleted. However, the iteration process is continued for sex and animal effects until
convergence is achieved. The convergence criterion can be defined as in Section 17.3.1.
In this example, solutions were said to have converged when the sum of squares of
differences between the current and previous solutions divided by the sum of squares
−7
of the current solution was less than 10 . The solutions for all effects in the first
round of iteration and at convergence at the 20th iteration are as follows:

Solutions
Effects At round 1 At convergence
Sex
Male 4.333 4.359
Female 3.400 3.404
Animal
1 0.333 0.098
2 −0.083 −0.019
3 −0.021 −0.041
4 −0.119 −0.009
5 −0.376 −0.186
6 0.392 0.177
7 −0.364 −0.249
8 0.282 0.183

Solving Linear Equations 281