Page 240 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The vectors of solutions in Eqn 13.4 for the example data are:
t′ = (t t ), since there are two thresholds
1 2
b′ =(h h h h )
1 2 1 2
u′ = (u u u u )
1 2 3 4
where h and h represent solutions for level i of herd–year and the sex of calf effects,
i i
respectively; and u is the vector of solutions for sires.
The inverse of the relationship for the assumed pedigree is:
⎡ 1.3333 0.0000 −0.6667 0.0000⎤
⎢ ⎥
−1 = ⎢ 0.0000 1.0000 0.0000 0.0000 ⎥ ⎥
A
⎢ −0.66667 0.0000 1.6667 − 0.6667⎥
⎢ ⎥
⎣ 0.0000 0.0000 − 0.6667 1.3333 ⎦
For the example data, the transpose of matrix X, which relates subclasses to herd–
year and sex of calf effects, and that of matrix Z, which relates subclasses to sires, are:
⎡ 111111111 00000000000⎤
⎢ ⎥
X′= ⎢ 000000000 11111111111 ⎥
⎢ 10 10 10 10 10 1 1 0 11 0 11 00 1⎥
⎢ ⎥
⎣ 01 01 01 01 01 0 0 1 0 1 0 01 1 0 ⎦
and:
⎡ 111 000000 111 00000000⎤
⎢ ⎥
Z′= ⎢ 000 111 000000 11 000000 ⎥
⎢ 000000 111 000000 11 0000⎥
⎢ ⎥
⎣ 000000000000000 11111 ⎦

Starting values for t, b and u are needed to commence the iterative process. Let b =
u = 0, but starting values for t can be computed from the proportion of records in all
i
categories of response preceding t. In this example, there is only one category before t
i 1
and 0.679 of the records are in this category. The first two categories precede t and 0.857
2
of the records are observed in both categories. Using these proportions, the values of t can
be obtained from the usual table of standardized normal deviates of the normal distribu-
tion. From these proportions, t = 0.468 and t = 1.080 and these were used as starting
1 2
values. However, using various starting values of t, Gianola and Foulley (1983) demon-
strated that the system of equations converged rapidly. It seems, therefore, that the system
of equations is not very sensitive to starting values for t. The calculations of the various
matrices in the equations have been illustrated below using solutions obtained after the
first iteration. The solutions obtained at the end of the first iteration and the updated
estimates for the effects (which are now the starting values for the second iteration) are:

a
Solutions at the end of iteration one Updated estimates after iteration one
Δt = −0.026992 t = 0.441008
1 1
Δt = −0.035208 t = 1.044792
2 2
Δh ˆ = 0.000000 h ˆ = 0.000000
1 1
Continued
224 Chapter 13