Page 312 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

After processing all animals with records, the contribution for animals in the
pedigree without records is accumulated. The equations for accumulating contribu-
tions for these animals is the same as shown above except that the coefficient for d(i) in
Eqns 17.13 and 17.14 is zero instead of one, indicating no contribution from records.
For example, for animal 4 with only the sire known:
v(4) = v(4) + d(1) + M (d(anim )) + (−2/3)a(d(anim )) = 8.833
4,4 4 1
Add contribution from progeny when processing the record for animal 7:

v(4) = 8.833 + −1.0a(d(anim )) + 0.25a(d(anim )) = 7.917
7 5
The vector v for all effects is:
v′ = [15.664 7.933 −2.586 −2.267 −2.317 7.917 5.864 5.300 4.938 8.033]
Next w is computed using matrix multiplication and scalar division as:

w = 95.1793/120.255 = 0.7915
(1)
The solution vector is then computed as b = b + wd . The vector b is:
(0)
(1)
(0)
b′ = [3.430 2.691 0.0 0.0 0.0 0.763 0.383 0.514 0.554 0.791]
(1)
(0)
The updated vector of residuals e is computed as e − wv. For the example data e (1)
(1)
is:
(1)
e′ =[0.602 0.521 2.047 1.794 1.834 −1.766 −1.741 −0.295 −0.408 −1.358]
−1 (1)
The vector v is then computed as M e . For the example data, v is:
v′= [0.201 0.260 0.558 0.449 0.458 −0.378 −0.290 −0.049 −0.082 −0.272]
Next, compute the scalar b. The denominator of b is equal to the numerator of
w and this has already been computed. Using the example data:
b = 4.634/95.179 = 0.0487
(1)
Finally, d , the search-direction vector for the next iteration is computed as
(0)
v + bd . This vector for the example data is:
(1)
d′ = [0.412 0.426 0.558 0.449 0.458 −0.331 −0.267 −0.017 −0.048 −0.223]
The next cycle of iteration is continued until the system of equations converges.
Convergence can either be monitored using the criteria defined in Example 17.1 or
the relative difference between the right-hand and left-hand sides:
−
yCb ( r+ )1
r
c () =
d
y
where:
i ( ) 1 2
2
x =Σx
i
Using the convergence criteria used in Example 17.1, the iteration was stopped at
the 10th iteration when equations converged to 8.3 −07 . Some intermediary and final
solutions are shown in the following table.

296 Chapter 17