Page 223 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

and its inverse is:
é 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0000 0.000 ù
ê ú
ê 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0 0.000 0.000 ú
ê 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0..000 0.000 0.000 ú
ê ú
ê 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.0000 0.000 0.000 0.000 ú
ê 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0 0.000 0.000 0.000 0.000 ú
ê ú
ê 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.0000 0.000 0.000 0.000 0.000 ú
-1
D = ê ú
8
ê 0.000 0.000 0.000 0.000 0.000 0.000 1.028 0.000 - 0.032 - 0.032 - 0.096 - 0.096 ú
ê 0.000 0.000 0.000 0.000 0.000 0.0000 0.000 1.000 0.000 0.000 0.000 0.000 ú
ê ú
ê 0.000 0.000 0.000 0.000 0.000 0.000 - 0.032 0.000 1.084 - 0.249 - 0.080 - 0.080 ú
0
ê ú
ê 0.000 0.000 0.000 0.0000 0.000 0.000 - 0.032 0.000 - 0.249 1.084 - 0.080 - 0.080 ú
ê 0.080 0.241 ú
ê 0.000 0.000 00.000 0.000 0.000 0.000 - 0.096 0.000 - 0.080 - 1.092 - ú
ë 0.0000 0.000 0.000 0.000 0.000 0.000 - 0.096 0.000 - 0.080 - 0.080 - 0.241 1..092 û
−1
The matrices A a and D a are added to Z′Z and W′W in the MME. The MME
−1
1 2
are of the order 26 by 26 and are too large to be presented. However, the solutions
to the MME by direct inversion of the coefficient matrix are:
Effects Solutions
Sex
Female 16.980
Male 20.030
Animal BV a DV a
1 −0.160 0.000
2 −0.160 0.000
3 0.059 0.000
4 0.819 0.000
5 −0.320 0.136
6 1.259 0.705
7 0.555 0.237
8 −0.998 −0.993
9 −0.350 0.000
10 −1.350 −1.333
11 1.061 1.428
12 −0.039 −0.038

a BV, DV, solutions for random animal and dominance effects,
respectively.

The results indicate that males were heavier than females by about 3.05 kg
ˆ
at weaning. The breeding value for animal i, a, from the MME can be calcu-
i
lated using Eqn 3.8, except that yield deviation is corrected not only for fixed effects
but also for dominance effect. Thus the solution for animal 6 can be calculated as:
ˆ
ˆ
a = n ((aˆ + aˆ )/2) + n (y − b − d ) + n (2aˆ − aˆ ) + n (2aˆ − aˆ ) + n (aˆ − aˆ )
ˆ
6 1 3 4 2 6 1 6 3 12 8 3 11 8 3 7 5
= n (0.059 + 0.819)/2 + n (20 − 16.980 − 0.705) + n (2(−0.039) − (−0.998))
1 2 3
+ n (2(1.061) − (−0.998)) + n (2(0.555) − (−0.320))
3 3
= 1.259
Non-additive Animal Models 207