Page 244 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The matrices in Eqn 13.4 can now be obtained by matrix multiplication and A −1
−1
−1
is added to Z′WZ. The matrix Z′WZ + A G is illustrated below:
⎡ 29.783773 0.000000 −12.666731 0.000000⎤
⎢ 0.000000 244.572445 0.000000 0.000000 ⎥
+
′
−1
Z WZ A G −1 = ⎢ ⎥
− ⎢ 12.666731 0.000000 35.566685 − 12.6666731⎥
⎢ ⎥
⎣ 0.000000 0.000000 − 12.666731 29.278162 ⎦
Then Eqn 13.4 is:
é 25.073 - 12.567 - 5.733 - 6.773 - 6.366 - 6.140 - 3.123 - 3.977 - 2.699 - 2.7707ù
ê ú
ê - 12.567 17.928 - 2.146 - 3.215 - 3.220 - 2.141 - 1.327 - 1.595 - 1.201 1 - 1.238 ú
ê - 5.733 - 2.146 7.879 0.000 4.595 3.284 1.796 3.550 2.533 0.0000ú
ê ú ú
ê - 6.773 - 3.215 0.000 9.989 4.992 4.997 2.655 2.022 1.367 3.945 ú
ê - 6.3366 - 3.220 4.595 4.992 9.586 0.000 2.698 2.062 2.710 2.117 ú
ê ú
ê - 6.140 - 2..141 3.284 4.997 0.000 8.281 1.753 3.511 1.190 1.828ú
ê ú
ê - 3.123 - 1.327 1..796 2.655 2.698 1.753 29.784 0.000 - 12.667 0.000 ú
ê - 3.977 - 1.595 3.5550 2.022 2.062 3.511 0.000 24.572 0.000 0.000ú
ê ú
ê - 2.699 - 1.201 2.533 1.3367 2.710 1.190 - 12.667 0.000 35.567 - 12.667 ú
ê ë - 2.707 - 1.238 0.000 3.9945 2.117 1.828 0.000 0.000 - 12.667 29.278 û ú
é D ˆ t ù é -0.289ù
ê 1 ú ê ú
ê D ˆ t 2 ú ê 0.459 ú
ê ˆ ú ê -0.021ú
ê ê Dh 1 ú ê ú
ê ˆ ú ê -0.149 ú
ê Dh 2 ú ê ú
ê D ˆ h ú -0..099 ú ú
1 = ê
ê ú ê - 0.071ú
ê D ˆ h 2 ú ê ú
ê D ˆ u ú ê - 0.104 ú
ê 1 ú ê - 0.021ú
ê D ˆ u ú ê ú
2
ê ú ê 0.031 ú
ê D ˆ u 3 ú ê ú
ê ë D ˆ u ú ë - 0.076 û
4 û
ˆ
The equations were solved with the solutions for Dh and Dh set to zero. The
1 1
equations converged rapidly, and solutions at various different iteration numbers
and the final solutions are given below. Solution from an analysis using a linear model
with an a value of 19 are also shown:
Iteration number
Solutions from
Effects 1 2 3 7 linear models
Threshold
1 0.4410 0.4375 0.4378 0.4378 ± 0.44 a –
2 1.0448 1.0661 1.0675 1.0675 ± 0.47 –
Herd–year
1 0.0000 0.0000 0.0000 0.0000 ± 0.00 0.0
2 0.2869 0.2763 0.2774 0.2774 ± 0.49 1.0604
Continued

228 Chapter 13