Page 122 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

−1
–1
in the second row of Eqn 6.11 is replaced by Z * ′R Z * + (D ⊗ A ), where D is a
n n
diagonal matrix of eigenvalues. Again, the last row of Eqn 6.11 is omitted. It there-
fore involves decomposing G to a matrix of eigenvectors (Z * ) and corresponding
eigenvalues (D). Thus Z * and D, respectively, are:
⎛ 0.7710 0.3896 −0.02940 −0.5029⎞
⎜ −0.5983 0.7139 −0.0268 −0.3628 ⎟
Z = ⎜ ⎟ ⎟ and
*
⎜ 0..0865 0.2427 0.9288 0.2664⎟
⎜ ⎝ 0.2000 0.5288 − 0.3685 0.7379⎠ ⎟
D = diag (8 8159 67 6963 22 8286 17 6592. . . . )
−1
*
*
Thus Z ′R Z for animal i is:
i i
⎛ 0.042 −0.009 −0.003 0.004⎞
⎜ −0.009 0.018 −0.004 −0.007 ⎟
ZR Z = ⎜ ⎟
′
−1
*
*
i i
⎜ −0..003 − 0.004 0.018 0.001⎟
⎜ ⎝ 0.004 − 0.007 0.001 0.030⎠ ⎟
The MME are set up as usual. Similar again to the FA model, the PC has 40 equations
and 388 non-zero elements. The solutions for the various effects from solving the
MME are:
Solutions for sex of calf effects

WWG PWG MSC BFAT
M 4.352 6.795 9.412 0.231
F 3.488 5.959 7.095 0.535

Animal solutions
Untransformed solutions Transformed solutions
WWG PWG BFAT MSC WWG PWG MSC BFAT

1 −0.032 0.287 0.303 −0.032 0.094 0.227 0.340 0.010
2 −0.009 −0.038 0.314 0.118 −0.090 −0.073 0.313 −0.050
3 0.031 −0.163 0.047 0.089 −0.086 −0.169 0.030 −0.032
4 −0.002 0.015 −0.844 −0.279 0.170 0.136 −0.855 0.113
5 0.045 −0.511 −0.473 0.078 −0.190 −0.407 −0.539 −0.029
6 −0.062 0.496 1.276 0.186 0.014 0.290 1.350 −0.083
7 0.007 −0.571 −0.732 0.022 −0.207 −0.400 −0.812 −0.015
8 −0.018 0.413 1.362 0.252 −0.019 0.178 1.431 −0.101

6.4.3 Analysis with reduced rank PC model

The diagonal matrix D with the full PC model in Section 6.4.2 indicates that the first prin-
cipal component accounts for about 8.82% of the total genetic variance. Deleting the first

106 Chapter 6