Page 123 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

*
*
eigenvalue gives a diagonal D of order 3 as D = diag(67.6963 22.8286 17.6592).
*
*
Then G , a new genetic covariance matrix, can be computed as M¢D M, where M is
equivalent to Z in Section 6.4.2 with a full PC model fitted but with the first column
*
deleted. Thus:
⎛ 14 759 22 067 3 412 7 640⎞
.
.
.
.
⎜ 22 067 36 844 9 456 21 055 ⎟
.
.
.
.
∗
∗
′
G = M D M = ⎜ ⎟ with
⎜ 3412 9 456 24 934 4 347⎟
2
.
.
.
.
⎜ ⎝ 7 640 21 055 4 347 31 647⎠ ⎟
.
.
.
.
⎛ 0 0.3896 − 0.02940 − 0.5029⎞
⎜ 0.7139 − 0.0268 − 0.3628 ⎟
M = ⎜ ⎟
⎜ 0.2427 0.9288 0.26664 ⎟
⎜ ⎝ 0.5288 − 0.3685 0.7379⎠ ⎟
The application of reduced rank PC is similar to the full PC analysis with Z *
replaced by M and D by D *. Thus for animal i, M′ R −1 M is:
i i
⎛ ⎛ 0.018 −0.004 −0.007⎞
⎜
′
−1
MR M = −0.004 0.018 0.001 ⎟
i i ⎜ ⎟
⎝ −0.007 0.001 0.030⎠
The MME for the reduced PC has 32 equations and 284 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 4.349 6.798 9.412 0.230
F 3.480 5.963 7.093 0.533
Solutions for animal effects
Untransformed solutions Transformed solutions a
WWG PWG MSC BFAT
1 0.295 0.305 −0.033 0.123 0.214 0.346 0.019
2 −0.037 0.314 0.118 −0.083 −0.078 0.314 −0.048
3 −0.170 0.046 0.090 −0.113 −0.156 0.025 −0.041
4 0.017 −0.844 −0.279 0.171 0.136 −0.854 0.115
5 −0.523 −0.476 0.080 −0.230 −0.390 −0.548 −0.042
6 0.511 1.279 0.185 0.069 0.263 1.362 −0.066
7 −0.576 −0.734 0.022 −0.215 −0.400 −0.816 −0.018
8 0.419 1.364 0.251 −0.003 0.171 1.435 −0.096
a Transformed solutions = vector of solutions multiplied by M

The deletion of the first eigenvalue in the reduced PC analysis had very little effect
in terms of the EBVs of animals for traits 3 and 4. Thus there was no ranking for MSC

Methods to Reduce the Dimension of Multivariate Models 107