Page 121 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Analysis with FA model

*
The main differences with fitting an FA model with no specific effects is that Z for
an animal in Eqn 6.11 is of order n by n and the last row of Eqn 6.11 is omitted.
*
Note that Z is now a product of the eigenvectors of G and the square root of a diago-
*
nal matrix of eigenvalues (see Section 6.4.2). Thus Z is:
⎛ 2.2974 3.2100 −0.1259 −2.0983⎞
⎜ −1.7761 5.8683 −0.1348 −1.5333 ⎟
Z = ⎜ ⎟ ⎟ and for animal i:
*
i
⎜ 0..2453 2.0018 4.4348 1.1272⎟
⎜ ⎝ 0.5878 4.3506 − 1.7658 3.0977⎠ ⎟
⎛ 0.372 −0.226 −0.042 0.054⎞
⎜ −0.226 1.236 −0.141 −00.246 ⎟
′
−1
*
ZR Z * = ⎜ ⎟ ⎟
i i
⎜ − 0.042 − 0.141 0.410 0.019⎟
⎜ ⎝ 0.054 − 0.246 0.019 0.537⎠ ⎟
Setting the MME follows the usual rules and the MME has 40 equations for the
example but with only 388 non-zero elements. The low number of non-zero elements
−1
2
is due to the fact that only n elements of A are contributed compared with n for
the MBLUP. Solving the equations gives the following solutions.
Solutions for calf sex effects
WWG PWG MSC BFAT

M 4.352 6.795 9.412 0.231
F 3.487 5.959 7.095 0.535
Animal solutions

Untransformed solutions Transformed solutions a
WWG PWG BFAT MSC WWG PWG MSC BFAT
1 −0.011 0.035 0.063 −0.008 0.095 0.227 0.340 0.010
2 −0.003 −0.005 0.066 0.028 −0.089 −0.073 0.313 −0.050
3 0.010 −0.020 0.010 0.021 −0.086 −0.169 0.031 −0.032
4 0.000 0.002 −0.177 −0.067 0.168 0.136 −0.855 0.113
5 0.015 −0.062 −0.099 0.019 −0.191 −0.407 −0.539 −0.029
6 −0.022 0.061 0.267 0.045 0.017 0.290 1.350 −0.082
7 0.003 −0.069 −0.153 0.005 −0.208 −0.399 −0.813 −0.015
8 −0.007 0.050 0.285 0.060 −0.017 0.178 1.431 −0.101
a Transformed solutions = vectors of solutions multiplied by Z*

6.4.2 Principal component analysis

Analysis with full PC model

The application of a full PC model with no rank reduction is similar to the FA analysis
*
except that Z is now a matrix of eigenvectors of order n by n and Z ′R Z + (I ⊗ A )
−1
*
*
–1
m
Methods to Reduce the Dimension of Multivariate Models 105