Page 120 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Assuming that G and R, respectively, are:

⎛20 18 4 9 ⎞ ⎛ 40 11 16 9⎞
⎜ 18 40 9 20 ⎟ ⎜ 11 30 112 14 ⎟
G = ⎜ ⎟ and R = ⎜ ⎟
⎜ 4 9 25 4. ⎟ 5 ⎜ 16 12 70 10⎟
⎜ ⎟ ⎜ ⎟
.
⎝ 920 4 5 32 ⎠ ⎝ 91410 55⎠
Applying Eqn 6.8 to G using the function factanal in the R package (The R Development
Core Team, 2010) gives:
F′ = (2.8532 6.3056 1.4250 3.1678) and S′ = (11.860 0.200 22.975 21.952)
This implies that the number of common factors, m, is equal to 1 for the example G
above. Thus the column vector z* for animal i in the matrix Z * in Eqn 6.11 equals F.
i
−1
Therefore, for animal i with a record, z*′r z* is 1.361. However, for animal i, W is
i
i
i
−1
a diagonal matrix and therefore W ′R W is computed as described for the MBLUP
i i
−1
model in Section 5.2. Thus for animal i, W ′R W is:
i i
⎛ 0 0297 ⎞
.
⎜ −0 0079. 0 0419 symmetric ⎟
.
′
−1
WR W i = ⎜ ⎟
i
⎜ −0 0052. − 0 0041. 0 0163 ⎟
.
⎜ ⎝ −0 0019. − 0 0086 − 0 0011 0 0209⎠ ⎟
0
.
.
.
Although there were 48 equations in the MME defined in Eqn 6.11 for this
example compared with 40 in the usual MBLUP, there were only 502 non-zero ele-
ments in the XFA compared with 620 in MBLUP, illustrating the increased sparsity
of the MME with the XFA model. Solving the MME gave the following solutions.
The results from the usual MBLUP gave exactly the same solutions and these have
not been presented.
Solutions for sex of calf effects
WWG PWG MSC BFAT
M 4.352 6.795 9.412 0.231
F 3.487 5.959 7.095 0.535
Animal and specific solutions
Specific effects solutions Transformed solutions b
COM a WWG PWG BFAT MSC WWG PWG MSC BFAT
1 0.036 −0.008 0.095 0.000 0.005 0.095 0.227 0.340 0.010
2 −0.012 −0.001 −0.073 0.021 0.000 −0.089 −0.073 0.313 −0.050
3 −0.027 0.068 0.031 0.208 0.000 −0.086 −0.169 0.031 −0.032
4 0.021 0.046 0.113 −0.021 0.000 0.168 0.136 −0.855 0.113
5 −0.064 −0.191 0.000 0.005 0.000 −0.191 −0.407 −0.539 −0.029
6 0.046 0.290 0.021 0.000 0.000 0.017 0.290 1.350 −0.082
7 −0.063 −0.813 0.208 0.000 0.029 −0.208 −0.399 −0.813 −0.015
8 0.028 −0.101 −0.021 0.000 0.000 −0.017 0.178 1.431 −0.101
b
a COM, solutions for common factor. Transformed solutions from Eqn 6.12.

104 Chapter 6