Page 200 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The matrix X in Eqn 11.7 is the same as X in Example 11.1 and Z computed as
Z = M – P is:
⎛ 1 357 − 0 357 0 286 0 286 − 0 286 − 1 214 − 0 143 0 071 − 0 143 1 214 ⎞
.
.
.
.
.
.
.
.
.
.
⎜ 0.357 − . 0 357 − . 0 714 − . 0 714 − . 0 286 . 0 786 − . 0 143 . 0 071 − . 0 143 − . 0 786 ⎟
.
⎜ ⎟
⎜ 0.357 . 0 643 . 1 286 . 0 286 . 0 714 − . 1 214 − . 0 143 . 0 071 − . 0 143 . 1 214 ⎟ ⎟
.
⎜ − . 0643 − 0 357 1 286 0 286 − 0 286 − 0 214 − 0 143 0 071 0 857 0 214 ⎟
.
.
.
3
.
.
.
.
.
.
Z = ⎜ − ⎜ − . 0 286 − − − . 1 214 ⎟ ⎟
.
.
⎜ 0 643 0.643 . 0 286 . 1 286 . 1 214 . 0 143 . 0 071 . 0 143 ⎟
.
.
.
.
.
3
.
.
.
⎜ . 0 357 . 0 643 − 0 714 0 286 − 0 286 0 786 − 0 143 0 071 0 857 0 214 ⎟
− ⎜ 0 643 − 0 357 0.286 . 0 286 − . 0 286 . 0 786 − . 0 143 . 0 071 . 0 857 − . 0 786 ⎟
.
.
.
⎜ ⎟
0 786 ⎠
− ⎝ . 0 643 . 0 643 . 0 286 − . 0 286 − . 0 143 0 071 0 857 − .
.
.
0 714 − .
0 214 − .
6
The MME in Eqn 11.7 can then be easily set up. The solutions for the mean and
SNP effects from solving the MME, either using weights or no weights, are shown in
ˆ
Table 11.1. The DGV for the reference animals is then computed as Zg. The results
are shown in Table 11.2.
Similarly, the DGV of the validation animals are computed as Z gˆ, where Z
2 2
contains the centralized genotypes for the selection candidates. Thus for the
unweighted analysis:
⎛ 0 087⎞
.
⎜ ⎜ −0 311 ⎟
.
ˆ a ⎡ ⎤ ⎜ ⎟
.
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.
.
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.
.
.
⎢ 21 ⎥ ⎛ . 1 357 − . 0 357 − − 0 714 − 0 714 − 0 286 − 0 214 1 857 0 071 − 0 143 1 214 ⎞ ⎜ 0 262⎟
ˆ a ⎢ 22 ⎥ ⎜ − 0 643 − 0 357 − − 0 714 0 286 0 714 0 786 − 0 143 0 071 − 1 143 − 0 786 ⎟ ⎜ −0 080 ⎟
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⎟ ⎜
⎢ ˆ a ⎥ ⎜ 0 214 ⎟ ⎜ 0 110 ⎟ ⎟
.
⎢ 23 ⎥ ⎜ − 0 643 0 643 0.2286 − 0 714 − 0 286 − 0 214 − 0 143 0 071 0 857 . .
.
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.
⎢ ˆ a ⎥ = ⎜ 0 357 − 0 357 − 0 7114 − 0 714 0 714 − 0 214 − 0 143 0 071 − 1 143 − 0 786 ⎟ ⎜ 0 139 ⎟ ⎟
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⎢ 24 ⎥ ⎜ ⎟ ⎜
ˆ a ⎢ ⎥ − ⎜ 0 643 − 0 357 − 0.7714 0 286 0 714 0 786 − 0 143 0 071 − 0 143 − 0 786 ⎟ ⎜ 0 000 ⎟
.
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.
⎟ ⎟ ⎜
⎢ 25 ⎥ ⎜ ⎝ 0 357 − 0 357 0 286 0 0 286 − 0 286 0 786 − 0 143 − 0 929 − 1 143 − 0 786 ⎠ ⎜ 0 000⎟ ⎟
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⎣ ˆ a ⎢ 26 ⎦ ⎥ ⎜ ⎟
.
⎜ −0 061 ⎟
⎜ ⎝ −0 016⎠ ⎟
.
⎛ 0 027⎞
.
⎜ 0 114 ⎟
.
⎜ ⎟
⎜ −0 240⎟
.
= ⎜ ⎟
.
⎜ 0 143 ⎟
⎜ 0 054⎟
.
⎜ ⎟
⎝ 0 0 354⎠
.
11.5.2 Equivalent models: GBLUP
An equivalent model to Eqn 11.6 is the application of the usual BLUP MME but with
−1
the inverse of the numerator relationship matrix (A ) replaced by the inverse of the
−1
genomic relationship matrix (G ) (Habier et al., 2007; Hayes et al., 2009). This tends
to be referred to generally as GBLUP. The DGVs are computed directly from the
MME as the sum of the SNP effects (a = Zg), with the assumption that SNP effects
are normally distributed. Assume the following mixed linear model:
y = Xb + Wa + e (11.8)
where y is the vector of observations, a is the vector of DGVs and W is the design
matrix linking records to breeding value (random animal or sire effect if an animal
or sire model has been fitted) and e is random residual effect. Given that a = Zg, then:
2
var(a) = ZZ′s g
184 Chapter 11