Page 204 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Selection index equations to predict DGV (a ˆ ) are constructed as the covariance
between y and a multiplied by the inverse of the variance of y and the deviation of y
from fixed effects solutions. Thus:

⎛ 2 ⎞⎞ −1
⎜
ˆ a = GG + R ⎛ s e 2 ⎟⎟ ( − X ˆ ) b (11.10)
y
⎜
⎝ ⎝ s a ⎠⎠
The vector of estimates of SNP effects (g) can be obtained from Eqn 11.10 as:
ˆ
⎛ 1 ⎞ ⎛ 2 ⎞⎞ −1
ˆ g = ⎜ ⎟ ZG + R ⎜ ⎛ s e ⎟⎟ ( − X ˆ ) b (11.11)
′
⎜
y
⎝ ∑ p j ( − p j ) ⎠ ⎝ ⎝ s a 2 ⎠⎠
1
2
The DGV of validation candidates without records can then be computed with the
selection index approach as:
⎛ 2 ⎞⎞ −1
ˆ a = CG + R ⎛ s e 2 ⎟⎟ ( − X ˆ ) b (11.12)
⎜
y
⎜
⎝ ⎝ s a ⎠⎠
where C is the genomic covariance between animals with and without records com-
puted as:
ZZ′
2
2 ∑ p ( 1 − p )
j
j
with Z being the matrix of centralized genotypes for the validation animals (see
2
Example 11.3).
Example 11.4
The data in Example 11.1 is again analysed using Eqn 11.10 and the same genetic
parameters to compute DGVs for the reference animals without using weights. The
solution of 9.994 has been assumed for the mean.
The X matrix in Eqn 11.10 equals X in Example 11.1, the G matrix is of order
8 for the reference animals only and corresponds to the first eight rows and columns
2
of G computed in Example 11.3 and R = Is , assuming no weights are used in the
e
analysis.
Solutions from solving Eqn 11.10 are shown in Table 11.2. Similarly, the DGV
of the selection candidates were obtained by Eqn 11.12 and these are also shown in
Table 11.2. The same solutions were obtained for both reference and validation ani-
mals as obtained from the SNP or GBLUP models.

11.6 Mixed Linear Models with Polygenic Effects

The genomic BLUP model used to estimate SNP effects in most livestock populations
is based on chips with densities of about 60K, and it is usually assumed that these
SNPs explain all the genetic variation for the traits analysed. However, fitting a
residual polygenic effect (RP) may account for the fact that SNPs may not explain all
the genetic variance and it has also been found to render SNP effects less biased

188 Chapter 11