Page 219 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

For the ith SNP, this can be expressed (Mrode et al., 2010) as:
−1
−1
−1
g = (z ′r z + a) z ′r z (yd ), j = 1, n (the number of animals)
ˆ
i ji ji ji ji i
ˆ
g = wt (yd ) (11.29)
i i i
where yd is the SNP deviation for the ith SNP, i.e. data information for that SNP
i
corrected for all effects apart from the SNP and the SNP deviation can be defined as
ˆ
−1
−1
−1
−1
−1
−1
yd = (z¢ r z ) z¢ r (y – z gˆ − x b), k ≠ i and wt = (z¢ r z + a) z ′r z . The DGV
i ji ji ji j jk k j i ji ji ji ji j
of animal j therefore is:
j ∑
DGV = z wt (yd )
i
ji
i
i
ˆ
For illustration purposes, the SNP solution for SNP 1, g in Example 11.2, can be
1
computed using Eqn 11.29 as follows:
The Z in Example 11.2, (z¢ z ) = 3.878, and (z¢ z + a) = 28.476.
j1 j1 j1 j1
The SNP deviation, yd = 0.638; therefore, wt = 3.878/28.476 = 0.136 and gˆ =
1 1 1
0.136 (0.638) = 0.087. Similar calculations indicated that for SNP 7, yd = −0.001,
7
wt = 0.007, and gˆ = 0.00.
7 7
In the case of Bayesian methods, there is an additional component as a result of
sampling from the conditional posterior distribution of g, such that:
2
−1
ˆ
−1
g = wt (yd ) + N(gˆ , (z ′r z + a ) s e) (11.30)
i i i i i i i
The second term on the right-hand side of Eqn 11.30 tends towards zero averaged
over all samples after the burn-in period.
Equation 11.29 indicates that with the SNP-BLUP model, the SNP solutions are
a function of the SNP deviations, which could be regarded as the unregressed SNP
allele substitution effects and the weight. Given that a is constant for the SNP-BLUP
model, the weight is therefore very dependent on the allele frequencies. Thus alleles
of lower frequencies will have a lower weight on their SNP deviations. In the calcula-
tions above, the weight for SNP1 with an allele frequency of 0.312 was much higher
than that for SNP 7. Mrode et al. (2010) obtained a correlation between the weights
and allele frequencies of 0.99 from the SNP-BLUP model. However, for BayesA and
BayesB, the estimation of individual variances meant that a, and therefore weights,
were different for each SNP. Thus SNP deviations were differentially weighted not
only on the basis of their allele frequencies but also on the basis of their genetic vari-
ance, i.e. by the amount of available information.

Computation of Genomic Breeding Values and Genomic Selection 203