Page 207 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The combined evaluations are called genotypic breeding values (GEBV) and these are
usually the published values for selection. The combination of DGVs and the conven-
tional evaluations is based on some sort of selection index approach. The selection index
presented by VanRaden et al. (2009) was:

GEBV = wt DGV + wt PTA + wt PTA
1 2 1 3 2
for animals in the reference population. Similarly, for selection candidates with no
daughter information:

GEBV = wt DGV + wt PA + wt PA
1 2 1 3 2
where PTA and PTA are predicted transmitting abilities from the official evalu-
1 2
ations based on all records and the evaluations of only the bulls in the reference
population using the A matrix, respectively. Correspondingly, PA and PA are
1 2
parent averages from the respective evaluations. The weights (wt ) were computed
i
as c′V . The matrix V is of order 3 × 3 with diagonal elements equal to the reli-
−1
abilities for DGV, PTA (PA ) and PTA (PA ), respectively. The off-diagonal ele-
1 1 2 2
ments were calculated as v = v , v = v and v = v + (v – v )(v − v )/
12 22 23 22 13 22 11 22 33 22
(1 – v ). The vector c has elements v , v and v .
22 11 22 33
Misztal et al. (2010) presented a method called the single-step approach that
combines conventional and DGVs in one step, resulting in the direct prediction of
EBVs for non-genotyped and GEBV for genotyped animals.
Assume the following mixed linear model:
y = Xb + Wa + e (11.17)
where y = vector of phenotypes or de-regressed breeding values, a = vector breeding
values and W is a design matrix that relates records to all animals including genotyped
and ungenotyped animals. Suppose a is portioned as a for ungenotyped animals and
1
a for genotyped animals, then:
2
a ⎛ 1 ⎞ ⎛ A 11 A ⎞ ⎛ 0 0 ⎞
12
2
var ⎜ ⎟ = ⎜ ⎟ s = A + ⎜ ⎟ s 2 a (11.18)
a ⎝
a
22
2 ⎠ ⎝ A 21 G ⎠ ⎝ 0 G − A ⎠
where A is the relationship matrix for only the genotyped animals.
22
2
It has already been shown in Section 11.5.2 that a = Zg and var(a ) = Gs .
2 2 a
Based on selection index theory, a can be predicted from the genotyped animals
1
(Legarra et al., 2009) as:
a = A A −1 Zg + ω
1 12 22
where ω is the residual term, such that:
−1
var(a ) = A A G A −1 A + A - A A −1 A
1 12 22 22 21 11 12 22 21
and this reduces to:
var(a ) = A + A A (G - A ) A −1 A
−1
1 11 12 22 22 22 21
Finally, cov(a , a ) = A A G.
−1
1 2 12 22
Putting all terms together into a matrix H, a covariance matrix of breeding
values including genomics information (Legarra et al., 2009; Christensen and Lund,
2010) is:
Computation of Genomic Breeding Values and Genomic Selection 191