Page 208 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

⎛ H H ⎞ ⎛ A + A A −1 ( G − A ) A −1 A A A −1 G⎞
H = ⎜ 11 12 ⎟ = ⎜ 11 12 22 22 22 21 12 22 ⎟
⎝
H
A
22
21 H ⎠ ⎝ GA −1 A 21 G ⎠
22
The matrix H could be regarded as a matrix that combines pedigree and genomic
relationships.
The single-step methodology involves the use of matrix H, and Aguilar et al. (2010)
and Christensen and Lund (2010) found the inverse of H has the following simple form:
⎛ 0 0 ⎞
H −1 = A −1 + ⎜ ⎝ 0 G − A − ⎟ ⎠
−1
1
22
where A is inverse of the relationship matrix for genotyped animals.
-1
22
−1
−1
This implies that by replacing A with H in the usual MME, direct prediction of EBVs
and genomic evaluations can be obtained for ungenotyped and genotyped animals.
Therefore, the MME for the single-step procedure (Eqn 11.17) are:
′
′
−1
−1
−1
⎛ XR X X ′R W ⎞ ⎛ ⎞ ˆ b ⎛ ⎛ XR y ⎞
⎜ −1 −1 −1 ⎟ ⎜ ⎟ = ⎜ −1 ⎟ (11.19)
′
′
⎝ WR X W ′R W + H a ⎠ ⎝ ⎠ ˆ a ⎝ WR y⎠
where:
2 2
e
a = s /s a
The main advantage of the single-step approach is that existing software for
genetic predictions can easily be modified to implement this method. However, the
−1
−1
computation of H requires efficient computation of G . Thus this could be a major
limitation, with large numbers of animals genotyped, since there are no simple rules
for computing the inverse of G. Another complication is that G must be on exactly
the same scale (e.g. scaled to the same base animals) as A, otherwise animals with
genotypes will have biased GEBV.
Example 11.6
The data in Example 11.1 is analysed using Eqn 11.17 assuming the same genetic
parameters, but the data is modified as follows. The first five animals (13 to 17) are
treated as ungenotyped animals with records, the next five animals (18 to 22) are
regarded as genotyped animals with records, while the remaining four animals (23 to 26)
are regarded as genotyped animals with no records. A weighted analysis was carried out
using the EDCs.
Therefore, the A matrix for the nine genotyped animals was extracted from the
22
last nine rows of A given in Example 11.3.
The G matrix is computed as:
ZZ′
2 ∑ p j 1 ( − p )
j
for the nine genotyped animals. However, due to the small size of the data, the G
used in the analysis was computed as G = 0.95G + 0.05A (Misztal et al., 2010), to
enable inversion of the matrix (see Section 11.5.2). The matrix G then is:

192 Chapter 11