Page 202 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

semi-definite but can be singular if two individuals have identical genotypes or the
number of markers (m) is less than genotyped individuals (n). If number of markers are
limited (m < n), an improved non-singular matrix G can be obtained as wtG + (1 – wt)A.
wt
VanRaden (2008) indicated that wt = 0.90, 0.95 and 0.98 gave good results.
Another method for computing G involves scaling ZZ′ by the reciprocals of
the expected variance of marker loci (VanRaden, 2008). Thus G = ZDZ′, where
D is diagonal with:
d = 1
21-
ii mp ( p )]
[
j j
The MME for Eqn 11.8 are:
′
−1
−1
−1
′
⎛ XR X X ′R W ⎞ ⎛ ⎞ ˆ b ⎛ ⎛ XR y ⎞
⎜ −1 −1 −1 ⎟ ⎜ ⎟ = ⎜ −1 ⎟ (11.9)
′
′
⎝ WR X W ′R W + G a ⎠ ⎝ ⎠ a ˆ ⎝ WR y⎠
2
where a now equals s /s .
2
e a
This approach for genomic evaluation has the advantage that existing software
for genetic evaluation can be used by replacing A with G and the systems of equations
are of the size of animals, which tend to be fewer than the number of SNPs. In pedi-
gree populations, G discriminates among sibs, and other relatives, allowing us to say
whether these sibs are more or less alike than expected, so we can capture informa-
tion on Mendelian sampling. Also, the method is attractive for populations without
good pedigree, as G will capture this information among the genotyped individuals
(Hayes and Daetwyler, 2013).
2
Note that Eqn 11.9 assumes all the additive genetic variance (s ) is captured by
a
the SNP, but this may not be the case if the linkage disequilibrium between SNP and
QTL is not perfect. Later, in Section 11.6, a model is discussed that might capture any
residual polygenic variance not captured by the SNPs. Another possible limitation is
−1
that there are no direct rules for computing G and in large populations the compu-
tation may not be feasible.
Example 11.3
The data in Example 11.1 is analysed using Eqns 11.8 and 11.9 and the same genetic
parameters to compute DGVs for both the reference and validation animals without
using weights.
The matrix X in Eqn 11.9 is the same X as in Example 11.1, W is a diagonal
matrix for the eight reference animals with records and a = 245/35.25 = 6.950.
The G matrix constructed from Z for the ten SNPs as:
ZZ′
2Σp ( 1 p )−
j j
with 2Σp (1 − p ) = 3.5383 is:
j j
13 1.472
14 −0.446 0.746
15 0.988 −0.930 1.634
16 0.059 −0.446 0.422 0.907
17 0.685 −0.950 1.048 0.402 1.593
18 −0.163 0.180 −0.365 −0.163 −0.102 0.746
19 −0.708 0.201 −0.627 0.423 −0.365 0.201 0.786
20 −0.547 0.079 −0.183 0.301 −0.203 0.079 0.382 0.826

186 Chapter 11