Page 213 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

ˆ
e = e + z g with i = 1, n
ˆ
ˆ
i i ij j
−1
−1
ˆ
Thus for the first SNP effect, g = (z′ z + a) z′ ˆe = (7 + 393.201) (−2.775) = −0.007.
1 i1 i1 i1 i
Assuming the random number generated from a normal distribution is 1.692
[1]
then g can be sampled as g [1] = −0.007 + 1.692 (12.218/400.201) = 0.289. After
j 1
ˆ
computing g [1] , the residual vector is updated as (e = e − z g [1] , i = 1, n) before
ˆ
1 i i i1 1
computing the next SNP effect. The estimates of g [1] to g [1] are given in Table 11.5.
2 8
The next cycle of sampling then begins again with sampling residual variance without
setting up of the vector of residuals.
For this example, the Gibbs sampling chain was ran 10,000 times, with the first
3000 considered as burn-in period. The posterior means computed from the remain-
ˆ
2
2
ing 7000 samples for b and s were 9.890 kg and 33.119 kg , respectively. The esti-
e
2
mates for g and s are given in Table 11.4.
ˆ
gi
The DGV of animals in the validation set can then be predicted using the solu-
tions for the SNP effects in Table 11.5 as Z gˆ, where Z is a matrix of genotypes for
2 2
the validation of test animals given in Example 11.2.
11.8.2 BayesB
The basic assumption in BayesA is that there is genetic variance at every loci or chromo-
some segment. It is possible that some SNPs will have zero effects as they are in genomic
regions with no QTL. The prior density of BayesA does not account for such SNPs
with zero effects as BayesA density peak at s = 0; in fact its probability of s = 0 is
2
2
gi gi
infinitesimal (Meuwissen et al., 2001). It is possible that genetic variance may be
observed in relatively few marker loci containing QTL. Meuwissen et al. (2001)
introduced BayesB to address this situation. Thus the prior distribution of BayesB is
a mixture distribution with some SNPs with zero effects and the rest with a t-distribution
(Hayes and Daetwyler, 2013). BayesB, therefore, uses a prior that has a high density,
2
2
π, at s = 0 and has an inverted chi-squared distribution for s > 0. Thus the prior
gi gi
distribution for BayesB is:
s = 0 with probability π
2
gi
−2
2
s ~ χ (v, S) with probability (1 − π) (11.24)
gi
where S is the scaling parameter, v the degrees of freedom and p is assumed known.
They set S to be to 0.0429 and computed it as in Eqn 11.22 while v was set to 4.234.
While the Gibbs sampling algorithm used for BayesA can also be used for BayesB,
it will not, however, move through the entire sampling space, as the sampling of
2
2
s = 0 is not possible if (g ′g ) is greater than zero. Also, if s = 0, the sampling of g
gi i i gi i
2
has an infinitesimal probability. This problem is overcome by sampling s and g
gi i
simultaneously from the distribution:
2
2
2
p(s ,g |y*) = p(s |y*) × p(g |s , y*) (11.25)
gi i gi i gi
where y* is the data vector y corrected for the mean and all genetic effects apart
2
from g . The first term in Eqn 11.25 implies sampling s without conditioning on g
i gi i
2
and then sampling from the second term of Eqn 11.25 for g conditional on s and y*
i gi
2
as in BayesA. The distribution p(s |y*) cannot be expressed in the form of a known
gi
distribution, therefore Meuwissen et al. (2001) used the Metropolis–Hastings (MH)
Computation of Genomic Breeding Values and Genomic Selection 197