Page 215 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Similarly, b [1] = (9.456 + 0.873 (.1 527 ) = 10.535, as in Example 11.7.
1
2
Using the steps outlined for the MH cycle for BayesB, s for the ith SNP effect is
gi
2
2
then sampled, which could result in either s = 0 or s > 0. In this example, 20 MH
gi gi
samples were evaluated per each round of Gibbs sampling, and for the first SNP, the
2
estimate of s = 1.105. Therefore, gˆ was sampled from the normal distribution using
g1 1
Eqn 11.23 as described in Example 11.7 but with a = 12.218/1.105 = 11.057. In this
1
2
example, s and gˆ for SNP, with i = 7, 9 and 10 were zero in the first round of iteration.
gi i i
2
The solutions for s and gˆ for the first round of iteration are presented in Table 11.5.
gi i
The Gibbs sampling was run for 10,000 cycles, with the first 3000 regarded as the
burn-in period. The posterior means computed from the remaining 7000 samples for b ˆ
2
2
2
and s were 9.792 kg and 34.930 kg , respectively. The estimates for gˆ and s are given
e gi
in Table 11.5. The DGV of animals in the validation set can then be predicted using the solu-
tions for the SNP effects in Table 11.2 as Z gˆ, where Z is defined as in Example 11.7.
2 2
11.8.3 BayesC
Habier et al. (2011) indicated the estimation of individual SNP variances in BayesA
and BayesB has only one additional degree of freedom compared with its prior,
and so the shrinkage of SNP effects is largely dependent on the scale parameter, S.
To overcome this limitation, they proposed BayesC, which involves estimating a sin-
gle variance that is common to all SNPs, thereby reducing the influence of the scale
parameter. Similar to BayesB, BayesC allows for some SNPs to have zero effects with
probability π while the remaining SNPs have non-zero effect with probability (1 – π).
Habier et al. (2011) indicated that since the priors of all SNP effects have a common
variance, the effect of an SNP fitted with probability (1 − π) comes from a mixture of
multivariate Student’s t-distributions.
In BayesC, it is assumed that π is known and the decision to include SNP depends
i
on the full conditional posterior of an indicator variable δ . This indicator variable
i
equals 1 if SNP is fitted, otherwise it is zero. Thus the decision to include the ith SNP
i
involves computing the probability k of δ = 1 as k = 1/{1 + (p(y* | δ = 0, )/
i
i
2
p(y* | δ = 1, s , ))}, where (p(y* | δ = 1, )) denotes the likelihood of the data given
i g i
2
that SNP is fitted with common variance s , refers to accepted values for all other
i g
parameters, (p(y* | δ = 0, )) denotes the likelihood of the data model without the
i
ith SNP and where y* is the data vector y corrected for the mean and all genetic
effects apart from g .
i
The computation of the required likelihood is easier to implement in a log-likelihood
form. Fernando (2010) presented such an algorithm based on the log-likelihood.
2
2
Given current estimates of s , and s , logLH1 with δ = 1 is computed as:
g e i
−1
logLH1 = −0.5(log(V)) + (z′y*)′V z′y* + log(1 − π) with
i i
2 2
i i g i i i i e
V = (z′z Is ,z′z ) + z′z *s
Similarly, the log-likelihood when δ = 0 is computed as logLH0 = −0.5(log(V)) +
i
−1
2
(z′y*)′V z′y* + log(π) but with V = z′z *s .
i i i i e
Then compute probability k of δ = 1 as k = 1/(1 + exp(logLH0 – logLH1)).
i
If k is greater than r, where r is a random drawn from a uniform distribution,
then SNP is fitted and g is sampled from the normal distribution using Eqn 11.23,
[j]
i i
otherwise g = 0.
[j]
i
Computation of Genomic Breeding Values and Genomic Selection 199