Page 217 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

11.8.4 BayesCp

In BayesC there is the implicit assumption that the probability, π > 0, i.e. a SNP has
zero effect, is regarded as known. Habier et al. (2011) argued that the shrinkage of
SNP effects is affected by π and should be estimated from the data and proposed
BayesCπ, which incorporates this estimation step. Thus compared to BayesC,
the additional feature of BayesCπ is estimating π from the data. The sampling
procedure for parameters in BayesCπ is therefore the same as BayesC apart from
the additional step of sampling for π. Thus only the procedure for sampling π is
described.
The parameter π is sampled from a beta distribution, with shape parameters
[j]
[j]
(m − k + 1) and (k + 1), with m equal to the total number of SNPs in the analysis
[j]
and k is the number of SNPs with non-zero effects fitted in the jth iteration.

Example 11.10
The application of BayesCπ is illustrated using the data in Example 11.1. The refer-
ence animals are analysed by applying the model in Eqn 11.6 using residual updat-
ing. The initial parameters are the same as outlined for BayesA in Example 11.7.
2
The starting values of π and s were set at 0.30 and 0.702, respectively.
g
2
The sampling procedure for s and b were as outlined in BayesA and therefore
e
with the same solutions in the first iteration. Then for the ith SNP, the probability of
ˆ
g having a zero effect or otherwise was computed as described earlier in this section.
i
In the first iteration, the first SNP has a non-zero effect; therefore, g = (z′ z + a) −1
ˆ
1
i1 i1
−1
ˆ
z′ e = (7 + 17.045) (−2.775) = −0.115, with a = 12.218/0.702. Assuming the ran-
i1 i
[1]
dom number generated from a normal distribution is 0.748, g was sampled using
j
Eqn 11.23 as g [1] = −0.115 + 0.748 (12.218/24.045) = 0.418. In the first round of
1
iteration, two SNPs (5 and 10) had zero effects and the solutions for g were the same
as obtained for BayesC (Table 11.6).
The sampling of common variance follows the same procedure for BayesC,
again with the degrees of freedom equal to the number of SNPs with non-zero
effects. For this example, eight SNPs had non-zero effects in the first iteration; there-
2
2
fore, s in the first iteration was sampled from the inverted χ distribution with
g
2
degrees of freedom now equal to 8 + 4.012 = 12.012, S = 0.352 and S gˆ =1.435.
i i
2
Thus given the value of 16.294 sampled from the inverted χ distribution, then in
2[1] 2
ˆ
1 i i
the first iteration s = (S + S g )/16.294 = 0.110.
[1]
Then π was sampled from the beta distribution with shape parameters
[1]
[1]
((m – k + 1) = 3) and ((k + 1) = 9), given eight SNPs had non-zero effects.
A value of 0.339 was sampled for π.
A total of 10,000 cycles was implemented for the Gibbs sampling and the first
3000 were discarded as the burn-in period. The posterior means computed from the
ˆ
2
2
2
remaining 7000 samples for b, s , s and π were 9.898 kg, 32.343 kg , 0.162 kg 2
e g
ˆ
and 0.51, respectively. The estimates for g were given in Table 11.6.
ˆ
2
The estimates for b and s were very consistent for the Bayesian models consid-
e
2
ered. Similarly, BayesC and BayesCπ gave very similar estimates of s , which were
g
consistent with estimates for BayesA but SNP solutions were different from the dif-
ferent models. The estimates of s for BayesB were almost double those from the
2
gi
other models.
Computation of Genomic Breeding Values and Genomic Selection 201