Page 216 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

2
After sampling the vector g, s is sampled from the following conditional poste-
g
rior distribution as:
−
2
j []
2
gg
s | g ~ c ( v + k , S + ′ ) (11.28)
g i i i
[ j]
with terms defined as in Eqn 11.22 but with degrees of freedom equal to v + k ,
[j]
where k is the number of SNPs with non-zero effects fitted in the jth iteration.
Example 11.9
The data in Example 11.1 is used to illustrate BayesC by applying the model in
ˆ
Eqn 11.6. The assumptions and the starting values for b, gˆ and a were the same as outlined
for BayesA in Example 11.7. The starting value of π was assumed at 0.30 while the
2
starting value of s was set at 0.702.
g
The sampling procedure for s and b were as outlined in BayesA and therefore
2
e
with the same solutions in the first iteration. Then for the ith SNP, the probability of
g having a zero effect or otherwise was computed as described earlier in this section.
ˆ
i
ˆ
In the first iteration, the first SNP has a non-zero effect; therefore, g = (z′ z + a) −1
1 i1 i1
−1
z′ e = (7 + 17.045) (−2.775) = −0.115, with a = 12.218/0.702. Assuming the ran-
ˆ
i1 i
dom number generated from a normal distribution is 0.748, g was sampled using
[1]
j
Eqn 11.23 as g [1] = −0.115 + 0.748 (12.218/24.045) = 0.418. In the first round of
1
ˆ
iteration, two SNPs (5 and 10) had zero effects. The solutions for g in the first itera-
i
tion are presented in Table 11.6.
The sampling of common variance was done using Eqn 11.28. For this example,
2
eight SNPs had non-zero effects in the first iteration; therefore, s in the first iteration
g
2
was sampled from the inverted χ distribution with degrees of freedom now equal to
8 + 4.012 = 12.012, S = 0.352 and Σ ˆ g = 1.435. Thus given the value of 16.294 sampled from
2
i
.
2 i 21[] = S ( + Σˆg 2 ) / 16 294 = .
the inverted χ distribution, then in the first iteration s g i i 0 110.
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
ˆ
2
2
2
2
for b, s and s were 9.828 kg, 32.377 kg and 0.184 kg , respectively. The estimates
e g
for g are given in Table 11.6.
ˆ
Table 11.6. Solutions for SNP effects from BayesC and BayesCπ.
BayesC BayesCπ
SNP First iteration Posterior means Posterior means
1 0.416 0.015 0.010
2 −0.360 −0.045 −0.029
3 −0.590 0.044 0.028
4 0.465 −0.014 −0.018
5 0.000 0.014 0.013
6 0.360 0.025 0.010
7 −0.586 −0.002 0.004
8 −0.307 0.009 0.003
9 −0.041 −0.013 −0.011
10 0.000 −0.002 −0.006
200 Chapter 11