Page 211 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

where e = (y – x b − z g); i = 1, n with n equal to the number of records or animals.
i i i i
2
Similarly, s is sampled from the following conditional posterior distribution:
gi
−
2
2
v
,
s ⏐ g ~ c ( + k S + ′ g g ) (11.22)
gi i i i i
with v = 4.012 and S derived as:
s (v − 2)
2
v
2
2
where s is the a prior value of s and k equals 1 for the ith SNP.
gi i
Other researchers (Xu, 2003; Ter Braak et al., 2005) have published similar
2
approaches with different priors for estimating s .
gi
Finally, g for the ith SNP is sampled from the following distribution as:
ˆ
i
g | b,g , s , s , y ~ N(gˆ , (z′z + a) s ); i ≠ j (11.23)
−1
2
2
2
i j gi e i i i e
with:
ˆ −1 2 2
e
i
i i
i
j j
g = (z′z + a) z′(y − Xb − z g ) and a = s /s gi
2
The Gibbs sampling procedure then consists of setting initial values for b, g, s and
e
s , and iteratively sampling successively from Eqns 11.20 to 11.23, using updated
2
g
values of the parameters from the i round in the i + 1 round. Assuming that p rounds
of iteration were performed, then p is called the length of the chain. The first j samples
are usually discarded as the burn-in period. This is to ensure that samples saved are
not influenced by the priors but are drawn from the posterior distribution. Posterior
means are then computed from the saved samples.
Example 11.7
Using the data in Example 11.1, the application of BayesA is illustrated using
residual updating (Legarra and Misztal, 2008). The data for the reference animals
is analysed by fitting the model in Eqn 11.6. Thus n, the number of records, is 8
and a flat prior has been assumed for b. It is also assumed that v = 4.012 and S is
derived as:
s (v − 2) =
2
.
v 0 352
where s = 0 702 . Note that the matrix of genotypes Z used in the computation
2
.
below has not been centralized and there Z equals M in Section 11.2.
ˆ
ˆ
−1
The starting value for b was computed as b = (X′X) X′y = 79.1/8 = 9.888 and
2
those for gˆ and s were 0.05 and 0.702, respectively, for all SNPs. The starting value
gi
2
2
2
for s was set as 2.484, thus s = s /2Σp (1 – p ) = 0.702. The starting values for
a gi a j j
ˆ
DGV for animals in the reference population were computed as a = Zg. Thus:
a′ = (0.45 0.30 0.55 0.45 0.45 0.50 0.40 0.35)
ˆ
Initially, a vector of residuals eˆ was computed as eˆ = y − Xb − Zg. Thus:
ˆ
Computation of Genomic Breeding Values and Genomic Selection 195