Page 212 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

ˆ e æ 1 ö æ 9.0 ö æ 9.888ö æ 0.45ö æ -1.388 ö
ç ÷ ç ÷ ç ÷ ç ÷ ç ÷
ç ˆ e 2 ÷ ç 13.4 ÷ ç 9.888 ÷ ç 0.30 ÷ ç 3.213 ÷
8
ˆ e ç ÷ ç 1 12.7 ÷ ç 9.888÷ ç 0.55÷ ç 2.263÷
ç 3 ÷ ç ÷ ç ÷ ç ÷ ç ÷
ˆ e ç 4 ÷ ç 15.4 ÷ ç 9.888 ÷ ç 0.45 ÷ ç 5.063 ÷
- -
-
ç ÷ = ç ÷ ç ÷ ç ÷ = ç ÷
ç ˆ e 5 ÷ ç 5.9 ÷ ç 9.888 ÷ ç 0.45 ÷ ç -4.438 ÷
ˆ e ç ÷ ç 7.7 ÷ ç 9.888÷ ç 0.50 ÷ ç ç -2.688 ÷
ç 6 ÷ ç ÷ ç ÷ ç ÷ ç ÷
ç ˆ e 7 ÷ ç 10.2 ÷ ç 9.888 ÷ ç 0.40 ÷ ÷ ç -0.088 ÷
ç ÷ ç ÷ ç ÷ ç ÷ ç ÷
è ˆ e 8 ø è 4.8 ø è 9.888 ø è 0.35 ø è -5.437 ø
ˆ
From the above, e′eˆ = 99.345, and thus given the value of 8.131 sampled from the
2 2[1]
e
inverted χ distribution with n – 2 degrees of freedom, s = 99.345/8.131 = 12.218,
using Eqn 11.21. The superscript in brackets denotes the iteration number.
ˆ
−1
[1]
Then sample b using Eqn 11.20, with b calculated as (x ′x) 1′eˆ = 9.456 after
j
initially updating eˆ, the vector of residuals to include information on b as:
ˆ
e = eˆ + Xb with i = 1, n
ˆ
i i
Assuming the random number generated from a normal distribution is 0.873 and
2
−1
(x ′x) s = 12.218/8 = 1.527, then b [1] = (9.456 + 0.873 (.527 ) = 10.535.
1
j e 1
After sampling for b, the ê is updated to exclude the information on b as:
ˆ
e = eˆ − Xb with i = 1, n
ˆ
i i
2
2
Using Eqn 11.22, s for the ith SNP effect is sampled from the inverted χ distribu-
gi
tion with degrees of freedom 5.012 and S = 0.352 computed earlier. For the first SNP,
2
g = 0.003, thus given the value of 11.422 sampled from the inverted χ distribution
2
ˆ
1
ˆ
2
2 [1] = (S + g )/11.422 = 0.031. The variance estimates for other SNPs in the first
s g1
iteration are shown in Table 11.5.
Finally, estimates of gˆ are sampled from the normal distribution using Eqn 11.23.
First update the vector of residuals to include information on the jth SNP. Thus for
the jth SNP effect:
Table 11.5. SNP solutions and variances from BayesA and BayesB.
BayesA BayesB
First iteration Posterior means First iteration Posterior means
SNP Effects Var Effects Var Effects Var Effects Var
1 0.289 0.031 0.018 0.170 2.187 1.105 0.038 0.316
2 0.279 0.049 −0.064 0.179 −1.565 0.516 −0.107 0.319
3 −0.010 0.070 0.058 0.179 −0.156 0.124 0.067 0.293
4 0.023 0.097 −0.023 0.176 −0.309 0.118 −0.034 0.300
5 0.045 0.052 0.022 0.167 0.413 0.363 0.047 0.328
6 −0.321 0.050 0.025 0.171 −0.521 0.161 0.031 0.283
7 0.411 0.256 −0.006 0.186 0.000 0.000 0.009 0.335
8 0.408 0.056 −0.008 0.168 −0.010 0.431 0.008 0.261
9 0.115 0.034 −0.003 0.162 0.000 0.000 −0.006 0.294
10 −0.578 0.152 −0.008 0.165 0.000 0.000 −0.017 0.286
Var, SNP variances.
196 Chapter 11