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Single-step approach

Model

We consider the standard animal model \[ \mathbf{y}=\mathbf{Xb}+\mathbf{Wu}+\mathbf{e}. \] If some animals are genotyped, their additive relationships are described with the genomic relationship matrix (\(\mathbf{G}\)). When the genotyped and the non-genotyped animals are simultaneously considered in the relationship matrix, the resulting matrix is \(\mathbf{H}\). Its inverse falls into a simple form. \[ \mathbf{H}^{-1} = \mathbf{A}^{-1} + \left[ \begin{array}{cc} \mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{G}^{-1}-\mathbf{A}_{22}^{-1} \end{array} \right] \] This \(\mathbf{G}^{-1}\) is usually blended with the pedigree matrix (\(\mathbf{G}^{-1}_{w}\) shown in the previous section). The system of mixed model equations is the same as the standard animal model with \(\mathbf{H}^{-1}\) instead of \(\mathbf{A}^{-1}\): \[ \left[ \begin{array}{ll} \mathbf{X}'\mathbf{R}^{-1}\mathbf{X} & \mathbf{X}'\mathbf{R}^{-1}\mathbf{Z}\\ \mathbf{Z}'\mathbf{R}^{-1}\mathbf{X} & \mathbf{Z}'\mathbf{R}^{-1}\mathbf{Z}+\mathbf{H}^{-1}/\sigma_u^2 \end{array} \right] \left[ \begin{array}{c} \mathbf{\hat{b}}\\ \mathbf{\hat{u}} \end{array} \right] = \left[ \begin{array}{l} \mathbf{X}'\mathbf{R}^{-1}\mathbf{y} \\ \mathbf{Z}'\mathbf{R}^{-1}\mathbf{y} \\ \end{array} \right] \] where \(\sigma_u^2 = 35.241\) and \(\sigma_e^2 = 245.0\) in this case. BLUPF90 fully supports ssGBLUP with a minimal description in the parameter file.

Files

The data file is different from the previous ones (data_mr11e.txt). The pedigree information is common to the previous analysis (pedigree_mr11e.txt).

The SNP marker file is unique for this analysis.

 18 11010202210000000000000000000000000000000000000000
...

The corresponding cross-reference file is as follows.

18 18
19 19
20 20
21 21
22 22
23 23
24 24
25 25
26 26

The parameter file is shown as follows.

DATAFILE
data_mr11e.txt
NUMBER_OF_TRAITS
1
NUMBER_OF_EFFECTS
2
OBSERVATION(S)
6
WEIGHT(S)
5
EFFECTS:
4  1 cross
1 26 cross
RANDOM_RESIDUAL VALUES
245.0
RANDOM_GROUP
2
RANDOM_TYPE
add_animal
FILE
pedigree_mr11e.txt
(CO)VARIANCES
35.241
OPTION SNP_file snp_mr11e.txt snp_mr11e_XrefID.txt
OPTION no_ quality_control
OPTION AlphaBeta 0.95 0.05
OPTION tunedG 0
OPTION thrStopCorAG 0.10
OPTION solv_method FSPAK

BLUPF90 (actually the embedded genomic routine) may stop because of the very low correlation between diagonals from \(\mathbf{G}\) and \(\mathbf{A}_{22}\). The correlation should be usually high enough; otherwise, there may be a problem in the quality of the genotypes or the pedigree. It is low in this case due to the small data set. The option thrStopCorAG prevents the program from stopping from the low correlation.

Solutions

Unfortunately, the solutions are totally different from the reference values in the textbook (p.193).

trait/effect level  solution
   1   1         1          8.38509553
   1   2         1         -0.27072327
   1   2         2          2.90677899
   1   2         3         -0.27072327
   1   2         4          2.58838142
   1   2         5         -2.59488845
   1   2         6         -1.88195674
   1   2         7         -0.99299119
   1   2         8         -1.02617193
   1   2         9         -3.14377983
   1   2        10         -1.69066025
   1   2        11         -3.31615787
   1   2        12          0.81555256
   1   2        13          0.63918948
   1   2        14          4.85991512
   1   2        15          4.20216687
   1   2        16          6.46125192
   1   2        17         -1.79124924
   1   2        18         -0.39297755
   1   2        19          1.47720048
   1   2        20         -2.90484503
   1   2        21         -0.54144654
   1   2        22          0.89069967
   1   2        23         -2.54427924
   1   2        24         -0.10603281
   1   2        25          0.94047078
   1   2        26          3.65328640

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