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Directly predicting the additive genetic merit with QTL

Model

There is an approach to directly predict an animal’s marker genetic merit. The QTL relationship matrix (\(\mathbf{G}_v\)) can be reduced to a relationship matrix among animals (\(\mathbf{A}_v\)). The mathematical model contains fixed effects, additive polygenic effects and additive genetic effects related to the marker. The system of mixed model equations is \[ \left[ \begin{array}{lll} \mathbf{X}'\mathbf{R}^{-1}\mathbf{X} & \mathbf{X}'\mathbf{R}^{-1}\mathbf{Z} & \mathbf{X}'\mathbf{R}^{-1}\mathbf{W}\\ \mathbf{Z}'\mathbf{R}^{-1}\mathbf{X} & \mathbf{Z}'\mathbf{R}^{-1}\mathbf{Z} + \mathbf{A}_{u}^{-1}/\sigma_u^{2} & \mathbf{Z}'\mathbf{R}^{-1}\mathbf{W}\\ \mathbf{W}'\mathbf{R}^{-1}\mathbf{X} & \mathbf{W}'\mathbf{R}^{-1}\mathbf{Z} & \mathbf{W}'\mathbf{R}^{-1}\mathbf{W} + \mathbf{A}_{v}^{-1}/\sigma_q^{2}\\ \end{array} \right] \left[ \begin{array}{c} \mathbf{\hat{b}}\\ \mathbf{\hat{u}}\\ \mathbf{\hat{q}} \end{array} \right] = \left[ \begin{array}{l} \mathbf{X}'\mathbf{R}^{-1}\mathbf{y} \\ \mathbf{Z}'\mathbf{R}^{-1}\mathbf{y} \\ \mathbf{W}'\mathbf{R}^{-1}\mathbf{y} \end{array} \right]. \] The author assumes \(\sigma_u^2 = 0.30\), \(\sigma_q^2 = 0.10\), and \(\sigma_e^2 = 0.60\).

Files

We use the same data set as the previous example except for removing the paternal and maternal QTL effects (data_mr10b.txt). An explanation for each column is given as follows.

1. Animal ID (calf)
2. Sex (1=male and 2=female)
3. Sire ID
4. Dam ID
5. Post weaning weight (kg)

The pedigree file is also the same as before (pedigree_mr10b.txt). It has the 4th column with the inb/upg code.

In this case, we should prepare \(\mathbf{A}_{v}^{-1}\) as an user-supplied file. The following file contains its diagonal and upper-triangular elements.

  1 1  4.966
  1 2  0.286
  1 3 -0.148
...
  4 4  5.978
  4 5 -2.971
  5 5  4.836

The parameter file is as follows.

DATAFILE
data_mr10b.txt
NUMBER_OF_TRAITS
1
NUMBER_OF_EFFECTS
3
OBSERVATION(S)
5
WEIGHT(S)

EFFECTS:
2 2 cross    # fixed effect
1 5 cross    # additive polygenic effect
1 5 cross    # additive QTL effect
RANDOM_RESIDUAL VALUES
0.60
RANDOM_GROUP
2
RANDOM_TYPE  # considering inbreeding
add_an_upginb
FILE
pedigree_mr10b.txt
(CO)VARIANCES
0.30
RANDOM_GROUP
3
RANDOM_TYPE  # reading user-supplied file
user_file
FILE         # its file name
userinverse_mr10b.txt
(CO)VARIANCES
0.10
OPTION solv_method FSPAK

Solutions

You can confirm the solutions are identical to the textbook.

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