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Animal model

Model

Mrode (2014) considered the pre-weaning gain (WWG) as a target trait, and he applied a linear mixed model to the trait with a fixed effect (sex), a random effect (animal), and the residual effect. Assume that the genetic variance was \(\sigma_u^2\) = 20 and the residual variance was \(\sigma_e^2= 40\). This is a typical animal model which have been already introduced in the previous chapter.

There are two possible systems of equations: 1) each variance component is explicitly involved in the equations, or 2) \(\alpha\), the variance ratio, is used. Two systems of equations should result in the same solutions.

Files

We now prepare the data file (data_mr03a.txt). It includes 5 observations just from 5 animals.

The data file has 5 columns as follows. This is the exact copy of the original table in the textbook.

1. Animal ID (calves)
2. Sex (1 for male and 2 for female)
3. Sire ID
4. Dam ID
5. Observations (WWG, kg)

Column 3 and 4 are not actually used in this analysis.

A pedigree file is also prepared. The 1st column is animal ID, the 2nd column for sire ID and the 3rd column for dam ID.

The parameter file is following. To obtain the exact solutions, we have OPTION solve_method FSPAK. With this option, we can calculate the reliability of a solution using the diagonal elements i.e., prediction error variance (PEV)) of the inverse of the left-hand side of mixed model equations. Additional option OPTION sol se is needed to calculate PEV.

DATAFILE
data_mr03a.txt
NUMBER_OF_TRAITS
1
NUMBER_OF_EFFECTS
2
OBSERVATION(S)
5
WEIGHT(S)

EFFECTS:
2 2 cross
1 8 cross
RANDOM_RESIDUAL VALUES
40.0
RANDOM_GROUP
2
RANDOM_TYPE
add_animal
FILE
pedigree_mr03a.txt
(CO)VARIANCES
20.0
OPTION solv_method FSPAK
OPTION sol se

Solutions

Invoking BLUPF90 with above parameter file, we immediately see the solution in the file solutions.

trait/effect level  solution          s.e.
   1   1         1          4.35850233          4.88082357
   1   1         2          3.40443010          5.66554023
   1   2         1          0.09844458          4.34094096
   1   2         2         -0.01877010          4.43664612
   1   2         3         -0.04108420          4.27297922
   1   2         4         -0.00866312          4.13608581
   1   2         5         -0.18573210          4.13814812
   1   2         6          0.17687209          4.20610397
   1   2         7         -0.24945855          4.20407502
   1   2         8          0.18261469          4.11029997

The solutions are identical to solutions shown in the textbook (pp.39). The above s.e. is the square root of diagonal elements of the inverse of the left-hand side. Note that the above s.e. is actually SEP (standard error of prediction) in the textbook (pp.45) not PEV. This happened because BLUPF90 created general mixed model equations explicitly containing \(\sigma_e^2\) and \(\sigma_u^2\) rather than the variance ratio \(\alpha=\sigma_e^2/\sigma_u^2\). So we don’t need to multiply extra \(\sigma_e^2\) by the inverse element to obtain PEV. Below, we will demonstrate the same left-hand side shown in the textbook and confirm its inverse equals to results in the textbook.

Alternative parameter file

The textbook uses a simplified form of mixed model equations in the single-trait analysis. BLUPF90 can handle this form with a tricky (not recommended) way. In this form, only the variance ratio matters. The ratio is \(\alpha=\sigma_e^2/\sigma_u^2=2.0\) and it is equivalent to assuming \(\sigma_e^2=1.0\) and \(\sigma_u^2=0.5\). The parameter file has these “equivalent” variance components.

DATAFILE
data_mr03a.txt
NUMBER_OF_TRAITS
1
NUMBER_OF_EFFECTS
2
OBSERVATION(S)
5
WEIGHT(S)

EFFECTS:
2 2 cross
1 8 cross
RANDOM_RESIDUAL VALUES
1.0
RANDOM_GROUP
2
RANDOM_TYPE
add_animal
FILE
pedigree_mr03a.txt
(CO)VARIANCES
0.5
OPTION solv_method FSPAK
OPTION sol se

The solutions are the following.

trait/effect level  solution          s.e.
   1   1         1          4.35850233          0.77172597
   1   1         2          3.40443010          0.89580057
   1   2         1          0.09844458          0.68636303
   1   2         2         -0.01877010          0.70149535
   1   2         3         -0.04108420          0.67561734
   1   2         4         -0.00866312          0.65397259
   1   2         5         -0.18573210          0.65429867
   1   2         6          0.17687209          0.66504343
   1   2         7         -0.24945855          0.66472263
   1   2         8          0.18261469          0.64989549

This parameter file will provide the same solutions as before. An advantage of this method is to possibly reduce the numerical error because of \(\sigma_e^2=1\), which doesn’t add the noise to the equations. We square the s.e. to obtain a diagonal element of the inverse of the left-hand side matrix. You can calculate the reliability or accuracy of the estimated breeding value by hand.

For example, the diagonal element for animal 1 is \(d_1 = 0.68636303^2 \approx 0.471\) which is the same as the reference value in the textbook (pp.45). The \(\mathrm{PEV}\) for animal 1 is \(\mathrm{PEV}_1 = d_i\sigma_e^2= 0.471 \times 40 = 18.84\). The standard error of prediction is \(\mathrm{SEP}_1 = \sqrt{\mathrm{PEV}_1} = \sqrt{18.84} = 4.341\) which is also the same as the textbook. The reliability for animal 1 is \(r^2 = 1 - \mathrm{SEP}^2 /\sigma_u^2 = 1 - \mathrm{PEV}/\sigma_u^2 = 1 - 18.84/20 = 0.058\). You can confirm that this rule is true for the other animals.

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