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Multiple-trait model with equal design matrix and no missing observations

Model

Here, we consider a multiple-trait model with equal design matrices, which means that the same model is applied to all traits. We consider a situation without missing observations.

In this example, the author considers a two-trait model with pre-weaning gain as the first trait (WWG) and post-weaning gain as the second trait (PWG). Both traits have the same model: a fixed effect for sex and two random effects, the additive genetic effect and the residual effect.

We also consider the genetic covariance between the two traits, as well as the genetic variance for each trait. The same applies to the residual covariance components. The genetic covariance matrix (\(\mathbf{G}_0\)) and the residual covariance matrix \(\mathbf{R}_0\) are both symmetric (\(2 \times 2\)) matrices. The values are defined in the textbook as \[ \mathbf{G}_{0} = \left[ \begin{array}{rr} 20&18\\ 18&40 \end{array} \right] \quad \text{and} \quad \mathbf{R}_{0} = \left[ \begin{array}{rr} 40&11\\ 11&30 \end{array} \right] \]

The matrix notation for the above model and its mixed model equations has already been introduced in the previous chapter. Here, we show only some relevant equations. The model can be written as \[ \mathbf{y} = \mathbf{Xb} + \mathbf{Zu} + \mathbf{e}. \] There are two ways to order the observations in \(\mathbf{y}\) (within-trait or within-animal). As discussed in the previous chapter, BLUPF90 orders the observations within-animal (see the previous chapter for details). The variance of \(\mathbf{y}\) is \[ \mathrm{var}(\mathbf{y}) = \mathbf{Z}\left(\mathbf{A} \otimes \mathbf{G}_{0}\right)\mathbf{Z}' + \mathbf{I} \otimes \mathbf{R}_{0} \] where \(\otimes\) is the Kronecker product operator. The mixed model equations can be written as \[ \left[ \begin{array}{ll} \mathbf{X}'(\mathbf{I}\otimes\mathbf{R}_{0}^{-1})\mathbf{X} & \mathbf{X}'(\mathbf{I}\otimes\mathbf{R}_{0}^{-1})\mathbf{Z} \\ \mathbf{Z}'(\mathbf{I}\otimes\mathbf{R}_{0}^{-1})\mathbf{X} & \mathbf{Z}'(\mathbf{I}\otimes\mathbf{R}_{0}^{-1})\mathbf{Z} + \mathbf{A}^{-1}\otimes\mathbf{G}^{-1}_{0} \end{array} \right] \left[ \begin{array}{c} \mathbf{\hat{b}}\\ \mathbf{\hat{u}} \end{array} \right] = \left[ \begin{array}{l} \mathbf{X}'(\mathbf{I}\otimes\mathbf{R}_{0}^{-1})\mathbf{y} \\ \mathbf{Z}'(\mathbf{I}\otimes\mathbf{R}_{0}^{-1})\mathbf{y} \end{array} \right] \]

Files

The data file (mr61.dat) is an extension of Example 3.1. It has an additional observation in the last column.

1. Animal ID (calves)
2. Sire ID
3. Dam ID
4. Pre-weaning gain (WWG; kg)
5. Post-weaning gain (PWG; kg)

The pedigree file (mr61.ped) is also the same as the previous one.

The parameter file contains two covariance matrices. We will compute the standard error of each estimate by taking the square root of the corresponding diagonal element of the inverse of the left-hand side of the mixed model equations.

DATAFILE
mr61.dat
NUMBER_OF_TRAITS
2
NUMBER_OF_EFFECTS
2
OBSERVATION(S)
5 6
WEIGHT(S)

EFFECTS:
2 2 2 cross
1 1 8 cross
RANDOM_RESIDUAL VALUES
40.0 11.0
11.0 30.0
RANDOM_GROUP
2
RANDOM_TYPE
add_animal
FILE
mr61.ped
(CO)VARIANCES
20.0 18.0
18.0 40.0
OPTION solv_method FSPAK
OPTION sol se
OPTION store_accuracy 2

Solutions

The file solution contains the solutions for the first and second traits. The solutions are identical to those shown in the textbook (pp. 101–102).

The reliability of an animal’s estimated breeding value can be calculated from the solutions produced by BLUPF90+. The reliability for trait \(j\) of animal \(i\) (\(r_{ij}^2\)) is \[ r_{ij}^2=1-\frac{\mathrm{PEV}_{ij}}{\sigma_{u_j}^{2}} \] where \(\mathrm{PEV}_{ij}\) is the diagonal element of the inverse of the left-hand side of the mixed model equations for the effect of interest. In the above solutions, the column s.e. contains the square root of the PEV (\(\sqrt{\mathrm{PEV}_{ij}}\)). For example, the reliabilities for animal 1 can be calculated as follows: \[ r_{11}^2=1-\frac{4.313320612^2}{20}\approx 0.070 \quad \text{and} \quad r_{21}^2=1-\frac{5.991809902^2}{40}\approx 0.102. \]

The additional option OPTION store_accuracy 2 calculates the reliability (not accuracy) of the breeding-value solutions and stores the results in acc_bf90.

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