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Multiple-trait model with equal design matrix and missing records

Model

Here we apply the same model described in the previous example to the data with missing observations. Model descriptions and mixed model equations are identical as before.

With a missing observation, \(\mathbf{R}_0\) and its inverse should be altered. For example, assuming a 2 trait model, if the observation of the first trait is missing, the first row and column in \(\mathbf{R}_0\) should be zeroed out. The corresponding inverse is the generalized inverse of this altered \(\mathbf{R}_0\). Illustrating this situation with the previous example, the result is \[ \mathbf{R}_{0} = \left[ \begin{array}{rr} 0&0\\ 0&30 \end{array} \right] \quad \text{and} \quad \mathbf{R}_{0}^{-} = \left[ \begin{array}{rr} 0&0\\ 0&30 \end{array} \right]^{-} = \left[ \begin{array}{rr} 0&0\\ 0&1/30 \end{array} \right]. \]

The generalized inverse of this zeroed matrix is equivalent to the inverse of a matrix containing only nonzero elements in the zeroed matrix (Searle, 1971). BLUPF90 can detect a missing observation and prepares an appropriate \(\mathbf{R}_{0}\) and its generalized inverse.

Files

One animal is added to the previous example and 2 observations are marked as missing. The missing observation is indicated as 0, which is the default missing code used in the BLUPF90 family (data_mr05b.txt). We can use an extended pedigree file as the previous one by adding the animal 9 (pedigree_mr05b.txt).

The parameter file is also identical except for omitting an option for standard error calculations.

DATAFILE
data_mr05b.txt
NUMBER_OF_TRAITS
2
NUMBER_OF_EFFECTS
2
OBSERVATION(S)
5 6
WEIGHT(S)

EFFECTS:
2 2 2 cross
1 1 9 cross
RANDOM_RESIDUAL VALUES
40.0 11.0
11.0 30.0
RANDOM_GROUP
2
RANDOM_TYPE
add_animal
FILE
pedigree_mr05b.txt
(CO)VARIANCES
20.0 18.0
18.0 40.0
OPTION solv_method FSPAK

Solutions

You can confirm the results are identical to the values in the textbook (p.80).

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