Variance Components Model
VCM Variance Components Model
The user may wish to de ne relationships between particular variance parameters. For
example, consider an experiment in which two or more separate trials are sown adjacent
to one another at the same trial site, with trials sharing a common plot boundary. In
this case it might be sensible to fit the same spatial parameters and error variances for
each trial. In other situations it can be sensible to define the same variance structure
over several model terms. ASReml 3 catered for equality and multiplicative relationships
among variance parameters. In ASReml 4 linear relationships among variance structure
parameters can be defined through a simple linear model and by supplying a design matrix
for a set of parameters. The design matrix is supplied as an ascii file containing a row for
each parameter in a set of contiguous parameters and a column for each new parameter.
This design matrix is associated with the job through a statement after the residual model
defnition line(s), of the form:
VCM first last [middle] new filename
where
first is the parameter number of the first parameter in the set,
last is the parameter number of the last parameter in the set,
new is the number of new parameters, and
filename is the name of the file containing the design matrix.
middle is the list of parameters between first and last,
required if not all parameters between first and last are involved.
For example, the Wolfinger rats example involves modelling a 5 cross 5 symmetric residual
matrix.
Wolfinger Rat data
treat !A
wt0 wt1 wt2 wt3 wt4
subject * !=V0
wolfrat.dat !skip 1 !ASUV !VCC 2
wt0 wt1 wt2 wt3 wt4 Trait treat Trait.treat
residual units.us(Trait)
Wolfinger (1996) reports the fitting of the HuynhFeldt variance structure to this data.
This structure is of the form
σii = σni
σij = (σni + σnj)/2 - σno, j < i <= p
In the rats example, the relationship between the original and new parameters is Σ =
Mσn where σ and σn are 15 cross 1 and 6 cross 1 vectors respectively, and M is a 15 cross 6 matrix:
1 0 0 0 0 0
0.5 0.5 0 0 0 -1
0 1 0 0 0 0
0.5 0 0.5 0 0 -1
0 0.5 0.5 0 0 -1
0 0 1 0 0 0
0.5 0 0 0.5 0 -1
0 0.5 0 0.5 0 -1
0 0 0.5 0.5 0 -1
0 0 0 1 0 0
0.5 0 0 0 0.5 -1
0 0.5 0 0 0.5 -1
0 0 0.5 0 0.5 -1
0 0 0 0.5 0.5 -1
0 0 0 0 1 0
A way of fitting this model would be to put the matrix values in a file HuynhFeldt.vcm
and use the statement
VCM 5 19 6 HuynhFeldt.vcm #parameters 5 to 19 explained in terms of 6 parameters
Note that if the user fits another model with differing numbers of variance structure
parameters so that the variance structure parameters are renumbered, then all the user needs
to do to continue with the same relationships is to change the first and last parameters
on the VCM line.
Creating the design matrix
The
!DESIGN
qualifier causes ASReml to write out the deisgn matrix.
The following ASReml job will create the design matrix used above.
Create VCM Design for H-F model
Row * Col * Off
Y !=V0
vcmdes.asd !DESIGN
Y ~ Row and(Row,-0.5) and(Col,0.5) Off
where the data file vcmdes.asd is displayed on the left,
and the design matrix produced in vcmdes.des is displayed
in the right.
1 1 0 1 0 0 0 0 0
1 2 -1 0.5 0.5 0 0 0 -1
2 2 0 0 1 0 0 0 0
1 3 -1 0.5 0 0.5 0 0 -1
2 3 -1 0 0.5 0.5 0 0 -1
3 3 0 0 0 1 0 0 0
1 4 -1 0.5 0 0 0.5 0 -1
2 4 -1 0 0.5 0 0.5 0 -1
3 4 -1 0 0 0.5 0.5 0 -1
4 4 0 0 0 0 1 0 0
1 5 -1 0.5 0 0 0 0.5 -1
2 5 -1 0 0.5 0 0 0.5 -1
3 5 -1 0 0 0.5 0 0.5 -1
4 5 -1 0 0 0 0.5 0.5 -1
5 5 0 0 0 0 0 1 0
The HuynhFeldt variance structure can then be fitted to the
Wolfinger rat data after renaming vcmdes.des to HuynhFeldt.vcm with the ASReml job
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