Dear Luc and others,
Luc's example of REML (ASREML) getting an answer for a covariance
when there is no information is an instructive one.
His data consisted of weights of surviving animals, missing values
for non surviving animals, and he was doing a bivariate analysis
of weight with survival.
I report what ASREML produces for a range of starting values.
The DATA file is
14.53 1
10.12 1
16.30 1
15.54 1
13.36 1
12.77 1
14.06 1
12.23 1
* 0
* 0
* 0
* 0
The basic ASREML job is
Luc Janss example # Header
W S # data fields
ljanss.dat # data file name
W S ~ Trait # bivariate linear model
1 2
12 # 12 records
2 0 US 3 .2 .3 # Error variance matrix (input)
For the two cases Luc reported, the LogL values from ASREML
were -8.476 and -9.01 which agree with his values except for a factor
of -0.5.
I ran ASREML to convergence with a range of covariances. In every case
the LogL converged to the same value but the WT variance
and the Covariance estimate changed depending on the starting value.
The cases were
Initial W Fitted S
Covariance Variance Covariance Variance
-.1 3.8644 -.0808 0.24242
0.0 3.8375 0.0000 0.24242
0.05 3.8442 0.0404 0.24242
0.1 3.8644 0.0808 0.24242
0.2 3.9452 0.1616 0.24242
0.4 4.2685 0.3232 0.24242
All these final models had the same LogL and the determinant of the matrix
is the same as well (0.9303).
If ASREML detects a singularity in the Average Information matrix,
it will not update the corresponding parameter but will flag it
as having no information. This typically occurs with variances
but for covariances, a singularity does not appear in
the information matrix so all parameters are updated.
In this case, there is no maximum value to the likelihood; just
a ridge of maximum likelihood and so all solutions have
equal likelihood. I.e. ASREML is combining the information
in the model (and initial values) with the data but once it
makes the variance of W compatible with the Covariance,
it has no information with which to further change the covariance.
In other words, if you supply a covariance and there is no
information in the data on the value of that covariance, REML
cannot contradict your initial value except so far as it
is inconsistent with the variance of W in this case.
This agrees with Dan's conclusion (erratum)
I trust thus explanation is helpful.
Arthur
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Arthur Gilmour PhD email: Arthur.Gilmour@agric.nsw.gov.au
Senior Research Scientist (Biometrics) fax: <61> 2 6391 3899
NSW Agriculture <61> 2 6391 3922
Orange Agricultural Institute telephone work: <61> 2 6391 3815
Forest Rd, ORANGE, 2800, AUSTRALIA home: <61> 2 6362 0046
ASREML is still free by anonymous ftp from pub/aar on ftp.res.bbsrc.ac.uk
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