Lmer and variance-covariance matrix

On Fri, Mar 11, 2011 at 2:37 PM, Rolf Turner<r.turner at auckland.ac.nz>  wrote:

On 12/03/11 02:56, Jarrod Hadfield wrote:
Hi,

In addition, each trait is only measured once for each id (correct?)
which means that the likelihood could not be optimised even if the
data-set was massive. If you could fix the residual variance to some
value (preferably zero) then the problem has a unique solution given
enough data, but I'm not sure this can be done in lmer?.
<SNIP>

I think that it ***CANNOT*** be done.  I once asked Doug about
the possibility of this, and he ignored me.  As people so often
do. :-) Especially when I ask silly questions .....

Did Doug really ignore you or did he say that the methods in lmer are
based on determining the solution to a penalized linear least squares
problem so they can't be applied to a model that has zero residual
variance.  Also the basic parameterization for the variance-covariance
matrix of the random effects is in terms of the relative standard
deviation (\sigma_1/\sigma) which is problematic when \sigma is zero.

(My apologies if I did ignore you, Rolf.  I get a lot of email and
sometimes such requests slip down the stack and then get lost.  I'm
very good at procrastinating about the answers to such questions.)

Yes, you really did ignore me.  But not to worry; I'm used to it! :-)
I also (more recently) asked Ben Bolker about this issue.  He
ignored me too!  At that stage I kind of took the hint ......

Your explanation of why it can't be done makes perfect sense.

However I find this constraint sad, because I like to be able to
fit ``marginal case'' models, which can also be fitted in a more
simple-minded manner and compare the results from the
simple-minded procedure with those from the sophisticated
procedure.  If they agree, then this augments my confidence
that I am implementing the sophisticated procedure correctly.

An example of such, relating to the current discussion, is a
simple repeated measures model with K (repeated) observations
on each of N subjects, with the within-subject covariance matrix
being an arbitrary positive definite K x K matrix.

This could be treated as a mixed model (if it were possible to
constrain the residual variance to be 0).  It can also be treated
as a (simple-minded) multivariate model --- N iid observations
of K-dimensional vectors, the mean and covariance matrix of
these vectors to be estimated.

I would have liked to be able to compare lmer() results with the
(trivial) multivariate analysis estimates.  To reassure myself that
I was understanding lmer() syntax correctly.

     cheers,

         Rolf