[R-sig-ME] Lmer and variance-covariance matrix

[Rolf Turner]

On 12/03/11 09:45, Douglas Bates wrote:
> 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