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

Ben Bolker bbolker at gmail.com
Sat Mar 12 00:50:59 CET 2011

On 03/11/2011 05:11 PM, Rolf Turner wrote:
> 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 ......

  I want to distinguish two cases here.
  It would be reasonable (although I think not currently feasible) to
fix any variance parameter *other than the residual variance* to zero,
refitting the model with that constraint, to do a test of the effect of
that parameter (keeping in mind the various limitations of finite sample
sizes/unknown null distributions, testing on the boundary, blah blah
blah).  This is a special case of the machinery that has to be built in
order for profiling of the random effects to work, and at a pinch you
can get the answer (if you can get lme4a to run on your system) by
fitting the profile and looking at the value at the boundary.
  It is reasonable, but not within the framework of lme4, to set one
particular random effect -- the residual variance -- to zero, because
(as Doug points out) it has a special role.

    Ben Bolker

> 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
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