[R-sig-ME] Lmer and variance-covariance matrix
Rolf Turner
r.turner at auckland.ac.nz
Fri Mar 11 23:11:40 CET 2011
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
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