[R-sig-ME] Distribution of deviance residuals

[Ben Bolker]

Roelof Coster <roelofcoster at ...> writes:

> 
> Hello,
> 
> What is the theoretical distribution of the residual deviances of a
> well-fitting logistic regression mixed model?
> 
> The background of my question is as follows: I am looking for a way to
> combine the ideas from regression tree modelling (aka model trees) and
> mixed models. I have a data set to which I want to fit a logistic
> regression model. My data come in groups, so I need a random effect to
> account for those groups.
> 
> The mob function in the party package does what I need, but only for
> fixed-effects models. As I understand it, that function arranges the data
> according to the levels of a certain categorical predictor. Next, it looks
> at the sequence obtained by cumulating the deviance residuals. Then a
> hypothesis test is done to assess whether this sequence can plausibly be a
> Brownian motion. If it isn't a Brownian motion, that is an indication that
> the data set should be splitted in two and that two separate models should
> be fitted. This process is repeated so that a binary tree is produced,
> with
> a logistic regression model for a part of the
> data in each leaf of the tree.
> 
> I would be grateful for any advice on how this technique can be made to
> work for my problem.


  Do you mean the deviance residuals (i.e. the per-observation contributions
to the deviance, or the signed square root of the contribution) or the total
residual deviance (i.e., the sum of squared deviance residuals)?
   If the former, I think you're probably in trouble -- empirically,
the distribution of residuals is usually pretty ugly, and theoretically
I can't think of a reason it should be nice.  If the latter, then
you can presumably just rely on asymptotic theory which would say
(I'm being sloppy here of course) that the sum of squares of lots of
iid things should be chi-square distributed (and then eventually
Normal).
  For what it's worth, a great deal of the theory of GLMMs is inherited
from GLM theory, so if you can solve your problem or find a solution
for a plain old *non*-mixed logistic regression, it is likely to work
reasonably well for a mixed logistic regression as well (provided you
have a reasonable number of levels of the random effect/your estimate
isn't singular).  (Conversely if it's known to be nasty for ordinary
logistic regression you're probably screwed.)

  As always I'm happy to be corrected by more sensible/knowledgeable
people.