Checking modeling assumptions in a binomial GLMM

Dear Ben,

Thank you for the helpful response.  I had posted the question to

r-sig-mixed last week, but I did not hear from anyone.  Perhaps, the
  moderator never approved my post.  Hence, the post to r-help.

[cc'ing to r-sig-mixed-models now]

My example has repeated binary (0/1) responses at each visit of a

clinical trial (it is actually the schizophrenia trial discussed in
  Hedeker and Gibbons' book on longitudinal analysis).  My impression
  was that diagnostics are quite difficult to do, but was interested
  in seeing if someone had demonstrated this.

I have some related questions: the glmer function in "lme4" does not

handle nAGQ > 1 when there are more than 1 random effects.  I know
  this is a curse of dimensionality problem, but I do not see why it
  cannot handle nAGQ up to 9 for 2-3 dimensions.  Is Laplace's
  approximation sufficiently accurate for multiple random effects?  Is
  mcmcGLMM the way to go for binary GLMM with multiple random effects?


To a large extent AGQ is not implemented for multiple random effects
(or, in lme4 >= 1.0.0, for vector-valued random effects) because we
simply haven't had the time and energy to implement it.  Doug Bates has
long felt/stated that AGQ would be infeasibly slow for multiple random
effects.  To be honest, I don't know if he's basing that on better knowledge
than I (or anyone!) have about the internals of lme4 (e.g. trying to
construct the data structures necessary to do AGQ would lead to a
catastrophic loss of sparsity) or whether it's just that his focus
is usually on gigantic data sets where multi-dimensional AGQ truly
would be infeasible.

  Certainly MCMCglmm, or going outside the R framework (to SAS
PROC GLIMMIX, or Stata's GLLAMM
<http://www.stata-press.com/books/mlmus3_ch10.pdf>), would be my first
resort when worrying about whether AGQ is necessary.
Unfortunately, I know of very little discussion about how to determine
in general whether AGQ is necessary (or what number of quadrature
points is sufficient), without actually doing it -- most of the examples
I've seen (e.g. <http://www.stata-press.com/books/mlmus3_ch10.pdf>
or Breslow 2003) just check by brute force (see
http://rpubs.com/bbolker/glmmchapter for another example).  It would
be nice to figure out a score test, or at least graphical diagnostics,
that could suggest (without actually doing the entire integral) how
much the underlying densities departed from those assumed by the
Laplace approximation.  (The zeta() function in
http://lme4.r-forge.r-project.org/JSS/glmer.Rnw might be a good
starting point ...)

  cheers
    Ben Bolker