glmer Z-test with individual random effects

Dear list,

As I have read (Bolker et al. 2009 TREE), the Wald Z test is only
appropriate for GLMMs in cases without overdispersion.

Assuming we use family=poisson with lmer and tackle overdispersion by
incorporating an individual random effect AND this adequately "reduces"
the overdispersion, is it then OK to use the Wald z test as reported by
lmer?

In other words, are the p-values reported by lmer in those cases
useful/"correct"? Or do they suffer from the usual problems with
figuring out the number of parameters used by the random effects?

They are equivalent to assuming an infinite/large 'denominator degrees
of freedom'.  If you have a large sample size (both a large number of
total samples relative to the number of parameters, and a large number
of random-effects levels/blocks) then this should be reasonable -- if
not, then yes, the 'usual problems with figuring out the number of
parameters' is relevant.  On the other hand, if you're willing to assume
that the sample size is large, then likelihood ratio rests
(anova(model1,model2)) are probably better than the Wald tests anyway.

Secondly, is it good practice to judge lmer's capability of "reducing"
the overdispersion by summing the squared residuals (pearson) and
compare this to a chi square distribution (with N-1 degrees of freedom)?

I would say this is reasonable, although again it's a rough guide
because the true degrees of freedom are a bit fuzzy -- it should
probably be at most N-(fixed effect degrees of freedom)?

   Would be happy to hear any conflicting opinions.

  Ben Bolker