p-values glmer in lme4

This diagnosis sounds correct, and I agree that calling these numbers
"z values" is probably the best way to make the reviewers happy.

It opens an interesting terminological can of worms.  My initial
reaction to John's post was "oh, I guess glmer should print 'z value'
rather than 't value' even for fits using families with an estimated
dispersion parameter". Then I thought "but if that's true shouldn't lmer
also print 'z value' rather than 't value', since it provides
essentially the same numbers?" Then I thought "if we switch lmer to
printing 'z value' will everyone start asking 'why does lmer provide z
values rather than t values?"  Sigh.

  The point is that most of this, while unfairly confusing, is just
convention.  "z values" and "t values" are the same thing - MLEs (or
REML estimates) of the parameters divided by their estimated standard
deviations. Of the common (G)LMM applications, the *only* case in which
these values are actually known to follow a t distribution exactly is
for linear mixed models (Gaussian conditional distribution), in the
classic case of a balanced, nested design (and, implied by John below,
that the fit uses REML). Otherwise it becomes a question of which
approximations you're happy with.

  And the sampling distributions of these values are never Normal (even
in the perfect theoretical world where all model assumptions are true),
except asymptotically.

p-values glmer in lme4

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