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p-values glmer in lme4

2 messages · John Fox, Ben Bolker

Original
John Fox
# 
Dear Leen Catrysse,

I'm going to assume that you used the glmer() function in the lme4 package to fit your gamma GLMM. I notice that the summary() for a gamma model fit by glmer() reports a "t value" for each fixed-effect coefficient  -- simply the Wald statistics given by the ratio of the estimated coefficient to its estimated asymptotic standard error -- followed by a "Pr(>|z|)". 

I suspect that the Wald statistic is labelled as a "t value" because the gamma GLMM has an estimated dispersion parameter, but because there are no degrees of freedom calculated for the estimated dispersion (as there could be, for example, for a LMM fit by REML), I think that it would probably be preferable to call the Wald statistic a "z value." In any event, the notation "Pr(>|z|)" suggests that the standard normal distribution is used to obtain a p-value.

So, to satisfy the reviewer, why not just call the Wald statistics "z-values" rather than a "t-values"?

I hope this helps,
 John

-----------------------------
John Fox, Professor Emeritus
McMaster University
Hamilton, Ontario, Canada
Web: socserv.mcmaster.ca/jfox
Ben Bolker
# 
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.
On 17-07-19 02:50 PM, Fox, John wrote: