p-values glmer in lme4

Hi Ben,

I'm glad that you chimed in on this question.

Of course, what you say about (virtually) all the p-values being
approximations is correct. My own preference would be to use
"t-value" when you look up a p-value (approximate or not) for a Wald
statistic in a t-distribution and "z-value" when you look up (an
asymptotic approximation to) a p-value in the standard-normal
distribution. Frankly, however, this seems a bit like splitting
hairs, and so I think that what you do now is fine.

Best, John

-----Original Message----- From: R-sig-mixed-models
[mailto:r-sig-mixed-models-bounces at r-project.org] On Behalf Of Ben
Bolker Sent: Wednesday, July 19, 2017 3:29 PM To:
r-sig-mixed-models at r-project.org Subject: Re: [R-sig-ME] 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.


On 17-07-19 02:50 PM, Fox, John wrote:

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

-----Original Message----- From: R-sig-mixed-models 
[mailto:r-sig-mixed-models-bounces at r-project.org] On Behalf Of
Catrysse Leen Sent: Wednesday, July 19, 2017 7:21 AM To:
r-sig-mixed-models at r-project.org Subject: [R-sig-ME] p-values
glmer in lme4

Dear,

I used GLMM to analyse eye-tracking data with the gamma
distribution and the log link. P-values were automatically
computed in the output (based on the asymptotic Wald tests). We
received a comment of a reviewer that the output of our GLMM
is inconsistent, as we report a t-value from the output and the
p-value based on the asymptotic Wald tests. Does anyone has
some feedback on how we can deal with this comment?

Thanks in advance, Leen Catrysse

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