[R-sig-ME] p-values glmer in lme4

Ben Bolker bbolker at gmail.com
Wed Jul 19 23:18:18 CEST 2017

   I agree that the label should go according to the inference used. The
funny (??) thing is that in base lmer (not in the lmerTest extension
package) *the summary doesn't print p-values at all*, so according to
the rubric above we can't tell whether the (mean/stderr) column should
be labeled 'z' or 't'. (We could label that column 'mean/std.err.', but
then some users would ask "why doesn't the summary print either z or t
values?" :-( )

On 17-07-19 03:58 PM, Fox, John wrote:
> 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
>>>> [[alternative HTML version deleted]]
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