[forwarding to r-sig-mixed-models as well]

---------- Forwarded message ----------
From: Ben Bolker <bbolker@gmail.com>
Date: Sat, Aug 2, 2014 at 6:09 PM
Subject: Re: [R-sig-ME] Standard errors in generalised linear mixed models?
To: Felix Suessenbach <felix.suessenbach@googlemail.com>


A lot of this is discussed at http://glmm.wikidot.com/faq ...

 I've given some answers below, invite further explication from other list
readers ...


On Sat, Aug 2, 2014 at 12:13 PM, Felix Suessenbach <
felix.suessenbach@googlemail.com> wrote:

> Dear Prof. Bates and colleagues,
>
>
>
> I have read your comments about the problem of significance testing in
> generalized linear mixed models (GLMM). If I understood it correctly
> significance testing of a predictor in GLMMs is tricky because when
> performing a likelihood ratio test (model comparison with and without the
> predictor) one cannot be sure that the underlying chi-square distribution
> is approximated correctly. More importantly, one would need the degrees of
> freedom in both models which cannot be obtained since a calculation like df
> = N-q-1 (N = number of groups, q = number of independent parameters) would
> not be possible as the parameters are not independent from each other. Is
> this brief depiction of the problem correct?
>

   More or less.  The problem is not specifically that the parameters "are
not independent from each other" (this is actually typical of any linear
model that is *not* balanced with orthogonal contrasts, and any
non-Gaussian generalized linear model).  It is that the sampling
distribution of the summary statistic that would correspond to an F
statistic is not actually F-distributed.

>
>
>
> I am currently working with a logistic multi-level model with a binary
> dependent variable, this is why I was confronted with this question.
>
>
>
> In the model summary I can see that glmer provides p-values not on the
> basis of a likelihood ratio test but from simply dividing the coefficient
> by its SE and comparing this value to a z distribution. I have the nagging
> feeling that I cannot trust this p value either. But why?
>
>
  These are Wald statistics (again see glmm FAQ URL referenced above),
which involve a different approximation (quadratic log-likelihood surface).


> Is this because we cannot interpret the estimate of the standard error (SE)
> as we would interpret ordinary SEs? How can it be interpreted?
>
> I understand the SE is estimated by ML or REML in glmer but does this
> estimation already involve knowing how many degrees of freedom one has in
> the model? Can the estimate of the SE be trusted?
>
>
It's not estimated in REML for GLMMs because we don't think we know how to
do that (again see the GLMM FAQ).  The estimate of the SE should be a
reasonable estimate of the uncertainty of the parameter if the sample size
is large enough, parameters are not too extreme, etc..  If you are
concerned, estimate the confidence intervals by profiling or parametric
bootstrap (see ?confint.merMod), and compare with the Wald estimate.



> Does the interpretation of the SE change between different kinds of GLMMs
> because, for example in the logistic GLMM the residual variance is
> restricted?
>
>
  Not really


>
>
> I would be more than thankful if you could explain this to me. Feel free to
> restrict the explanation to logistic multilevel models with a binary
> outcome variable.
>
>
>
> Kind regards,
>
> Felix Süßenbach
>
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>
>
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