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?



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?



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?



Does the interpretation of the SE change between different kinds of GLMMs
because, for example in the logistic GLMM the residual variance is
restricted?





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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