Which assumptions do I need to check for a GLMM with a binary response (and how?)

Question

I am modeling binomial responses using Generalized Linear Mixed Models with a nested random effect (not of interest, simply a control: year nested within location) and both categorical, count, and continuous fixed effects.

Which assumptions do I need to check for this kind of model and how do I check them?

So far I have gathered the following:

1) Overdispersion can not be detected in binomial response data. Solution: assume the data are not overdispersed.

2) Checking for heteroscedasticity is complicated and there is no good fix if you detect it. Solution: assume homoscedasticity

3) Check for outliers that are over-influencing the model. Solution: plot residuals against fitted values and look for outliers.

4) Make sure that residual variance does not differ across groups. Still looking for a way to do this.

What here is right/wrong?

Answer 1

In principle, a binomial can have all the problems any GLMM can have, in particular

1. misfit
2. distribution problems, including dispersion problems such as the four points you mention
3. spatial / temporal / ... residual correlation
4. RE problems, e.g. non-normality of REs

For problems 1-3, I would recommend to use the DHARMa R package (disclaimer: I'm the developer).

With DHARMa, misfit (e.g. nonlinear effect of a predictor) can be quantified as for any normal regression model, by plotting res ~ pred.

Regarding dispersion problems: while it is true that 0/1 data does not have dispersion, k/n binomial data has. Thus, when you group your 0/1 data, you can indeed encounter dispersion problems such as OD, heteroskedasticity, outliers and variance, and they are usually indicative of model error. See section on binomial data in the DHARMa vignette for details.

Note that you can additionally have residual autocorrelation in 0/1 data (e.g. spatial, temporal, phylogenetic). DHARMa allows to check for that.

Regarding 4: standard mixed models assume that the REs are normally distributed, which does not need to be the case. Such problems can show up in DHARMa residuals, but I would recommend to just make a qq plot of the REs, and possibly also have a look if REs show autocorrelation or a correlation with a predictor (in which case you may want to modify your model structure).