I have a data set which consists of a binary response, two variables of random effects and a dozen predictors, most of which are categorical. The data set has some 1400 observations, one third of which have missing data (only in predictors). Missing data in each column varies from 0 to 18%, and about 3% of all data is missing.
My first idea was to use complete case analysis, since I would keep almost 1000 observations, and the proportion of the response variable keeps the same (~12% for all cases, only complete cases and only excluded cases). However, I'm not sure that data is MCAR and I didn't find a way to test it with categorical variables. Moreover, other options could provide better estimates.
I've seen that GLMM deals well with missing data, since it's based on maximum likelihood, but I'm not sure how it's different from a simple listwise/pairwise deletion followed by model selection. Does it really take observations with missing data into account and how does it work? (if it helps, I'm using glmer function on R).
At last, I've read a little about multiple imputation, but it doesn't seem to work well for categorical variables and I was not sure if it works with mixed models. Also, I've seen that there were no such big differences between results obtained with or without MI, at least for longitudinal mixed-model analysis (although my data is clustered, not longitudinal).
Is the GLMM a good option? If so, do I need to do something different do deal with the missing data? But if not, what should I do?