Poisson or Multinomial Logistic Regression (or something else)? There seems to be a lot of discussion about this on CV but none quite answer my question.
I have a variable y which represents number of adverse events occurring within a four-week period. There are three types of adverse event: 'mild', 'moderate' and 'severe' (and of course 'none').
There are three four-week periods. Thus each participant will have a count of number of each of the three incidents in each four-week period.
Now for my question. What regression model would be best here? A multilevel random slopes GLM with ID as the random factor and period (1 - 3) and experimental group (placebo vs treatment) as the fixed factors seems appropriate, but what sort?
Both poisson and multinomial logistic regression deal with incidences (i.e. odds of occurence) of discrete outcomes, but I am not really sure what category my analysis falls into. If the outcome was a binary one (i.e. a single event type and you either experience or don't) then it would be poisson/Negative binomial for certain. But the fact that there are three possible events has me at a loss for what analysis to run.
Any help appreciated. 
 A: If you have an ordinal response (discrete outcomes, no metric, ordered), you would normally use an ordered (mixed) regression, which can be comfortably fit with 
https://cran.r-project.org/web/packages/ordinal/index.html
You probably want to specify a random intercept per subject, and you might think about having a random slope time|subject to account for temporal trends per subject. 
I don't really see how this is changed by observing multiple events, as you allude to in your comments to the answer by Björn - multiple observations should not be a problem. Or could the same subject at the same time simultaneously have a mild AND a moderate outcome?
A: You would typically use Poisson or Negative Binomial (the latter if rates are assumed vary across patients), if the outcome is 0, 1, 2, 3, .... events in a time period - possibly with a log (observation time) offset. Logistic regression, if it's yes/no, but if you know when the event happened, time to event methods would typically preferable for a number of reasons (one group may have more/earlier drop-outs, if everyone has an event the only way treatments can differ is in the timing etc.).
That you have multiple observations per subject from different periods is indeed important (e.g. one patient may simply have a high chance of getting AEs due to being a bit frail). You account for this by the random effect that tries to take into account how much the outcomes for a patient in one period tells you about his/her oitcomes in another period.
