# Poisson or Multinomial Logistic Regression (or something else)?

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.

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.

• Thank you @Bjorn. Howver it is not 0,1,2,3 events in a time period. It is more like number of events of type 1, 2, or 3 (e.g. a single person in a single time period could experience two events of type 1, one of type 2, and three of type three). I agree that time to each type of event could be interesting, but it is not prinamrily what we are interested in. We are more interested in number of each type of event. – llewmills Nov 10 '17 at 15:42
• Different event types adds an extra level of complexity. Per-se, how you model each depends on what you have: occurred y/n or number of events in this time period (some integer in [0 to $\infty$)) Either you look at them separately (and have a multiplicity problem, which you may want to do something about) or you try to have some kind of hierarchical model that induces some kind of sensible shrinkage (e.g. à la Amy Xia, H., Ma, H., & Carlin, B. P. (2011). Bayesian hierarchical modeling for detecting safety signals in clinical trials. Journal of biopharmaceutical statistics, 21(5), 1006-1029.). – Björn Nov 10 '17 at 16:03
• Yes it did occur to me that a separate Poisson/NB GLMM for each event type would do it, but I wondered if it could be done in a single analysis. Thanks for the reference. I will have a look. – llewmills Nov 10 '17 at 16:06
• If they are very similar event types (e.g. event 1=cardiovascular death, event 2=MACE and event 3=MACE+), then a shared frailty model or some kind of copula model could be an option – Björn Nov 10 '17 at 16:11

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?

• No @Florian Hartig they can only have one observation at one time. Thank you for the advice. But it is not the fact that there can be multiple observations that has perplexed me, but rather that there are multiple types of observations that can each be observed multiple times. I suppose I am confused about what separates count data from categorical data observed multiple times. Both Poisson/NB and logistic/multinomial logistic deal test the likelihood of event occurrence, so how are they different? – llewmills Nov 15 '17 at 1:45
• Poisson/NB is simply not appropriate for a response with a fixed number of outcomes - you have outcomes 1-4, but a NB can also predict a 5 (no upper bound). Multinomial is used when the response is unordered. Logistic can be used for ordered data one factor at a time, but an ordered regression is easier / better. – Florian Hartig Nov 15 '17 at 9:01