# How to model counts of a categorical variable?

I'm interested in assessing the impact of various covariates (age, sex, Charlson comorbidity score, etc.) on the incidence rate of a pulmonary event. However, the event is not binary. Each patient could have one of three values: the first (0) indicating no event, the second (1) indicating a non-severe event, and the third (2) indicating a severe event.

I'm unsure which regression model I should use for these data.

Any insight would be greatly appreciated!

• It sounds like an ordinal regression problem. By the way, you always can combine categories 1 & 2 to make it binary, so I guess that you mean that you don't want to treat this as binary, rather that it isn't..? – Tim Apr 22 '19 at 19:50
• Yeah, I'd prefer it not to be binary. I'm now wondering if I could use a generalized linear model with whichever distribution is appropriate for original regression and the log of each patient's follow-up time as the offset. – jos0909 Apr 22 '19 at 20:52
• If you are collecting information on follow-up time there is presumably a start date for each patient and a time between start and event, so this seems better to be handled with survival analysis. The two different types of events could be considered competing events, which can be handled by survival analysis. Is there some reason why competing-events survival analysis can't be used here? – EdM Apr 22 '19 at 20:57

## 1 Answer

This sounds like an ordinal regression problem. Logistic regression is model for binary data, with more then two categories, you can use multinomial regression, or (if this makes any sense) treat them as continuous and use linear regression. The problem with multinomial regression for such data would be that it doesn't care about ordering of the categories, and models all the categories as separate entities. This may be a problem, for example, in cases like modelling Likert-style data in form of (1) I strongly disagree - (2) I disagree - (3) I have no opinion - (4) I agree - (5) I strongly agree, where you implicitly assume that there are are some incremental relationships between the answers (if someone strongly agrees, then she agrees as well and doesn't disagree etc.). What follows, in such cases people sometimes treat such variables as continuous and use linear regression, but the problem wit linear regression is that it assumes that your categories have numerical meaning, and arithmetic operations on them are also meaningful (agree + 1 = agree * 1.25 = strongly agree).

Because the standard regression models do not really work for such "incremental" data, there is a special model for it, the ordinal regression model. It assumes that your outcome is on ordinal scale, so the order of the categories matters, but they have no numerical meaning. This seems to be your case, as there seems to be obvious order: no event < non-severe event < severe event. It models cumulative probabilities

$$\Pr(Y \le i|X) = \sigma(\theta_i - X\beta)$$

where $$\sigma$$ is the links function (logit, or probit etc.). For more details, you can check also other questions tagged as .