Multinomial Logistic Regression Appropriate?

Suppose the outcome variable can take on 3 discrete values that are not ordinal: (e.g. 1 - weekly, 1- quarterly, and 3 - annually). If for each subject, there are multiple outcomes, would multinomial logistic regression still be favorable to use? For example, suppose the data looked like this:

    id    outcome
1        1
1        3
2        1
2        3
.        .
.        .

• "Weekly, quarterly, annually" is ordinal. Commented Mar 20, 2017 at 22:13

Yes, multinomial regression is still usable here, though you need to interpret the results accordingly. Perhaps you want to use ID to generate probabilities of different outcomes? If that's the case, then you actually must have numerous observations per ID, and the observations for each ID must have at least some variation in the outcome variable or you'll have perfect prediction for at least one of the IDs and the algorithm will lock up. Speaking of which, if that's your intention, you're going to construct dummy variables out of all the IDs, and omit one as your base variable. Or maybe you don't actually want to use ID in your model at all, and you're just worried that these observations don't all come from totally different entities (people, etc.), in which case you generally don't need to worry. You may still want to do a fixed-effects model, however, including those ID dummy variables, if, again, you have enough observations and variance, and have reason to think that constant unit-level properties/effects may bias your key inferences and theoretical conclusions. Again, it depends on your research question and the interpretation you're trying to arrive at, but there's nothing inherently wrong or inappropriate about your model choice.

(And re: the above comment, I wouldn't necessarily dub "Weekly, quarterly, annually" ordinal. They're ordinal in a certain sense, but everything is ordinal in a certain sense, and the same things are ordinal with different rankings in different ways - those three words can be ranked in terms of the number of letters they have, too. Yet again, it depends on your research question: is it relevant that weeks are shorter than quarters which are shorter than years? It might not be at all. And on top of that, treating something as ordinal involves assumptions that are simply not made when you treat it as categorical, so the worst you can do if you ignore relevant ordinality (or even continuity) is lose efficiency.)

• I would like to determine what factors are most associated with each of the categories of the outcome variables. So id 1 may have a lot of 1s, id 2 might have an equal number of 1s, 2s, and 3s etc. Given this, what factors would be most associated with levels 1, 2 or 3? Commented Mar 21, 2017 at 1:59
• I'm not 100% sure I follow - I can't say what other possible predictors would and wouldn't correlate with these different IDs - but I can say that it's not inherently a problem of any kind if obs. with different IDs tend to have different outcomes, or even every different frequencies of different outcomes. Obviously, these differences would reflect some sort of correlation between whatever ID measures and the outcome variable, but that too isn't an inherently bad thing. It may still be best to control for those fixed effects, though. Generally look into fixed effects and dummy variables.
– DHW
Commented Mar 21, 2017 at 2:29

Yes, if your outcome/dependent variable is categorical and you don't want to treat it as ordinal (i.e. you don't think a "one unit increase" is a meaningful statement about it) the a multinomial logit is the appropriate model to use. Almost all other types of models (OLS, binary logit, probit, ordered logit, .) all identify the association between a one unit increase in a given independent variable and some sort of "increase" in the dependent variable (e.g. moving from 0 to 1 in a binary logit model). So all of those models will give you garbage results if you try to use them to analyze a variable where a "one unit increase" doesn't mean anything.

A multinomial logit model can get around this limitation...although in practice it does so by running a bunch of different binary logit models - so when you run the model you won't get one set of coefficients but K-1 sets, where K is the number of categories, where each is basically a binary logit model on a different dependent variable.

In your case you have 3 categories for how frequently someone goes to the movies 1 - weekly, 1- quarterly, and 3 - annually. To specify a multinomial logit you need to choose one of these as your reference category. Let's say you choose "annually." And lets say you have two independent variables - age (continuous) and female (binary)

The model will generate two sets of coefficients. in the first set you will get coefficients for age and female that tell you how getting older or being female is associated with going to the movies quarterly as opposed to annually and then you will get a second set of coefficients that tells you the extent to which getting older or being female is associated with going to the movies weekly as opposed to annually. The model allows gender and age to have different effects for each of these two questions. Note that this model would not allow you to test whether (say) age was associated with going weekly as opposed to quarterly - if you wanted to answer that question you would need to run a separate model treating one of those two categories as the reference.

• Treating a variable as ordinal does not imply anything about a "one unit increase". Commented Sep 2, 2023 at 12:01