A great tool to have in your statistical tool belt is logistic regression.
It comes in many varieties and many of us are familiar with the variety for binary outcomes.
But multinomial and ordinal varieties of logistic regression are also incredibly useful and worth knowing.
They can be tricky to decide between in practice, however. In some — but not all — situations you could use either.
So let’s look at how they differ, when you might want to use one or the other, and how to decide.
Both multinomial and ordinal models are used for categorical outcomes with more than two categories.
The simplest decision criterion is whether that outcome is nominal (i.e., no ordering to the categories) or ordinal (i.e., the categories have an order).
It should be that simple.
Here’s why it isn’t:
1. While there is only one logistic regression model appropriate for nominal outcomes, there are quite a few for ordinal outcomes.
These models account for the ordering of the outcome categories in different ways. Most software, however, offers you only one model for nominal and one for ordinal outcomes.
2. The most common of these models for ordinal outcomes is the proportional odds model. It has a strong assumption with two names — the proportional odds assumption or parallel lines assumption.
It essentially means that the predictors have the same effect on the odds of moving to a higher-order category everywhere along the scale.
This assumption is rarely met in real data, yet is a requirement for the only ordinal model available in most software.
3. If you have a nominal outcome variable, it never makes sense to choose an ordinal model. Your results would be gibberish and you’ll be violating assumptions all over the place.
(That makes one choice simple!)
In contrast, you can run a nominal model for an ordinal variable and not violate any assumptions. But you may not be answering the research question you’re really interested in if it incorporates the ordering.
4. The names. Most software refers to a model for an ordinal variable as an ordinal logistic regression (which makes sense, but isn’t specific enough).
In contrast, they will call a model for a nominal variable a multinomial logistic regression (wait – what?).
It gets better.
Some software procedures require you to specify the distribution for the outcome and the link function, not the type of model you want to run for that outcome. Both ordinal and nominal variables, as it turns out, have multinomial distributions.
What differentiates them is the version of logit link function they use. So if you don’t specify that part correctly, you may not realize you’re actually running a model that assumes an ordinal outcome on a nominal outcome. Not good.
A link function with a name like “mlogit,” “multinomial logit,” or “generalized logit” assumes no ordering.
A link function with a name like “clogit” or “cumulative logit” assumes ordering, so only use this if your outcome really is ordinal.
If you have a nominal outcome, make sure you’re not running an ordinal model.
If you have an ordinal outcome and the proportional odds assumption is met, you can run the cumulative logit version of ordinal logistic regression.
If you have an ordinal outcome and your proportional odds assumption isn’t met, you can:
1. Run a different ordinal model
2. Run a nominal model as long as it still answers your research question