Ordinal or multinomial logistic regression?

When the outcome is a category with three or more levels, the usual binary logistic regression no longer fits. Two extensions take its place, and they answer to different data. Ordinal logistic regression is for categories that have a natural order, such as a four-point agreement scale or a grade of severity. Multinomial logistic regression is for categories with no order, such as the type of transport someone chooses or which of three habitats a species is found in.

The choice is mostly decided by the outcome itself, not by the p-values. If the categories are ordered, start with ordinal (proportional-odds) logistic regression, because it uses the order and summarises each predictor with a single odds ratio. If they are not ordered, use multinomial. The one real decision is what to do when an ordered outcome fails the proportional-odds assumption, and that is covered below.

Ordinal or multinomial logistic regression: the one question that decides

Ask whether you could put the categories in a sequence that everyone would agree on without looking at the data. “Strongly disagree, disagree, agree, strongly agree” passes. “Library, home, cafe” does not: no sequence is more natural than another, and reordering the labels would not change what they mean.

That question matters because the two models spend your data differently. An ordinal model estimates one coefficient per predictor, plus a set of cut points between neighbouring categories. A multinomial model estimates a separate set of coefficients for every category except one. With four outcome categories and two predictor terms, the ordinal model has five parameters in our example below and the multinomial model has nine. More parameters need more observations per category, and sparse categories make multinomial estimates unstable.

If you have a binary outcome, neither applies; see logistic regression and odds ratios.

Ordinal logistic regression and the proportional-odds assumption

The most common ordinal model is the proportional-odds (cumulative logit) model (McCullagh, 1980). It splits the ordered outcome at each cut point: strongly disagree versus everything above it, disagree or lower versus agree or higher, and so on. Each split is a binary logistic regression. The model then forces the effect of every predictor to be the same on all of those splits. The result is one odds ratio per predictor, read as the multiplicative change in the odds of being in a higher category rather than a lower one, wherever you place the cut.

That equal-effect restriction is the proportional-odds assumption, also called the parallel-lines assumption. It is what makes the model compact, and it is the thing to check. If a teaching format raises the odds of agreeing by a factor of 3 at the first split but only 1.2 at the last, a single odds ratio hides the real pattern.

Brant (1990) described a way to test the assumption, both for the model overall and for each predictor separately. A significant overall test is a sign that one odds ratio does not describe the whole scale. Two cautions apply. With a large sample the test will reject for tiny, harmless departures, and with a small sample it has little power to catch real ones, so look at the size of the differences between the split-specific estimates as well as the p-value.

If the assumption clearly fails, there are three honest options:

• Multinomial logistic regression on the same outcome. It makes no assumption about order, at the cost of more parameters and a loss of the single summary odds ratio.
• A partial proportional-odds model, which lets only the offending predictors have split-specific effects. This is a literature option that needs specialised software.
• Collapsing categories into a binary outcome, if that matches the research question. Say plainly that you did it, because it throws information away.

Multinomial logistic regression and the reference category

Multinomial logistic regression picks one outcome category as the reference and models the log-odds of every other category against it. For each predictor you get one odds ratio per non-reference category: how much the predictor changes the odds of being in that category rather than in the reference one.

Two consequences follow. First, the choice of reference category changes which comparisons you see, not the fitted model. Switching the reference rewrites the same probabilities as different contrasts, so choose a reference that makes the comparisons you care about direct (a baseline, a control, the most common category). Second, the model has no notion of distance between categories, so it cannot tell you that “agree” is closer to “strongly agree” than to “strongly disagree”.

Worked example: teaching formats and agreement

A synthetic survey asks 150 students, 50 in each of three teaching formats (lecture, flipped classroom, project-based), whether they agree with “This course helped me understand the material”. The outcome has four ordered levels. The data are synthetic, generated in R 4.6.1.

Format	Strongly disagree	Disagree	Agree	Strongly agree
Lecture	10	18	15	7
Flipped	5	12	20	13
Project	3	8	19	20

A proportional-odds model with lecture as the baseline gives one odds ratio per format. For flipped, the odds ratio is 2.33 (95% CI 1.14 to 4.85), and for project-based it is 4.32 (2.06 to 9.25). Read the first as: whichever cut point you pick, the odds of landing in a higher agreement category are about 2.3 times as large in the flipped format as in the lecture format. An overall proportional-odds test gives a chi-squared of 0.13 on 4 degrees of freedom, p above 0.99, so there is no sign that the effect differs across cut points.

A multinomial model on the same data, with “strongly disagree” as the reference, needs six odds ratios instead of two:

Format versus lecture	Disagree vs SD	Agree vs SD	Strongly agree vs SD
Flipped	1.33	2.67	3.71
Project	1.48	4.22	9.52

The pattern is the one the ordinal model summarised: the odds ratios grow the further the category is from the reference. They are also far less precise, because each rests on small cell counts (the confidence intervals are wide). Here the extra detail confirms the simple story rather than changing it, which is what you hope to see when the proportional-odds assumption holds. The fit also costs more: nine parameters against five for the same 150 students.

Now a genuinely unordered outcome. Suppose 100 students report their preferred study place (library, home, cafe), split into 50 first-years and 50 seniors. Among first-years, 25 pick the library, 20 home and 5 a cafe. Among seniors, 14 pick the library, 12 home and 24 a cafe. There is no order to exploit, so multinomial it is, with library as the reference. Seniors versus first-years: the odds ratio for home against library is 1.07 (95% CI 0.41 to 2.83), essentially no difference, and for cafe against library it is 8.57 (2.67 to 27.47). One predictor, two different stories, which a single ordinal odds ratio could not have expressed.

Which one to report

• Ordered outcome, proportional-odds check passes: ordinal logistic regression, reporting one odds ratio per predictor and the overall check.
• Ordered outcome, assumption clearly fails: multinomial logistic regression, or ordinal with the departures described openly. Report both when the choice matters to the conclusion, and say why you chose one.
• Unordered outcome: multinomial, with a stated reference category.
• Do not treat an ordered scale as unordered just to avoid the assumption without saying so, and do not pick between the two models by whichever gives the smaller p-value. Decide from the nature of the outcome first.
• Very small categories: merge neighbours only where the merge makes substantive sense, and report that you did.

Agresti (2013) covers both families, including the extensions for ordinal data when proportional odds fails.

How Celeus handles this

Celeus proposes ordinal logistic regression only when the outcome is a genuinely ordered category, and multinomial logistic regression for an unordered one. For an ordinal model it runs an overall proportional-odds test and reports the result with a note in the assumptions list. For a multinomial model it names the reference category in the results, and when it had to convert your column to an unordered factor it also tells you to recode the outcome in Prepare data to change the reference.