Ordinal and Multinomial Logit

Ordinary logistic regression handles a yes/no outcome. Real categorical outcomes are often richer than that: which of several unordered choices did someone make, or which of several ranked tiers does something fall into. Multinomial and ordinal logit extend the same logistic machinery to those two cases, and the distinction between them is exactly whether the categories have a natural order.

Multinomial logit handles unordered categories — think "which of four asset classes did this fund rotate into." It works by picking one category as a baseline and modeling the log-odds of every other category relative to it as a linear function of the predictors, giving a separate set of coefficients for each non-baseline category. There is no assumption that category 3 is "between" categories 2 and 4 in any sense — the categories are just labels.

Ordinal logit (often the proportional-odds model) handles ranked categories — a credit rating from AAA down to junk, or a survey response from strongly-disagree to strongly-agree. Rather than fitting separate coefficients per category, it assumes a single underlying continuous score, and models the probability of being at or below each cutoff along that score with logistic curves that share the same slope but different intercepts — one cutoff per boundary between adjacent categories. That shared-slope assumption (the "proportional odds" assumption) is what lets ordinal logit use far fewer parameters than multinomial logit would need for the same number of categories, but it is also the assumption most often violated in practice and worth checking.

Choosing the wrong one costs information: fitting a multinomial model to genuinely ordered categories throws away the ordering and needs more data to estimate more parameters, while forcing an ordinal model onto genuinely unordered categories imposes a ranking that does not exist and can produce a poor fit.