Multinomial Logistic Regression

What multinomial logistic regression models

Multinomial logistic regression (multinomial logit, MNL, also called polytomous logistic regression) predicts an outcome with three or more unordered categories. Typical examples in marketing and the social sciences are which brand a customer buys, which channel they use (store, web, app), which subscription plan they choose, how they commute, or which party they vote for. It extends binary logistic regression: instead of one equation comparing “yes” with “no”, it estimates one equation for each category compared with a chosen reference category.

The model

With K outcome categories and category 1 as the reference, the model estimates K − 1 sets of coefficients:

ln[P(Y = k) / P(Y = 1)] = β_0k + β_1kx₁ + … + β_pkx_p, k = 2, …, K

The probabilities of all categories follow from these log-odds and always add up to one. Coefficients are estimated by maximum likelihood with Newton-Raphson iterations, the same estimator used by SPSS NOMREG, Stata mlogit and R’s nnet::multinom; the results here match statsmodels’ MNLogit.

The exponentiated coefficient e^β is the relative risk ratio (RRR), sometimes loosely called an odds ratio: the factor by which the odds of choosing category k rather than the reference multiply for a one-unit increase in the predictor. An RRR of 1.04 for income in the Cobalt equation means each extra thousand of income multiplies the odds of choosing Cobalt over Aurora by 1.04.

Reading the output

• Model fit: the likelihood ratio χ² compares the full model with an intercept-only model; pseudo R² values (Cox and Snell, Nagelkerke, McFadden) summarise improvement in fit, and AIC/BIC help compare alternative specifications.
• Likelihood ratio tests: each predictor is removed in turn and the model refitted. A significant χ² (with (K − 1) × number of dummy columns degrees of freedom) means the predictor matters for the outcome overall, across all categories. This is the test to report first, because a predictor can be important even if no single Wald test is significant, and vice versa.
• Parameter estimates: one block per non-reference category, with B, SE, Wald z, p, RRR and its 95% confidence interval. Interpret each block as a comparison with the reference.
• Classification table: observed against predicted categories, with the percentage correctly classified. Compare it with the baseline of always predicting the most common category; multinomial models of consumer choice often improve on the baseline only modestly even when predictors are highly significant.

Worked example with the built-in data

The example contains 480 customers who each chose one of four brands, with age, income, gender and whether they shop online as predictors; Aurora is the reference. The model is significant, χ²(12) = 43.81, p < .001, Nagelkerke R² = .095. Likelihood ratio tests show that age (χ²(3) = 12.69, p = .005) and income (χ²(3) = 19.25, p < .001) affect brand choice, while gender and online shopping do not reach significance. In the Brava block, each extra year of age reduces the odds of choosing Brava rather than Aurora by about 3% (RRR = 0.971); in the Cobalt block, each thousand of income raises the odds of Cobalt over Aurora by about 4% (RRR = 1.038). The model classifies 48.3% of customers correctly against a baseline of 46.0%: predictors shift probabilities in meaningful ways but individual choices remain hard to predict, as is typical for brand choice.

Choosing the reference category

The overall fit, the likelihood ratio tests and the predicted probabilities do not depend on the reference category; only the parameter table changes. Choose a reference that makes comparisons meaningful: the market leader, the status quo, the most common category or a natural baseline such as “no purchase”. To compare two non-reference categories directly, rerun the analysis with one of them as the reference.

Assumptions and checks

• Independence of irrelevant alternatives (IIA). The odds between any two categories do not depend on which other categories exist. This can fail when two alternatives are close substitutes (the classic red bus / blue bus problem). If IIA is doubtful, nested logit or multinomial probit models are alternatives.
• No perfect separation. If a predictor perfectly predicts a category (for example, every customer over 70 chose the same brand), estimates diverge; the tool stops with a message. Merge rare categories or remove the offending predictor.
• Sample size. Multinomial models estimate many parameters: (K − 1) × (p + 1). Aim for at least 10 cases in the smallest category per predictor, and merge categories with very few cases.
• Multicollinearity among predictors inflates standard errors, as in any regression.

Multinomial logit vs related models

If the categories are ordered, ordinal logistic regression uses the ordering and needs far fewer parameters. If the outcome is binary, use binary logistic regression. When the predictors describe the alternatives (price, features) rather than the chooser (age, income), the appropriate model is the conditional logit used in choice-based conjoint analysis; this tool is for chooser characteristics. Discriminant analysis is an older alternative that assumes normally distributed predictors.

Preparing your data

Use one row per case with the outcome category and the predictors. Numeric predictors enter as they are; text predictors are dummy coded automatically with their first category as the reference. Rows with missing values are excluded listwise and the number is reported.

Reporting in APA style

Report the reference category, the overall model test and pseudo R², the likelihood ratio test for each predictor, and B, SE, p and RRR with 95% CIs for key comparisons. Example: “A multinomial logistic regression with Aurora as the reference category was significant, χ²(12) = 43.81, p < .001, Nagelkerke R² = .095. Age, χ²(3) = 12.69, p = .005, and income, χ²(3) = 19.25, p < .001, were significant predictors of brand choice. Higher income increased the odds of choosing Cobalt rather than Aurora (RRR = 1.04, 95% CI [1.02, 1.06]).”

Checklist before you report

• Check category sizes: every outcome category should have enough cases; merge or drop categories with very few observations.
• Report the reference category: every coefficient is a comparison with it, so readers need to know which one it is.
• Lead with the likelihood ratio tests: they show which predictors matter overall before you discuss individual coefficients.
• Interpret RRRs with confidence intervals: a wide interval that includes 1 means the direction of the effect is uncertain.
• Compare with the baseline accuracy: state the percentage that always predicting the most common category would achieve.
• Consider IIA: if two categories are close substitutes, mention the assumption or test a nested model.
• Show predicted probabilities: for key predictors, a table or chart of predicted probabilities across values is easier to read than coefficients.

Key references: Hosmer, Lemeshow and Sturdivant (2013), Applied Logistic Regression; Long and Freese (2014), Regression Models for Categorical Dependent Variables; McFadden (1974), on conditional logit models of choice.