Multinomial Logistic Regression

multinomial

regression

nnet

softmax

Regression for unordered categorical outcomes with three or more levels

Published

April 17, 2026

Research question

Multinomial logistic regression models an unordered categorical outcome with three or more levels, using one level as the reference. Biomedical example: in an emergency-department triage study, does age, sex, and chief-complaint category predict the disposition outcome (discharge, admit to ward, admit to ICU)?

Assumptions

Assumption	How to verify in R
Nominal outcome with >= 3 categories	data check
Independence of irrelevant alternatives (IIA)	Hausman-McFadden test; sensitivity analysis
Independent observations	design
No severe multicollinearity among predictors	`car::vif()` on linear-predictor equivalents

Hypotheses

For each coefficient per contrast (non-reference vs. reference category): \(H_0: \beta_{jk} = 0\) vs. \(H_1: \beta_{jk} \ne 0\).

R code

library(tidyverse); library(nnet); library(broom); library(gtsummary)
set.seed(42)

triage <- tibble(
  age       = round(rnorm(300, 55, 19)),
  sex       = factor(sample(c("F", "M"), 300, replace = TRUE)),
  complaint = factor(sample(c("Cardiac", "Respiratory", "Trauma", "Other"),
                             300, replace = TRUE, prob = c(0.25, 0.20, 0.15, 0.40)))
) |>
  mutate(
    lp_admit = -1 + 0.03 * age + 0.4 * (complaint == "Cardiac"),
    lp_icu   = -3 + 0.04 * age + 0.8 * (complaint == "Cardiac") + 0.5 * (sex == "M"),
    dispo    = factor(
      sapply(1:300, function(i) {
        p_admit <- plogis(lp_admit[i]) * (1 - plogis(lp_icu[i]))
        p_icu   <- plogis(lp_icu[i])
        sample(c("Discharge", "Ward", "ICU"), 1,
               prob = c(1 - p_admit - p_icu, p_admit, p_icu))
      }),
      levels = c("Discharge", "Ward", "ICU")
    )
  )

fit <- multinom(dispo ~ age + sex + complaint, data = triage, trace = FALSE)

broom::tidy(fit, conf.int = TRUE, exponentiate = TRUE)

tbl_regression(fit, exponentiate = TRUE) |>
  add_global_p()

Interpreting the output

The model returns two sets of coefficients (Ward vs. Discharge; ICU vs. Discharge). Each exponentiated coefficient is the relative-risk ratio (RRR) for that category vs. the reference. For example, every additional decade of age multiplies the RRR for ICU admission by \(\exp(10 \times 0.04) = 1.49\).

Effect size

RRRs per contrast (the multinomial analogue of odds ratios). Overall model fit: McFadden pseudo-\(R^2\) via performance::r2_mcfadden().

Reporting (APA 7)

In a multinomial logistic regression, each 10-year increase in age was associated with a higher relative risk of ICU admission (RRR = 1.49, 95 % CI 1.20-1.85, p < .001) and ward admission (RRR = 1.35, 95 % CI 1.14-1.59, p < .001) compared to discharge, after adjustment for sex and chief-complaint category.

Common pitfalls

Choice of reference category changes all coefficients; pick a clinically meaningful baseline.
multinom() uses a neural-network implementation and may have trouble with small samples; check for convergence warnings.
Reporting raw logits instead of RRRs makes interpretation harder.
IIA assumption is often violated; sensitivity analyses using the nested logit model are appropriate when it matters.

Parametric vs. non-parametric alternative

For ordered outcomes, prefer ordinal logistic regression for efficiency. For comparing several categorical variables without a modelled outcome, use the chi-squared contingency test.