Ordinal Logistic Regression

ordinal-regression

proportional-odds

polr

Proportional-odds regression for ordered categorical outcomes

Published

April 17, 2026

Research question

Use ordinal logistic regression when the outcome has ordered categories (e.g., disease stage I-IV, Likert satisfaction 1-5, NYHA class I-IV). The proportional-odds model fits a single set of coefficients across cumulative thresholds. Biomedical example: does a new rehabilitation programme improve mRS (modified Rankin Scale, 0-5) at 90 days after stroke, adjusting for age and baseline NIHSS?

Assumptions

Assumption	How to verify in R
Ordered categorical outcome	`is.ordered(y)`
Proportional odds (parallel lines)	`brant::brant()` test
Independent observations	design
No severe multicollinearity	`car::vif()` on the linear-predictor equivalent

If proportional odds fails, use partial-proportional-odds (VGAM::vglm(..., cumulative(parallel = FALSE))) or multinomial logistic regression.

Hypotheses

For each coefficient: \(H_0: \beta_j = 0\) vs. \(H_1: \beta_j \ne 0\). The coefficient represents the log odds of being in a higher category vs. lower, constant across cut-points under proportional odds.

R code

library(tidyverse); library(MASS); library(brant); library(broom); library(gtsummary)
set.seed(42)

stroke <- tibble(
  age    = rnorm(200, 70, 12),
  nihss  = sample(2:20, 200, replace = TRUE),
  arm    = factor(sample(c("Standard", "Enhanced"), 200, replace = TRUE)),
  lp     = -0.02 * age + 0.2 * nihss - 0.8 * (arm == "Enhanced")
) |>
  mutate(mrs = cut(plogis(lp + rnorm(200, 0, 1)),
                   breaks = c(-Inf, 0.15, 0.30, 0.50, 0.70, 0.85, Inf),
                   labels = 0:5)) |>
  mutate(mrs = factor(mrs, levels = 0:5, ordered = TRUE))

fit <- MASS::polr(mrs ~ age + nihss + arm, data = stroke, Hess = TRUE)

# Coefficients with OR and CIs
broom::tidy(fit, conf.int = TRUE, exponentiate = TRUE, p.values = TRUE)

# Proportional-odds assumption
brant::brant(fit)

tbl_regression(fit, exponentiate = TRUE)

Interpreting the output

Each coefficient is the log odds ratio for being in a higher mRS category per unit of predictor. An OR < 1 for armEnhanced indicates the enhanced programme reduces the odds of worse outcomes. Brant’s test returns a chi-squared statistic for the parallel-lines assumption; a non-significant overall \(p\) supports proportional odds.

Effect size

Per-predictor odds ratios. Model fit: McFadden or Nagelkerke pseudo-\(R^2\) via performance::r2_mcfadden().

Reporting (APA 7)

In a proportional-odds model, the enhanced rehabilitation programme reduced the odds of a higher 90-day mRS score (OR = 0.47, 95 % CI 0.28-0.79, p = .004) after adjusting for age and baseline NIHSS. Brant’s test gave no evidence against the proportional-odds assumption (p = .41).

Common pitfalls

Treating an ordinal outcome as continuous (linear regression) obscures the ordering and inflates Type I error in tails.
Treating an ordinal outcome as nominal (multinomial) loses efficiency.
Ignoring the proportional-odds assumption; always run Brant’s test.
Collapsing categories to binary throws away the ordering.

Parametric vs. non-parametric alternative

When proportional odds fails: partial-proportional-odds model (VGAM) or generalised ordered logit (oglmx).
Alternative: continuation-ratio or adjacent-category logit (VGAM).
Rank-based: Jonckheere-Terpstra for ordered groups without covariates.