Ordinal Logistic Regression

Predicts an ordered categorical outcome — e.g., low/medium/high engagement — respecting the ordering that multinomial models throw away.

Data requirements

Assumptions

• Independence of observations.
• Proportional odds (parallel lines): each predictor's effect is the same across all cumulative splits of the outcome — test it (Brant test / SPSS test of parallel lines).
• No severe multicollinearity.
• Adequate cell sizes across outcome categories.

Hypotheses

The concept

The model stacks all cumulative splits of the ordered outcome (low vs rest; low+medium vs high; …) and fits one common slope per predictor across them, with separate intercepts (thresholds) per split. The exponentiated slope is a cumulative odds ratio: the multiplicative change in the odds of being in a higher (vs lower) category per unit of the predictor.

The proportional-odds assumption is what buys the single-coefficient elegance; when the Brant/parallel-lines test rejects it for a predictor, that predictor's effect differs across the scale and needs freeing (partial proportional odds) or a multinomial fallback. Thresholds themselves are rarely of substantive interest.

Worked example

Predicting engagement band (low/medium/high) from job autonomy and workload among 350 employees.

Result: autonomy OR = 1.86, 95% CI [1.48, 2.35] — each autonomy point nearly doubles the odds of a higher engagement band; workload OR = 0.71. Test of parallel lines: χ²(2) = 2.1, p = .35 — assumption holds.

How to run it

library(MASS)
df$engage_band <- factor(df$engage_band, levels = c("low","medium","high"), ordered = TRUE)
model <- polr(engage_band ~ autonomy + workload, data = df, Hess = TRUE)
summary(model)
exp(cbind(OR = coef(model), confint(model)))

library(brant); brant(model)     # proportional odds test

from statsmodels.miscmodels.ordinal_model import OrderedModel
import numpy as np

model = OrderedModel(df["engage_band"],           # ordered categorical
                     df[["autonomy", "workload"]],
                     distr="logit").fit(method="bfgs")
print(model.summary())
print(np.exp(model.params[:2]))    # ORs for the predictors

Interpreting the output

• OR per predictor: odds of being in a higher outcome category multiply by OR per unit increase.
• Always report the parallel-lines/Brant test — the model's credibility rests on it.
• Model-level fit: likelihood-ratio χ² and pseudo-R² (named).
• Predicted category probabilities at illustrative predictor values communicate results best.

APA-style reporting

Common mistakes

Related methods