Ordinal Logistic Regression (Proportional Odds Model)

When the outcome is ordered

Many outcomes in survey and marketing research are ordered categories: satisfaction from “very dissatisfied” to “very satisfied”, purchase intent from “definitely not” to “definitely”, a 1–5 Likert rating, education level, or the severity of a condition. Treating such outcomes as continuous with linear regression assumes equal distances between categories, which is rarely justified, and predictions can fall outside the scale. Treating them as unordered with multinomial regression throws away the ordering and needs many more parameters. Ordinal logistic regression, also called the proportional odds model, ordered logit or cumulative logit model, is designed exactly for this case.

The proportional odds model

With J ordered categories, the model describes the cumulative probabilities of being at or below each category j:

logit[P(Y ≤ j)] = θ_j − (β₁x₁ + β₂x₂ + …), j = 1, …, J − 1

There are J − 1 thresholds (cut-points) θ_j, one between each pair of adjacent categories, and one coefficient β per predictor that applies to all thresholds. This is the parameterisation used by SPSS PLUM, R’s MASS::polr and Stata’s ologit, so results match those programs. A positive β means that higher values of the predictor shift people towards higher outcome categories.

The exponentiated coefficient e^β is a cumulative odds ratio: the factor by which the odds of being in a higher category (rather than at or below any given category) multiply for a one-unit increase in the predictor. An odds ratio of 1.5 for price fairness means each extra scale point multiplies the odds of being more satisfied by 1.5, whichever threshold you look at. The probit link replaces the logistic distribution with the normal distribution; coefficients are then on a latent standard-normal scale and have no odds-ratio interpretation, but substantive conclusions are usually the same.

The proportional odds assumption and the Brant test

The model’s key assumption is that each predictor has the same effect at every threshold, so the J − 1 cumulative logit curves are parallel (hence “test of parallel lines” in SPSS). The Brant test (Brant, 1990) checks this. It fits a separate binary logistic regression for each threshold (Y > 1, Y > 2, …), estimates the covariance of the coefficients across these models, and tests with a Wald χ² whether the coefficients are equal. The tool reports an omnibus test and a test for each predictor. A non-significant result supports the model. A significant result for a specific predictor means its effect differs between thresholds: consider a partial proportional odds model, a generalised ordered logit, or a multinomial model. With large samples the test can flag small, practically unimportant departures, so compare the binary coefficients before abandoning the model.

Worked example with the built-in data

The example contains 360 subscribers whose satisfaction is recorded in four ordered categories, from dissatisfied to very satisfied, with tenure, perceived price fairness, plan type and number of support tickets as predictors. The model is significant, χ²(5) = 37.69, p < .001, Nagelkerke R² = .106. Each extra point of price fairness multiplies the odds of a higher satisfaction category by 1.53 (95% CI 1.28 to 1.84). Premium plan customers have 1.73 times the odds of the Basic plan, and each support ticket reduces the odds by about 21% (OR = 0.79). Tenure is not significant. The Brant test supports proportional odds (omnibus χ²(10) = 15.53, p = .114). These estimates agree with statsmodels’ OrderedModel and R’s polr to three decimals.

Preparing your data

• Outcome: numeric codes (1–5) are ordered numerically. For text categories, type the order from lowest to highest in the category order box, for example Dissatisfied, Neutral, Satisfied, Very satisfied; otherwise text is ordered alphabetically, which is rarely what you want.
• Predictors: numeric variables enter as they are. Text predictors are dummy coded automatically, with the first category (alphabetically) as the reference.
• Sample size: as a rule of thumb, have at least 10 cases per predictor in the smallest outcome category. Merge very sparse categories (for example combine 1 and 2 on a five-point scale when only a handful of people chose 1).

Interpreting the output

• Model fit: the likelihood ratio χ² compares the model with a thresholds-only model. Pseudo R² values (Cox and Snell, Nagelkerke, McFadden) are typically much lower than OLS R² and should not be read as “variance explained”; McFadden values of .2 to .4 already indicate an excellent fit.
• Coefficients and odds ratios: use the Wald z-test and the confidence interval for each predictor.
• Thresholds: usually not of substantive interest, but they show where the cut-points between categories lie on the latent scale; widely spaced thresholds indicate categories that are clearly distinct.
• Predicted probabilities: the chart shows how the probability of each category changes across a numeric predictor, holding the others at their means. This is often the clearest way to communicate results.

Ordinal vs other regression models

Use binary logistic regression when the outcome has two categories. Use multinomial logistic regression when categories are unordered or when proportional odds clearly fails. Linear regression on a Likert outcome is a common approximation with seven or more categories and roughly symmetric distributions, but ordinal regression is more defensible, especially with skewed responses or few categories. For count outcomes, use the Poisson and negative binomial tool.

Reporting in APA style

Report the model test, a pseudo R², the odds ratios with 95% confidence intervals, and the test of parallel lines. Example: “An ordinal logistic regression showed that price fairness (OR = 1.53, 95% CI [1.28, 1.84], p < .001) and support tickets (OR = 0.79, 95% CI [0.68, 0.92], p = .003) predicted satisfaction, χ²(5) = 37.69, p < .001, Nagelkerke R² = .106. The Brant test indicated that the proportional odds assumption was met, χ²(10) = 15.53, p = .114.”

Checklist before you report

• State the outcome order: list the categories from lowest to highest so readers know what “higher” means.
• Check sparse categories: merge outcome categories with very few observations before fitting.
• Report the model test and a pseudo R²: the likelihood ratio χ² with its degrees of freedom and, for example, Nagelkerke R².
• Report odds ratios with 95% CIs: they are easier to interpret than log-odds coefficients.
• Report the test of parallel lines: give the Brant omnibus test and mention any predictor that violates the assumption.
• Show predicted probabilities: a chart of category probabilities across a key predictor makes the effect tangible.
• Mention the software and link function: for example “proportional odds model with logit link, FreeStatisticalTools.com”.

Key references: McCullagh (1980), Journal of the Royal Statistical Society B; Brant (1990), Biometrics; Agresti (2010), Analysis of Ordinal Categorical Data; Long and Freese (2014), Regression Models for Categorical Dependent Variables Using Stata.