Ordinal regression models outcomes with a natural order, such as satisfaction ratings or severity levels. This guide covers the latent-variable view, the proportional odds assumption and how to check it, how to read odds ratios and cut points, and how to run it in AnalyZ Solutions.
Ordinal regression is a method for studying outcomes with ordered categories, such as ratings from low to high. It shows which factors are linked to a higher or lower category, how strongly, and how likely each category is for a particular person.
Many outcomes are ordered: satisfaction from very dissatisfied to very satisfied, pain from mild to severe, agreement on a five-point scale. The order matters, but the distances between the categories are unknown. The thing you want to explain is the outcome, and the factors you use to explain it are the predictors.
Suppose 400 people rate a service as Low, Medium or High satisfaction. You also know each person's gender, their age, and whether they use the service online. You want to know three things. Do women and men rate it differently once age and use are taken into account? Does satisfaction rise with age? And what are the chances of Low, Medium and High for a woman aged 40 who uses the service online? Ordinal regression answers all three.
Linear regression treats the categories as numbers, such as 1, 2 and 3, and assumes the gap between each pair is equal. Is the gap between very dissatisfied and dissatisfied really the same as the gap between satisfied and very satisfied? Nobody knows. Multinomial regression avoids that assumption but throws away the order. Ordinal regression respects the order without assuming equal gaps.
Use ordinal regression when the outcome has three or more ordered categories and you want to relate it to predictors. Common examples are Likert items, income bands or severity categories, education levels, and severity ratings.
If the outcome has only two categories, use logistic regression. If the categories have no order, use multinomial regression. If the outcome is a sum of several Likert items and has many distinct values, linear regression is often acceptable. For comparing groups without predictors, the Kruskal-Wallis and Mann-Whitney tests are simpler options.
| Feature | Linear regression | Ordinal regression | Multinomial regression |
|---|---|---|---|
| Outcome | Numeric | Three or more ordered categories | Three or more unordered categories |
| Use of the order | Treats the codes as numbers with equal gaps | Uses the order without assuming equal gaps | Ignores the order |
| Coefficients per predictor | One | One | One for each category except the reference |
| Key assumption | Equal gaps and normal errors | Proportional odds | None about the order |
| Effect measure | Difference in the score | Odds ratio | Relative risk ratio |
| Number of parameters | Few | Few | Many |
Ordinal regression rests on two ideas: working with cumulative chances instead of the chance of each category on its own, and picturing the categories as slices of a hidden scale.
Instead of the chance of each category separately, the model works with the chance of being at or below each point on the scale. For a man aged 40 who does not use the service online, that might be a 51% chance of being at Low, a 77% chance of being at Low or Medium, and a 100% chance of being at High or below. The chance of a single category is the difference between neighbouring cumulative chances. Medium is 77% minus 51%, which is 26%, and High is 100% minus 77%, which is 23%.
Behind an ordered answer sits a hidden continuous scale, such as how satisfied a person really is. Cut points divide that scale into the categories people actually report: below the first cut point the person says Low, between the two cut points Medium, and above the second High. Predictors move a person's position on the hidden scale. Among people with the same predictors, positions follow a logistic curve.
The figure shows two profiles drawn over the same cut points. For the profile on the right, the whole curve has moved to the right, so more of it lies beyond the second cut point. The High category grows and the Low category shrinks. Only the position of the curve changes between people. The cut points are fixed and shared, which keeps the model simple and is exactly what the proportional odds assumption describes.
The most common version, and the one in AnalyZ Solutions, is the proportional odds model. It imagines a series of cut points that divide the scale, for example between dissatisfied and neutral, between neutral and satisfied, and so on. Each predictor is assumed to have the same effect at every cut point.
In practice, this gives you a single odds ratio for each predictor. If the odds ratio for women compared with men is 1.8, women have 80% higher odds of being in a higher satisfaction category than men, and this holds whether you are looking at the boundary between low and middle, or between middle and high.
This is a strong assumption, and it can fail. A change to a service might help people move from low to middle satisfaction but do nothing to move them from middle to high. When the assumption fails, a single odds ratio hides that pattern.
AnalyZ Solutions does not currently test the assumption, so check it yourself. Split the scale at each cut point into two groups, run a logistic regression for each split, and compare the odds ratios for your key predictors. In the example, satisfaction has three levels, so there are two splits: Medium or High against Low, and High against Low or Medium.
| Predictor | Ordinal model | Split 1: Medium or High against Low | Split 2: High against Low or Medium |
|---|---|---|---|
| Gender: Female (ref: Male) | 1.65 (1.14 to 2.39) | 1.81 (1.20 to 2.73) | 1.44 (0.94 to 2.20) |
| Age (per 10 years) | 1.20 (1.04 to 1.37) | 1.28 (1.10 to 1.50) | 1.09 (0.93 to 1.27) |
| Uses the online service (ref: No) | 1.42 (0.97 to 2.07) | 1.37 (0.89 to 2.09) | 1.40 (0.91 to 2.15) |
The ratios from the two splits are broadly similar, and every interval overlaps the ordinal estimate. Age shows the largest gap, 1.28 against 1.09, but the intervals overlap widely, so there is no clear evidence against proportional odds. A warning sign would be ratios that point in different directions, or intervals that do not overlap, for example a ratio of 3 for one split and 1 for the other.
These separate regressions are a check, and you should not report them in place of the ordinal model. If the assumption clearly fails, you can report the splits separately, merge categories, or use multinomial regression and give up the ordering. AnalyZ Solutions fits only the proportional odds version.
Ordinal regression models the cumulative chance of being at or below each level. For three levels (Low, Medium, High) there are two cumulative probabilities, the chance of Low and the chance of Medium or lower. Each cut point θ divides the scale, and the same predictor index is used for every cut point.
The minus sign means a higher index pushes people towards higher levels. The chance of a single level comes from subtracting neighbouring cumulative probabilities. The coefficients and the cut points are found together by maximum likelihood, and e^b is the odds ratio.
Take a woman aged 40 who uses the online service, with cut points θ₁ = 0.752 and θ₂ = 1.948.
For a man aged 40 who does not use the service, the index is 0.713, and the same steps give Low 51%, Medium 26%, and High 23%. The gap between the two profiles combines the effects of gender, age and use of the service across the whole scale.
The odds ratio for gender is e^0.500 = 1.649. Because one set of coefficients serves every cut point, this ratio is assumed to be the same at each step of the scale. That is the proportional odds assumption.
Ordinal regression rests on a few assumptions. Breaking them does not stop the model from running, but it can mislead.
Each predictor's effect is reported as an odds ratio with a 95% confidence interval and a significance level, with a toggle for raw coefficients on the log-odds scale. An odds ratio above 1 means higher values of the predictor go with a higher position on the outcome scale. Below 1 means a lower position. For a categorical predictor, the comparison is with the reference category.
Read the direction against the way the scale is coded. If your outcome is coded from 1 for very satisfied to 5 for very dissatisfied, an odds ratio above 1 means more dissatisfaction.
An odds ratio of 1.65 for women does not mean women are 65% more likely to be in the High category. It means their odds of being above any given cut point are 65% higher. The effect on the probability of each category depends on where the person starts.
The table lists cut points, also called thresholds, which mark the boundaries between adjacent categories. They are on the log-odds scale. In the example the cut points are 0.752 and 1.948, so a person whose index is 0 would have a 68% chance of the Low category. They are structural parts of the model and not results to interpret, and you do not need to report them in a findings summary. They are needed to calculate predicted probabilities, as in the worked prediction above.
When a predictor raises the index, probability moves out of the lowest category and into the highest. The middle category can rise or fall, because it gains people from Low and loses people to High. In the figure, the chance of High rises steadily with age in both profiles and the chance of Low falls, while Medium barely moves.
When your readers need to see this, report predicted probabilities for example profiles next to the odds ratios.
The summary gives the number of cases, the number of outcome levels, the log-likelihood, a likelihood-ratio test, a pseudo R-squared, and AIC.
Open Inferential Analysis, choose Regression, then Ordinal Regression.
The figures below are invented to show the steps, and they follow the results tables in AnalyZ Solutions, with a few columns left out to save space. Suppose 400 people rate a service as Low, Medium, or High satisfaction, in that order. The predictors are gender (reference: Male), age in decades, and use of the online service (reference: No).
| N | Log-likelihood | LR chi2 | df | Prob > chi2 | Pseudo R² | AIC | Levels |
|---|---|---|---|---|---|---|---|
| 400 | -427.4601 | 15.13 | 3 | 0.0017 | 0.017 | 864.92 | 3 |
| Variable | Odds ratio | 95% CI lower | 95% CI upper | p-value | Sig |
|---|---|---|---|---|---|
| Gender: Female (ref: Male) | 1.649 | 1.140 | 2.386 | 0.0079 | ** |
| Age (per 10 years) | 1.195 | 1.042 | 1.371 | 0.0108 | * |
| Uses the online service (ref: No) | 1.421 | 0.974 | 2.075 | 0.0685 | ns |
Women and older respondents tended to report higher satisfaction, and both differences are significant at the 5% level or better. Users of the online service also leaned towards higher satisfaction, but the evidence was too weak to be sure. The cut points that the model estimates describe the scale and are not findings. Check informally that key effects are similar at each step of the scale before relying on a single odds ratio.
Using it for unordered categories. The model assumes the order is meaningful.
Ignoring the coding direction. Read every odds ratio against the way the scale is coded.
Reading an odds ratio as a change in probability. It describes the odds of being above a cut point. The change in each category's probability depends on the starting point.
Assuming proportional odds without a check. If key effects differ across the scale, a single odds ratio can mislead.
Reporting the cut points as findings. They describe the scale, not the effects.
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