Ordinal Logistic Regression

When respondents rate satisfaction from 1 to 5, a 5 is a higher rating than a 4. But you cannot assume that a change from 1 to 2 is the same size as a change from 4 to 5.

Ordinal logistic regression uses the order information in such a categorical outcome to estimate how the explanatory variables relate to the level of satisfaction. The usual proportional odds model assumes that an explanatory variable acts with the same slope across the different cumulative splits.

It can also be used in studies whose outcomes have an order, such as “low · medium · high” or “mild · moderate · severe.” Conversely, if there is no natural order, as with car · bus · bicycle, multinomial logistic regression is appropriate.

Suppose a company asked 240 customers who used its service center to rate their satisfaction with the call from 1 to 5. Each customer responded only once, and the call duration and the customer's trust in the agent were recorded as well.

The regression coefficients and predicted probabilities below are hypothetical values for illustration meant to show the concepts. They are not the result of generating or fitting real raw data from 240 people.

If satisfaction is recorded from 1 to 5, you can think of four cumulative splits: “1 or below,” “2 or below,” “3 or below” and “4 or below.” The proportional odds model gives each split its own intercept (cutpoint) but estimates a common slope for the explanatory variables.

The same model may define the sign and the direction of the cumulative probability differently depending on the software. When interpreting odds ratios, always check the model's cumulative direction and the sign of the coefficients.

Customers with a short handling time may show a high probability of answering 5, while customers with a longer handling time may show higher probabilities of 1–2. For any given customer, the category probabilities from the model add up to 100%.

This table is the same illustrative distribution as the figure at the top. The probabilities are not calculated from specific regression coefficients or from fitting a real model.

When explaining ordinal logistic regression results, rather than presenting odds ratios alone, it helps readers if you also show predicted probabilities for each category under the conditions of interest.

Using the equation above, a positive β means that as the explanatory variable increases, the odds of being in a higher satisfaction category increase. For example, if β for a 1-point increase in trust is 0.60, then exp(0.60) ≈ 1.82. This means that, other conditions being equal, the cumulative odds of being in a higher category are about 1.82 times as large.

When satisfaction falls as handling time gets longer, β is negative in the same notation and exp(β) comes out smaller than 1.

Below are example regression coefficients for interpretation practice. Because no data were fitted, p-values and test statistics are not shown.

In a real results table, also check the coefficients' standard errors, 95% confidence intervals, p-values, the number of observations and model fit. The cutpoints serve as the thresholds that divide the categories and are not interpreted the same way as the effects of ordinary explanatory variables.

The standard ordinal logistic model assumes that the slope for handling time is the same across all cumulative splits, such as “1 vs. 2–5” and “1–2 vs. 3–5.” This is called the proportional odds assumption, or the parallel lines assumption.

If the assumption does not hold, you can consider other approaches such as a partial proportional odds model or a generalized ordered logit. Rather than deciding on the basis of a single test p-value, look at the amount of data in each category, how the effect changes and the purpose of the study together.

If similar customers are grouped within the same branch or agent, you also need to account for that cluster structure. Just because the outcome is ordinal does not mean every assumption of the standard model is automatically satisfied.

Linear regression is not always wrong just because 1–5 was stored as numbers. But if you want to focus on the ranking of the response categories and avoid the equal-spacing assumption, ordinal logistic regression is useful.

Writing “the odds ratio is 1.82, so the probability of a 5 rises by 82%” is a wrong interpretation. The odds ratio is a relative change in odds based on cumulative categories, and changes in the probability of individual categories must be calculated separately from the model.

Also, just because customers with high trust also reported high satisfaction, you cannot conclude that raising trust will necessarily increase satisfaction. An association observed in a cross-sectional survey is not the same as a causal effect.

In the write-up, explain the order of the outcome categories, the explanatory variables, the direction of the model used, the evaluation of the proportional odds assumption, and the regression coefficients with 95% confidence intervals, linking them together.

In a real study, you should then add the estimated odds ratios, 95% confidence intervals, p-values, model fit and the results of checking the proportional odds assumption. The β values and odds ratios given above are for illustration, so they must not be quoted as they are in a real analysis report.

Before starting the analysis, first check the order of the outcome and the number of responses in each category, and organize the coding and units of the explanatory variables. Then fit the proportional odds model, check its assumptions, and interpret the odds ratios together with the predicted probabilities; this is a natural order.