Ordinal Regression
The Ordinal Regression procedure (referred to as PLUM in the syntax) allows you to build models, generate predictions, and evaluate the importance of various predictor variables in cases where the dependent (target) variable is ordinal in nature.
Ordinal dependents and linear regression. When you are trying to predict ordinal responses, the usual linear regression models don't work very well. Those methods can work only by assuming that the outcome (dependent) variable is measured on an interval scale. Because this is not true for ordinal outcome variables, the simplifying assumptions on which linear regression relies are not satisfied, and thus the regression model may not accurately reflect the relationships in the data. In particular, linear regression is sensitive to the way you define categories of the target variable. With an ordinal variable, the important thing is the ordering of categories. So, if you collapse two adjacent categories into one larger category, you are making only a small change, and models built using the old and new categorizations should be very similar. Unfortunately, because linear regression is sensitive to the categorization used, a model built before merging categories could be quite different from one built after.
Generalized linear models. An alternative approach uses a generalization of linear regression called a generalized linear model to predict cumulative probabilities for the categories. With this method, you fit a separate equation for each category of the ordinal dependent variable. Each equation gives a predicted probability of being in the corresponding category or any lower category.
| Category | Probability of Membership | Cumulative Probability |
|---|---|---|
| Current | 0.80 | 0.80 |
| 30 days past due | 0.07 | 0.87 |
| 60 days past due | 0.07 | 0.94 |
| 90 days past due | 0.05 | 0.99 |
| Uncollectable | 0.01 | 1.00 |
For example, look at the distribution shown in the table. With no predictors in the model, predictions are based only on the overall probabilities of being in each category. The predicted cumulative probability for the first category is 0.80. The prediction for the second category is 0.80+0.07=0.87. The prediction for the third is 0.80+0.07+0.07=0.94, and so on. The prediction for the last category is always 1.0, since all cases must be in either the last category or a lower category. Because of this, the prediction equation for the last category is not needed.