When the response variable for a regression model is categorical, linear models don’t work. Logistic regression is one type of model that does, and it’s relatively straightforward for binary responses.
When the response variable is not just categorical, but ordered categories, the model needs to be able to handle the multiple categories, and ideally, account for the ordering.
An easy-to-understand and common example is level of educational attainment. Depending on the population being studied, some response categories may include:
1 Less than high school
2 Some high school, but no degree
3 Attain GED
4 High school graduate
You can see how there are qualitative differences in these categories that wouldn’t be captured by years of education. You can also see that the number of years isn’t equal from one category to another, but there is a definite ordering that we wouldn’t want to ignore.
Let’s say the model is interested in seeing how participation in a middle-school community program to help kids stay in school affected their later educational attainment, controlling for gender, race, size of high school, and middle school test scores.
One popular logistic regression model for ordered responses is called a proportional odds model.
The Proportional Odds Model
It’s a type of logistic regression in which you’re modeling the relationship between predictor variables and the propensity to be in each higher ordered category.
For example, the model would report how each predictor variable uniquely affects the odds of being in category 2 or higher compared to category 1; being in category 3 or higher compared to being in category 2 or 1; up to being in category 4 compared to being in categories 1, 2, or 3.
Each comparison has its own intercept, but the same set of regression coefficient estimates. The intercepts reflect the fact that some categories, like high school graduate, are just more likely, regardless of the predictors.
The regression coefficients represent the relationship of each predictor, each X, to the odds that an individual would be in each category or above compared to all lower categories.
(Note: different stat software procedures use different defaults on the ordering—some model being in a higher category, some model being in a lower category. Make sure you know which direction your software is using).
One nice thing about this model is it is relatively simple to interpret and report—there is just a single coefficient for each predictor.
But it also creates one extremely important assumption–that the relationship of predictors to the odds of a response being in the next higher order category is the same regardless of which categories you’re comparing.
So if the program doubled the odds of being a high school graduate, compared to all lower categories, it would have to also double the odds of attaining a GED or high school diploma, compared to none or some high school.
This is called the proportional odds assumptions or the parallel regression assumption. Unfortunately this assumption is hard to meet in real data.
In fact, it seems a middle-school program would have a much bigger effect on some of the lower categories—maybe getting kids to continue into high school–than it would on later categories.
The Generalized Ordered Logistic Regression Model
Luckily, there are alternatives. Here I focus on one, the generalized ordered logistic regression. It’s a more complicated model, because it has a unique set of regression coefficients for each comparison.
It does this by fitting a separate set of regression coefficients for each comparison. The comparisons are the same—we’re still measuring, for example, the odds of being in category 2 or higher compared to category 1 or the odds of being in category 4 compared to 3 and below.
The result is usually a much better-fitting, but complicated model. The result in this case would be 3 sets of regression coefficients. Having more response categories means having more sets of regression coefficients.
This can be hard to interpret—there is no single number for the effect of the program on the odds of attaining more education.
But if there really is no single effect, interesting patterns may emerge. For example, the program may have positive impacts on getting kids into high school (being in categories 2, 3, or 4 compared to 1) but no impact on attaining some sort of degree or GED (being in category 3 or 4 compared to 1 or 2).