Testing the Proportional Odds Assumption

1. What is the Proportional Odds Assumption? A Deeper Look

The proportional odds (PO) assumption states that the effect of any predictor variable is the same, regardless of where you "split" the ordinal outcome variable.

For example, if our outcome is "unlikely," "somewhat likely," and "very likely," the PO assumption means a predictor's effect on the odds of being "somewhat or very likely" (vs. "unlikely") is the same as its effect on the odds of being "very likely" (vs. "unlikely or somewhat likely").

The model that relies on this assumption is also called the parallel lines model. This name comes from the fact that if you were to plot the cumulative logit functions, they would be parallel to each other—having the same slope ($\beta$) but different intercepts (${\alpha }_j$).

To make this concrete, let's examine a document that uses hypothetical data to illustrate what it means for the assumption to be met versus violated.

As you saw, when the assumption holds, the odds ratio is constant. When it's violated, the odds ratio changes depending on the comparison being made. Using a single "average" odds ratio in the second case would be misleading, as it would hide the fact that the predictor's effect is much stronger at the lower end of the outcome scale.

2. How to Statistically Test the Assumption: The Brant Test

Visual inspection and hypothetical tables are good for understanding, but we need a formal statistical test to make a decision. The most common method is the Brant test (also called the Brant-Wald test).

The Brant test checks whether the coefficients are indeed equal across the different cumulative logit models. The hypotheses are:

• Null Hypothesis ($H_0$): The proportional odds assumption holds. The effects of the predictors are consistent across all cut-points (the "lines" are parallel).
• Alternative Hypothesis ($H_A$): The proportional odds assumption is violated. The effect of at least one predictor is not consistent across all cut-points.

This leads to a slightly counter-intuitive rule for interpreting the p-value:

• A large p-value (e.g., > 0.05) means we fail to reject the null hypothesis. This is good news! It suggests the assumption is reasonable, and we can confidently interpret our proportional odds model.
• A small p-value (e.g., < 0.05) means we reject the null hypothesis. This is bad news for our simple model. It provides evidence that the assumption is violated.

Let's watch a video that explains this logic.

3. Performing and Interpreting the Brant Test in Practice

Now that we understand the theory, let's see how it's done. Statistical software makes this test easy to perform. The output typically provides two key pieces of information:

1. An Omnibus Test (Overall Model): This gives a single chi-square statistic and p-value for the entire model, testing if the PO assumption is violated by any predictor.
2. Individual Variable Tests: These provide a separate test for each predictor variable, which is extremely useful for pinpointing the source of a violation if the omnibus test is significant.

The following video demonstrates how to run the Brant test in R and interpret the results.

Here is an example of what the output looks like when the assumption is violated.

In the output shown in the image Brant test output, the overall (omnibus) p-value is 0.000. This is a clear rejection of the null hypothesis, indicating a significant violation of the proportional odds assumption in the model. Looking at the individual variables would help identify which predictor(s) are causing this problem.

4. What To Do When the Assumption Fails?

Your goal of achieving a "functional understanding" means knowing what to do when things go wrong. If the Brant test is significant, you cannot confidently report the single odds ratio from your ordinal model. You have a few options:

1. Use a Multinomial Logistic Regression: This model does not assume any ordering in the outcome. It estimates a separate set of coefficients for each category relative to a baseline. This is a safe option but is less parsimonious (it estimates many more parameters) and gives up the information that your outcome variable is ordered.

2. Use a Generalized Ordinal Logit Model (GOLOGIT): This is often the best approach. It's a more flexible model that allows you to relax the proportional odds assumption for only those variables that violate it, while keeping the simpler structure for the variables that don't. This is also known as a partial proportional odds model.

3. Collapse Categories: If the violation seems to occur at a specific cut-point, you might consider collapsing the outcome categories around that point. This should be done with caution and strong theoretical justification, not just to make the test pass.

We will explore the first two options in more detail in the next lesson. For now, the key is to recognize that a significant Brant test requires you to change your modeling strategy.

Conclusion

In this lesson, we demystified the most important assumption in ordinal logistic regression. You learned not only what the proportional odds assumption means but also why it's crucial for correct interpretation and how to test it.

Key Takeaways:

• The Proportional Odds Assumption: States that a predictor's effect (its odds ratio) is consistent across all possible splits of the ordered outcome.
• The Brant Test: A formal statistical test for the PO assumption. The null hypothesis is that the assumption holds.
• Interpreting the p-value: A non-significant (large) p-value is desired, as it indicates the assumption is not violated. A significant (small) p-value suggests the assumption is violated.
• Pinpointing Violations: The Brant test provides an overall model test and individual tests for each predictor, helping you identify the source of a problem.
• When it Fails: A significant Brant test means your proportional odds model is likely misspecified. You should consider alternative models like multinomial regression or, preferably, a partial proportional odds model.

Preview of the Next Lesson:

Now that you know how to identify a problem with the proportional odds assumption, the next logical step is to learn what to do about it. In our next lesson, we will describe alternative modeling strategies for when the proportional odds assumption is violated, focusing on the practical differences between multinomial and partial proportional odds models.