# Lecture 6: Ordinal Regression

Guest lecture by Sunny Tseng

## Agenda

• Finish up Survival Analysis (Vincenzo)
• Ordinal regression discussion: notes (Sunny)
• Ordinal regression live coding: worksheet.R (Sunny)

## Learning Objectives

By the end of the lecture, students are expected to be able to:

(To be updated based on what was actually discussed)

• Identify whether a variance is ordinal.
• Identify the model assumptions in a proportional odds model.
• Interpret the regression coefficients in a proportional odds model.
• Fit a proportional odds model in R, extract relevant quantities, and make predictions.

## Resources

• Chapter 8 of Applied Logistic Regression by David W. Hosmer, Jr., Stanley Lemeshow.

## Concepts

• A variable is ordinal if its values have a natural ordering.
• For example, months have an inherent order.
• A proportional odds model is a commonly used model that allows us to interpret how predictors influence an ordinal response. Let’s consider lower levels as being “worse”.
• It models an individual’s odds of having an outcome “worse than” (less than or equal to) level `k` for all `k` as being some baseline odds, multiplied by `exp(eta)`, where `eta` is a linear combination of the predictors. Sometimes (like in R’s `MASS::polr()`) `eta` is a negative linear combination of predictors, so that the multiplicative factor is `exp(-eta)`.
• The coefficient `beta` on a predictor `X` (contained in `eta`) has the following interpretation (if `eta` is defined as a linear combination of predictors without a negative sign in front): an increase in `X` by one unit is associated with `exp(beta)` times the odds of being worse off. If `eta` is defined with a negative sign, the same interpretation follows with `exp(-beta)` instead of `exp(beta)`.