Chapter 22 Regression when data are ordinal

22.1 Concept list

  • 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).