Proportional-odds model
Definition taught in Categorical responses, part of Generalised linear models
Proportional-odds model
| Type | Definition |
| Taught in | Categorical responses |
| Part | Generalised linear models |
| Worked examples | 1 |
| Questions using it | 1 |
A proportional-odds model for ordered categories 1,\ldots,K and covariates {\boldsymbol{x}} is \mathrm{logit}\,\mathrm{P}(Y\ge k\mid {\boldsymbol{x}})=\beta_{0k}+{\boldsymbol{x}}^\top\boldsymbol{\beta}, \qquad k=2,\ldots,K, with \beta_{02}\ge\beta_{03}\ge\cdots\ge\beta_{0K}. A unit increase in a covariate multiplies the odds of being in category k or higher by e^{\beta_j}, and that factor does not depend on k.
Examples
C25. Proportional odds
Question page: C25. Proportional odds
What is the proportional-odds assumption in an ordinal regression model?
Definitions/notation. For ordered categories 1,\ldots,K, use the upper-tail convention \gamma_k(x)=\mathbb{P}(Y\ge k\mid x) and \operatorname{logit}\gamma_k(x)=a_k+x^\top\beta, k=2,\ldots,K. The slope vector \beta is common to all cutpoints, so a unit change in x_j multiplies every cumulative odds by e^{\beta_j}, holding the other covariates fixed.
Used in
- Taught in Categorical responses, where the derivation and examples are.
- Questions that use it (1) — C25
- All 3 questions on Categorical responses