Proportional Odds Model

Definition

The proportional odds model is a Generalized Linear Model used for modeling the ordinal response.

Consider the response $Y$ having $k$ possible categories. $\mathrm{logit}({\gamma }_j)=\ln \left(\frac{{\gamma }_j}{1-{\gamma }_j}\right)={\alpha }_j-{\mathbf{\beta }}^{⊺}\mathbf{x},j=1,\dots ,k-1$ And the cumulative probability of a response is ${\gamma }_j(\mathbf{x})=P(Y\le j|\mathbf{x})$.