Regression Models for Ordinal Data

TLDR

The paper develops a general class of regression models for ordinal data. The models exploit the ordinal nature by describing stochastic ordering, eliminate the need for score assignment, and employ iteratively reweighted least squares for estimation, with applications illustrated by examples. The proportional odds and proportional hazards models are emphasized as the most useful due to their simple interpretation, while the linear models extend generalized linear models and nonlinear extensions converge via iteratively reweighted least squares.

Abstract

Summary A general class of regression models for ordinal data is developed and discussed. These models utilize the ordinal nature of the data by describing various modes of stochastic ordering and this eliminates the need for assigning scores or otherwise assuming cardinality instead of ordinality. Two models in particular, the proportional odds and the proportional hazards models are likely to be most useful in practice because of the simplicity of their interpretation. These linear models are shown to be multivariate extensions of generalized linear models. Extensions to non-linear models are discussed and it is shown that even here the method of iteratively reweighted least squares converges to the maximum likelihood estimate, a property which greatly simplifies the necessary computation. Applications are discussed with the aid of examples.