Structured Mixture of Continuation-ratio Logits Models for Ordinal Regression

Abstract

Traditionally, ordinal responses are assumed to arise from discretizing a latent continuous distribution, with covariate effects entering linearly. This approach limits the covariate-response functional relationship and faces computational challenges when the number of categories is large. We develop a novel Bayesian nonparametric modeling approach to ordinal regression, based on priors placed directly on the discrete distribution of the ordinal responses. The nonparametric model is built from a structured mixture of multinomial distributions. We leverage a continuation-ratio logits representation and Polya-gamma augmentation to formulate the mixture kernel, while the mixing weights are defined through a stick-breaking process that depend on covariates. The regression functions under the mixture model can be expressed as weighted sums of regression functions under traditional parametric models, with weights dependent on covariates. Thus, the modeling approach achieves flexibility in ordinal regression relationships, avoiding linearity or additivity assumptions in the covariate effects. Moreoer, the model retains a conditional independent structure for category-specific parameters, which results in computational efficiency in posterior simulation by allowing partial parallel sampling. The methodology is illustrated with several data examples.