UCSC-SOE-22-01: Structured mixture of continuation-ratio logits models for ordinal regression

Jizhou Kang and Athanasios Kottas
01/10/2022 12:27 PM
Statistics
Traditionally, ordinal responses are assumed to arise through discretization of a latent continuous distribution, with covariate effects entering linearly. This approach limits the covariate-response relationship and faces computational challenges when the number of response 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 prior probability model is built from a structured mixture of multinomial distributions. We leverage a continuation-ratio logits representation and PĆ³lya-Gamma augmentation to formulate the mixture kernel, with mixture weights defined through the logit stick-breaking process that allows the covariates to enter through a linear function. The implied regression functions for the response probabilities can be expressed as weighted sums of regression functions under traditional parametric models, with covariate-dependent weights. Thus, the modeling approach achieves flexibility in ordinal regression relationships, avoiding linearity or additivity assumptions in the covariate effects. Moreover, the model features a conditional independence structure for category-specific parameters, which results in computationally efficient posterior simulation by allowing partial parallel sampling. The methodology is illustrated with several synthetic and real data examples.

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