Compared to the nominal scale, the ordinal scale for a categorical outcome variable has the property of making a monotonicity assumption for the covariate effects meaningful. This assumption is encoded in the commonly used proportional odds model, but there it is combined with other parametric assumptions such as linearity and additivity. Herein, the considered models are non-parametric and the only condition imposed is that the effects of the covariates on the outcome categories are stochastically monotone according to the ordinal scale. We are not aware of the existence of other comparable multivariable models that would be suitable for inference purposes. We generalize our previously proposed Bayesian monotonic multivariable regression model to ordinal outcomes, and propose an estimation procedure based on reversible jump Markov chain Monte Carlo. The model is based on a marked point process construction, which allows it to approximate arbitrary monotonic regression function shapes, and has a built-in covariate selection property. We study the performance of the proposed approach through extensive simulation studies, and demonstrate its practical application in two real data examples.
The work of Olli Saarela was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. Christian Rohrbeck was beneficiary of an AXA Research Fund postdoctoral grant. The work of Elja Arjas was supported by the Big Insight research programme, University of Oslo.
We are grateful to Arnoldo Frigessi for arranging for us a short visit at OCBE, during which some important steps were made towards completing this paper.
"Bayesian Non-Parametric Ordinal Regression Under a Monotonicity Constraint." Bayesian Anal. 18 (1) 193 - 221, March 2023. https://doi.org/10.1214/22-BA1310