Generalized Ordered Regression Models

We examine models that relax proportionality in cumulative ordered regression models. Something fundamental arising from ordered variables and stochastic ordering implies a partitioning. Efforts to relax proportionality also relax the ability to collapse an inherently multidimensional problem to a partitioning of the (unidimensional) real line. It is surprising and unfortunate to find that deviations from proportionality are sufficient to generate internal contradictions; undecidable propositions must exist by relaxing proportional odds without other relevant and significant changes in the underlying model. We prove a single theorem linking continuous support and partitions of a latent space to show that for these two characteristics to be simultaneously satisfied, the model must be the proportional-odds model. Conditioning on the adjacency that is closely related to the partitioning is fruitful, but at this point we join the class of continuation-ratio models. Alternatively, Anderson’s (1984) stereotype model is quite general and nests ordered and unordered choice models, but again we have left the domain of cumulative models. Adopting multidimensional cumulative models or imposing covariate-specific thresholds are the only certain methods for avoiding these troubles in the cumulative framework. It is generically impossible to generalize the cumulative class of ordered regression models in ways consistent with the spirit of generalized cumulative regression models. Monte Carlo studies also demonstrate the general principles.