  The use of the proportional odds (PO) model for ordinal regression is
ubiquitous in the literature. If the assumption of parallel lines does not hold
for the data, then an alternative is to specify a non-proportional odds (NPO)
model, where the regression parameters are allowed to vary depending on the
level of the response. However, it is often difficult to fit these models, and
challenges regarding model choice and fitting are further compounded if there
are a large number of explanatory variables. We make two contributions towards
tackling these issues: firstly, we develop a Bayesian method for fitting these
models, that ensures the stochastic ordering conditions hold for an arbitrary
finite range of the explanatory variables, allowing NPO models to be fitted to
any observed data set. Secondly, we use reversible-jump Markov chain Monte
Carlo to allow the model to choose between PO and NPO structures for each
explanatory variable, and show how variable selection can be incorporated.
These methods can be adapted for any monotonic increasing link functions. We
illustrate the utility of these approaches on novel data from a longitudinal
study of individual-level risk factors affecting body condition score in a dog
population in Zenzele, South Africa.
