Ordinal regression modelling between proportional odds and non-proportional odds

The proportional odds model has become the most widely used model
in ordinal regression. Despite favourable properties in
applications it is often an inappropriate simplification yielding
bad data fit. The more flexible non-proportional odds model or
partial proportional odds model have the disadvantage that common
estimation procedures as Fisher scoring often fail to converge.
Then neither estimates nor test statistics for the validity of
partial proportional odds models are available. In the present
paper estimates are proposed which are based on penalization of
parameters across response categories. For appropriate smoothing
penalized estimates exist almost always and are used to derive test
statistics for the assumption of partial proportional odds. In
addition, models are considered where the variation of parameters
across response categories is constrained. Instead of using
prespecified scalars (Peterson&Harrell 1990) penalized
estimates are used in the identification of these constrained
models. The methods are illustrated by various applications. The
application to the retinopathy status in chronic diabetes shows how
the proposed test statistics may be used in the diagnosis of
partial proportional odds models in order to prevent artefacts.