  In applications involving ordinal predictors, common approaches to reduce
dimensionality are either extensions of unsupervised techniques such as
principal component analysis, or variable selection procedures that rely on
modeling the regression function. In this paper, a supervised dimension
reduction method tailored to ordered categorical predictors is introduced. It
uses a model-based dimension reduction approach, inspired by extending
sufficient dimension reductions to the context of latent Gaussian variables.
The reduction is chosen without modeling the response as a function of the
predictors and does not impose any distributional assumption on the response or
on the response given the predictors. A likelihood-based estimator of the
reduction is derived and an iterative expectation-maximization type algorithm
is proposed to alleviate the computational load and thus make the method more
practical. A regularized estimator, which simultaneously achieves variable
selection and dimension reduction, is also presented. Performance of the
proposed method is evaluated through simulations and a real data example for
socioeconomic index construction, comparing favorably to widespread use
techniques.
