Supervised Dimension Reduction for Ordinal Predictors

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. A supervised dimension reduction method tailored to ordered categorical predictors is introduced which 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.