Regression with Ordered Predictors via Ordinal Smoothing Splines

Many applied studies collect one or more ordered categorical predictors, which do not fit neatly within classic regression frameworks. In most cases, ordinal predictors are treated as either nominal (unordered) variables or metric (continuous) variables in regression models, which is theoretically and/or computationally undesirable. In this paper, we discuss the benefit of taking a smoothing spline approach to the modeling of ordinal predictors. The purpose of this paper is to provide theoretical insight into the ordinal smoothing spline, as well as examples revealing the potential of the ordinal smoothing spline for various types of applied research. Specifically, we (i) derive the analytical form of the ordinal smoothing spline reproducing kernel, (ii) propose an ordinal smoothing spline isotonic regression estimator, (iii) prove an asymptotic equivalence between the ordinal and linear smoothing spline reproducing kernel functions, (iv) develop large sample approximations for the ordinal smoothing spline, and (v) demonstrate the use of ordinal smoothing splines for isotonic regression and semiparametric regression with multiple predictors. Our results reveal that the ordinal smoothing spline offers a flexible approach for incorporating ordered predictors in regression models, and has the benefit of being invariant to any monotonic transformation of the predictor scores.