Isotonic Additive Interaction Models

Abstract

This chapter considers parametric and nonparametric approaches to fitting isotonic additive models. The modeled relationship between a predictor vector and a response is assumed smooth, additive with low-order interactions, and monotone in some of the predictors. A parametric solution is to fit a monotone regression spline model. A nonparametric solution is to fit an unrestricted additive model first and then isotonize the fit. There are circumstances where the relationship is expected to be monotone only in some of the inputs. Additivity is often found to be a useful generalization of many simpler models. An additional advantage of using additive models is their interpretability that they largely share with the linear models. The I-splines are linearly independent and hence form a basis. In addition, they are nondecreasing, so a nonpositive linear combination is nonincreasing. It is found that not every nonincreasing spline can be represented in that form, but that loss is usually not significant. It is also found that when an interaction model is fitted, some main effects and lower order interactions may become aliased by the higher order interactions. It is suggested that if the fitting procedure allows for different amount on smoothing in different regions, more smoothing may be carried out near flat spots.