Dimension reduction for the conditional mean in regressions with categorical predictors

Consider the regression of a response Y on a vector of quantitative predictors X and a categorical predictor W. In this article we describe a first method for reducing the dimension of X without loss of information on the conditional mean E(Y\X, W) and without requiring a prespecified parametric model. The method, which allows for, but does not require, parametric versions of the subpopulation mean functions E(Y\X, W = w), includes a procedure for inference about the dimension of X after reduction. This work integrates previous studies on dimension reduction for the conditional mean E(Y\X) in the absence of categorical predictors and dimension reduction for the full conditional distribution of Y\(X, W). The methodology we describe may be particularly useful for constructing low-dimensional summary plots to aid in model-building at the outset of an analysis. Our proposals provide an often parsimonious alternative to the standard technique of modeling with interaction terms to adapt a mean function for different subpopulations determined by the levels of W. Examples illustrating this and other aspects of the development are presented.