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.