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Dimension reduction for the conditional mean in regressions with categorical predictors. (English) Zbl 1042.62037
Summary: Consider the regression of a response \(Y\) on a vector of quantitative predictors \({\mathbf X}\) and a categorical predictor \(W\). We describe a first method for reducing the dimension of \({\mathbf X}\) without loss of information on the conditional mean \(E(Y\mid{\mathbf X}, W)\) and without requiring a prespecified parametric model. The method, which allows for, but does not require, parametric versions of the subpopulation mean functions \(E(Y\mid{\mathbf X}, W= w)\), includes a procedure for inference about the dimension of \({\mathbf X}\) after reduction.
This work integrates previous studies on dimension reduction for the conditional mean \(E(Y\mid{\mathbf X})\) in the absence of categorical predictors and dimension reduction for the full conditional distribution of \(Y\mid({\mathbf X}, W)\). 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.

MSC:
62G08 Nonparametric regression and quantile regression
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G09 Nonparametric statistical resampling methods
Software:
ARC
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