an:02071024
Zbl 1042.62037
Li, Bing; Cook, R. Dennis; Chiaromonte, Francesca
Dimension reduction for the conditional mean in regressions with categorical predictors
EN
Ann. Stat. 31, No. 5, 1636-1668 (2003).
0090-5364 2168-8966
2003
j
62G08 62H05 62G09
analysis of covariance; central subspace; graphics; OLS; SIR; PHD; SAVE; diabetes
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.