Fisher Lecture: Dimension Reduction in Regression

TL;DR

This paper revisits principal components in regression, explores model-based extensions, and discusses broader roles and flexible approaches to dimension reduction beyond traditional conditioning on predictors.

Contribution

It introduces new perspectives on principal components in regression and proposes general model-based and model-free dimension reduction methods.

Findings

Principal components have a broader role in regression than traditionally thought.

Model-based extensions enhance the flexibility of dimension reduction techniques.

Conditioning on observed predictors may limit regression methodology options.

Abstract

Beginning with a discussion of R. A. Fisher's early written remarks that relate to dimension reduction, this article revisits principal components as a reductive method in regression, develops several model-based extensions and ends with descriptions of general approaches to model-based and model-free dimension reduction in regression. It is argued that the role for principal components and related methodology may be broader than previously seen and that the common practice of conditioning on observed values of the predictors may unnecessarily limit the choice of regression methodology.