A Note on High-Dimensional Linear Regression With Interactions

Ning Hao and Hao Helen Zhang

The American Statistician, 2017, vol. 71, issue 4, 291-297

Abstract: The problem of interaction selection in high-dimensional data analysis has recently received much attention. This note aims to address and clarify several fundamental issues in interaction selection for linear regression models, especially when the input dimension p is much larger than the sample size n. We first discuss how to give a formal definition of “importance” for main and interaction effects. Then we focus on two-stage methods, which are computationally attractive for high-dimensional data analysis but thus far have been regarded as heuristic. We revisit the counterexample of Turlach and provide new insight to justify two-stage methods from the theoretical perspective. In the end, we suggest new strategies for interaction selection under the marginality principle and provide some simulation results.

Date: 2017
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DOI: 10.1080/00031305.2016.1264311

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