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A Note on High-Dimensional Linear Regression With Interactions

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  • Ning Hao
  • Hao Helen Zhang

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.

Suggested Citation

  • Ning Hao & Hao Helen Zhang, 2017. "A Note on High-Dimensional Linear Regression With Interactions," The American Statistician, Taylor & Francis Journals, vol. 71(4), pages 291-297, October.
    • Handle: RePEc:taf:amstat:v:71:y:2017:i:4:p:291-297
    DOI: 10.1080/00031305.2016.1264311
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    References listed on IDEAS

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    1. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    2. Jiahua Chen & Zehua Chen, 2008. "Extended Bayesian information criteria for model selection with large model spaces," Biometrika, Biometrika Trust, vol. 95(3), pages 759-771.
    3. Ning Hao & Hao Helen Zhang, 2014. "Interaction Screening for Ultrahigh-Dimensional Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1285-1301, September.
    4. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    5. Ning Hao & Yang Feng & Hao Helen Zhang, 2018. "Model Selection for High-Dimensional Quadratic Regression via Regularization," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 615-625, April.
    6. Choi, Nam Hee & Li, William & Zhu, Ji, 2010. "Variable Selection With the Strong Heredity Constraint and Its Oracle Property," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 354-364.
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    Cited by:

    1. Haofeng Wang & Hongxia Jin & Xuejun Jiang & Jingzhi Li, 2022. "Model Selection for High Dimensional Nonparametric Additive Models via Ridge Estimation," Mathematics, MDPI, vol. 10(23), pages 1-22, December.
    2. Feng Li & Yajie Li & Sanying Feng, 2021. "Estimation for Varying Coefficient Models with Hierarchical Structure," Mathematics, MDPI, vol. 9(2), pages 1-18, January.
    3. Xiong, Wei & Chen, Yaxian & Ma, Shuangge, 2023. "Unified model-free interaction screening via CV-entropy filter," Computational Statistics & Data Analysis, Elsevier, vol. 180(C).
    4. Wang, Cheng & Chen, Haozhe & Jiang, Binyan, 2024. "HiQR: An efficient algorithm for high-dimensional quadratic regression with penalties," Computational Statistics & Data Analysis, Elsevier, vol. 192(C).

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