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Setwise Hierarchical Variable Selection and the Generalized Linear Step-Up Procedure for False Discovery Rate Control

Sarah Organ, Toby Kenney, Hong Gu

TL;DR

A new setwise variable-selection framework that identifies clusters of potential predictors rather than forcing selection of a single variable, supports strong predictive performance while maintaining rigorous FDR control and achieves higher power than existing approaches while preserving FDR control.

Abstract

Controlling the false discovery rate (FDR) in variable selection becomes challenging when predictors are correlated, as existing methods often exclude all members of correlated groups and consequently perform poorly for prediction. We introduce a new setwise variable-selection framework that identifies clusters of potential predictors rather than forcing selection of a single variable. By allowing any member of a selected set to serve as a surrogate predictor, our approach supports strong predictive performance while maintaining rigorous FDR control. We construct sets via hierarchical clustering of predictors based on correlation, then test whether each set contains any non-null effects. Similar clustering and setwise selection have been applied in the familywise error rate (FWER) control regime, but previous research has been unable to overcome the inherent challenges of extending this to the FDR control framework. To control the FDR, we develop substantial generalizations of linear step-up procedures, extending the Benjamini-Hochberg and Benjamini-Yekutieli methods to accommodate the logical dependencies among these composite hypotheses. We prove that these procedures control the FDR at the nominal level and highlight their broader applicability. Simulation studies and real-data analyses show that our methods achieve higher power than existing approaches while preserving FDR control, yielding more informative variable selections and improved predictive models.

Setwise Hierarchical Variable Selection and the Generalized Linear Step-Up Procedure for False Discovery Rate Control

TL;DR

A new setwise variable-selection framework that identifies clusters of potential predictors rather than forcing selection of a single variable, supports strong predictive performance while maintaining rigorous FDR control and achieves higher power than existing approaches while preserving FDR control.

Abstract

Controlling the false discovery rate (FDR) in variable selection becomes challenging when predictors are correlated, as existing methods often exclude all members of correlated groups and consequently perform poorly for prediction. We introduce a new setwise variable-selection framework that identifies clusters of potential predictors rather than forcing selection of a single variable. By allowing any member of a selected set to serve as a surrogate predictor, our approach supports strong predictive performance while maintaining rigorous FDR control. We construct sets via hierarchical clustering of predictors based on correlation, then test whether each set contains any non-null effects. Similar clustering and setwise selection have been applied in the familywise error rate (FWER) control regime, but previous research has been unable to overcome the inherent challenges of extending this to the FDR control framework. To control the FDR, we develop substantial generalizations of linear step-up procedures, extending the Benjamini-Hochberg and Benjamini-Yekutieli methods to accommodate the logical dependencies among these composite hypotheses. We prove that these procedures control the FDR at the nominal level and highlight their broader applicability. Simulation studies and real-data analyses show that our methods achieve higher power than existing approaches while preserving FDR control, yielding more informative variable selections and improved predictive models.
Paper Structure (22 sections, 7 theorems, 12 equations, 5 figures, 3 tables)

This paper contains 22 sections, 7 theorems, 12 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Example
  4. Preliminaries
  5. Contributions and Overview
  6. The Generalized Linear Step-Up Procedure (GLSUP)
  7. General Procedure
  8. Modified GLSUP Procedure
  9. Theory for the Generalized Linear Step-Up Procedure
  10. The SHRED Procedures for Setwise Variable Selection with FDR Control
  11. The SHRED Procedure
  12. Threshold Slope
  13. Simulations
  14. Simulation Design
  15. Results
  16. ...and 7 more sections

Key Result

Theorem 1

For the inputs from Definition SHRED:GLSUPData, suppose that for any $J\subseteq\{1,\ldots,m\}$, $\sigma\left(\overline{J}\right)\leqslant \sum_{i\in J}w_i$ for some weights $w_1,\ldots,w_m$. Then the generalized linear step-up procedure with cut-off where $\sigma_i=\sigma(\{i\})$, controls the gFDR at level $q$.

Figures (5)

  • Figure 1: Gaussian regression simulation results. Error bars indicate one standard error.
  • Figure 2: Logistic regression simulation results. Error bars indicate one standard error.
  • Figure 3: Poisson regression simulation results. Error bars indicate one standard error.
  • Figure 4: Difference in gPower between SHRED methods and BH, at various correlation cuts of the hierarchical tree, $p = 300, n = 1000, T = 100$.
  • Figure 5: Midas case study selected taxa from each method

Theorems & Definitions (14)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Definition 2: PRDS
  • Theorem 2
  • Definition 3: PPRDS
  • Theorem 3
  • Definition 4: SPRDS
  • Theorem 4
  • Corollary 1
  • ...and 4 more