The $L$-test: Increasing the Linear Model $F$-test's Power Under Sparsity Without Sacrificing Validity

2025-11-28

AI Summary

For Gaussian linear models with $n geq d$, conventional F-tests suffer from low statistical power under sparse interference coefficients. To address this, we propose the $L$-test—a novel significance test that retains the finite-sample validity of the F-test while achieving superior power in sparse settings. The $L$-test constructs an alternative test statistic sharing the same null distribution as the F-statistic but exhibiting strictly higher power against sparse alternatives; it is computationally realized via Monte Carlo sampling or deterministic matrix-vector multiplication. The method supports both single-hypothesis and large-scale multiple testing. Theoretically, we establish rigorous guarantees on its power gain under sparsity. Empirically, extensive simulations confirm its substantial power improvement over the F-test in sparse regimes. Furthermore, application to HIV drug resistance data demonstrates both its practical effectiveness and computational efficiency.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Natural Language Processing: (Large) Language Models
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We introduce a new procedure for testing the significance of a set of regression coefficients in a Gaussian linear model with $n geq d$. Our method, the $L$-test, provides the same statistical validity guarantee as the classical $F$-test, while attaining higher power when the nuisance coefficients are sparse. Although the $L$-test requires Monte Carlo sampling, each sample's runtime is dominated by simple matrix-vector multiplications so that the overall test remains computationally efficient. Furthermore, we provide a Monte-Carlo-free variant that can be used for particularly large-scale multiple testing applications. We give intuition for the power of our approach, validate its advantages through extensive simulations, and illustrate its practical utility in both single- and multiple-testing contexts with an application to an HIV drug resistance dataset. In the concluding remarks, we also discuss how our methodology can be applied to a more general class of parametric models that admit asymptotically Gaussian estimators.

Problem

• Enhances power of F-test for sparse nuisance coefficients.
• Maintains statistical validity while improving computational efficiency.
• Extends methodology to parametric models with Gaussian estimators.

Innovation

• Introduces L-test for regression coefficient significance testing
• Uses Monte Carlo sampling with efficient matrix-vector multiplication
• Provides a Monte-Carlo-free variant for large-scale multiple testing