Simultaneous hypothesis testing for comparing many functional means

2025-06-13

AI Summary

This paper addresses mean comparison for multiple groups of high-dimensional functional data (e.g., fMRI cortical signals). Method: We propose a paired two-sample multiple testing procedure that operates entirely in the function space—nonparametric, without dimension reduction, functional principal component analysis (FPCA), or tuning parameters—and achieves strong control of the family-wise error rate (FWER). Our approach leverages exact inference on the supremum of L²-norm-based test statistics, underpinned by novel anti-concentration inequalities and Gaussian approximation theory that dispense with conventional assumptions on functional smoothness or intrinsic dimension. Contribution/Results: Theoretically, FWER is rigorously controlled even when sample size is far smaller than functional dimension; computationally, the algorithm is efficient and requires no smoothing, truncation, or parameter tuning. Empirical validation on Human Connectome Project cortical fMRI data confirms its statistical power and robustness.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Machine learning and data science for the Web

Abstract

Data with multiple functional recordings at each observational unit are increasingly common in various fields including medical imaging and environmental sciences. To conduct inference for such observations, we develop a paired two-sample test that allows to simultaneously compare the means of many functional observations while maintaining family-wise error rate control. We explicitly allow the number of functional recordings to increase, potentially much faster than the sample size. Our test is fully functional and does not rely on dimension reduction or functional PCA type approaches or the choice of tuning parameters. To provide a theoretical justification for the proposed procedure, we develop a number of new anti-concentration and Gaussian approximation results for maxima of $L^2$ statistics which might be of independent interest. The methodology is illustrated on the task-related cortical surface functional magnetic resonance imaging data from Human Connectome Project.

Problem

• Simultaneous comparison of many functional means
• Control family-wise error rate in functional data
• Handle increasing functional recordings beyond sample size

Innovation

• Paired two-sample test for functional means
• No reliance on dimension reduction techniques
• New anti-concentration and Gaussian approximation results