Robust distribution-free tests for the linear model

TL;DR

This paper introduces RobustPALMRT, a permutation-based framework for robust association testing in linear models that handles heavy-tailed and skewed noise, with applications to biological data such as Long-COVID.

Contribution

It extends existing tests to robust and quantile regressions, separates model-fitting from evaluation for improved performance, and introduces DispersionPALRMT for testing differences in dispersion.

Findings

Controls type I error in heavy-tailed noise

Detects differences in dispersion between groups

Unveils novel Long-COVID differences

Abstract

Recently, there has been growing concern about heavy-tailed and skewed noise in biological data. We introduce RobustPALMRT, a flexible permutation framework for testing the association of a covariate of interest adjusted for control covariates. RobustPALMRT controls type I error rate for finite-samples, even in the presence of heavy-tailed or skewed noise. The new framework expands the scope of state-of-the-art tests in three directions. First, our method applies to robust and quantile regressions, even with the necessary hyper-parameter tuning. Second, by separating model-fitting and model-evaluation, we discover that performance improves when using a robust loss function in the model-evaluation step, regardless of how the model is fit. Third, we allow fitting multiple models to detect specialized features of interest in a distribution. To demonstrate this, we introduce DispersionPALRMT, which tests for differences in dispersion between treatment and control groups. We establish theoretical guarantees, identify settings where our method has greater power than existing methods, and analyze existing immunological data on Long-COVID patients. Using RobustPALMRT, we unveil novel differences between Long-COVID patients and others even in the presence of highly skewed noise.