permutation testing

This work addresses the challenge of finite-sample inference for individual regression coefficients in fixed-design linear models when errors exhibit dependence or heteroskedasticity. The authors propose a unified randomization testing framework based on group permutations, which rigorously controls Type I error under exchangeable errors and enhances power through design-dependent geometric separation. The approach is further extended to non-exchangeable settings, establishing quantitative robustness for approximately symmetric errors. The study proves that the resulting Type I error bound of level $2\alpha$ is tight and, by integrating a constructive algorithm, sub-Gaussian analysis, and conformal inference, achieves substantially improved power under heavy-tailed designs while preserving finite-sample validity.

This paper addresses the often-overlooked exchangeability assumption underlying permutation tests in multiple linear regression—a condition critical for valid statistical inference. We rigorously clarify the logical relationship between exchangeability and the null hypothesis, and systematically evaluate the robustness of common permutation schemes—response permutation, residual permutation, and design-matrix permutation—under both satisfied and violated exchangeability conditions, via theoretical analysis and simulation studies. We propose a novel, pedagogically integrated framework that unifies conceptual understanding with formal theory, and extend the analysis for the first time to hierarchical and clustered regression models, enhancing methodological generality. Results show that while standard regression settings yield consistent conclusions across permutation schemes, inference deteriorates markedly when exchangeability fails. Our framework significantly improves students’ conceptual grasp of resampling-based inference, offering a new paradigm for statistics education and applied practice.

This paper addresses the problem of testing exchangeability of random variables and invariance under compact groups. Methodologically, it introduces a novel e-value-based framework for posterior-valid p-values: (i) it derives the first exact analytic expression for posterior-valid p-values in group-invariance testing; (ii) it designs two data-dependent sampling schemes that unify and extend validity guarantees to arbitrary stopping times; and (iii) it integrates group representation theory, sequential analysis, and the likelihood ratio principle to establish new optimality characterizations under group invariance. Contributions include: (i) substantially improving the statistical power of the t-test in spherical symmetry testing; (ii) uncovering an intrinsic connection between exchangeability testing and the softmax function; and (iii) proposing a new sign-symmetry test whose power dominates existing approaches.

To address the computational bottleneck of performing thousands of hypothesis tests on high-dimensional genetic or neuroimaging data, this paper proposes an anytime-terminating Monte Carlo p-value construction method—the first to extend anytime-valid sequential testing theory to the multiple testing framework. The method is compatible with standard false discovery rate (FDR) control procedures such as Benjamini–Hochberg, guarantees finite-sample FDR control, and substantially reduces the average number of permutations required. Its core innovations integrate sequential Monte Carlo testing, arbitrary stopping time theory, randomized permutation mechanisms, and adaptive correction strategies. Experiments on both synthetic and real-world datasets demonstrate improved statistical power and over 50% reduction in computational time compared to state-of-the-art methods. The implementation is publicly available.

Causal effect estimation in high-dimensional, sparse compositional data—such as microbiome abundances—is challenging: standard parametric models struggle to respect the simplex constraint, and unbiased estimation of aggregate statistics (e.g., diversity indices) on response variables remains elusive. Method: We propose the Average Perturbation Effect (APE) framework, which defines interpretable statistical functionals directly on the simplex. By modeling perturbations under a reparameterization that accounts for perturbation-dependent confounding, APE inherently adjusts for confounding bias, yielding unbiased and identifiable causal effects. Unlike marginal analyses, APE circumvents inherent bias induced by compositional constraints. Contribution/Results: Integrated with semiparametric efficient estimation (e.g., doubly robust methods), APE outperforms existing approaches in simulations and semi-synthetic studies. It is successfully applied to real-world problems—including the association between racial diversity and academic performance, and microbiome–host phenotype relationships—demonstrating improved estimation stability and enhanced causal interpretability.

This study addresses the issue of size distortion in conventional robust standard errors and bootstrap methods under multi-way clustering when the number of effective clusters is small or the data exhibit heavy-tailed distributions. Building on an error condition of exchangeability that better reflects economic reality, the authors develop a finite-sample valid permutation inference framework for multi-way clustered linear regressions and extend it to settings with missing data. Simulation results demonstrate that the proposed method maintains correct test size while delivering substantially higher power than existing approaches. Notably, the analysis reveals a phase-transition-like phenomenon in testing power as the clustering structure varies, underscoring the method’s superior performance in realistic empirical scenarios.

This work addresses the challenge that, under non-exchangeability, the covariance structure of permutation statistics deviates from that of the original test statistics, rendering conventional studentization incapable of recovering the correct joint asymptotic distribution. To overcome this limitation, the authors propose a general and computationally efficient covariance correction method that requires no assumptions about specific parameters, test statistics, or permutation schemes, and remains valid even in singular covariance settings. Unlike existing approaches—such as pre-pivoting—which suffer from high computational costs, the proposed method accurately restores the asymptotic dependence structure of permutation statistics. Theoretical analysis and extensive simulations demonstrate that it achieves asymptotically valid and powerful multiple testing across diverse scenarios, significantly outperforming current methods in inferential accuracy and efficiency.

This study addresses the problem of testing conditional independence between random variables \( X \) and \( Y \) given a confounding variable \( Z \). It proposes a local permutation test based on data-adaptive binning—such as equal-count binning—where permutations of \( X \) and \( Y \) are performed within each subregion defined by \( Z \). The method provides, for the first time, finite-sample Type I error control guarantees for arbitrary test statistics. Under linear confounding models, it achieves power comparable to that of the oracle likelihood ratio test. Theoretical analysis shows that a constant bin size suffices to attain performance on par with increasing bin sizes, and numerical experiments confirm the method’s statistical efficiency and practical utility.

This work proposes a finite-sample inference framework based on the repro samples method to address the problem of mismatched response variables and covariates in linear regression caused by an unknown permutation. By constructing a permutation-space localization strategy with polynomially decaying coverage error, the approach effectively narrows the candidate set of permutations. Integrating conditional Monte Carlo testing, linear assignment optimization, and ridge regularization, the method enables, for the first time, rigorous hypothesis testing on the permutation structure and robust inference of regression coefficients within a finite-sample setting. Numerical simulations and real-world air quality data analysis demonstrate that the proposed procedure achieves high statistical power while properly controlling Type I error, alongside favorable computational scalability.

This study addresses the failure of conventional causal inference methods in group interaction experiments, where within-group interactions and interference effects violate standard assumptions. The authors develop a design-based causal inference framework that systematically characterizes identifiability under various scenarios—such as fixed or random group assignment and presence or absence of interference—and proposes corresponding inference strategies. Innovatively, they introduce a coupling strategy to handle complex dependence structures, integrating sparse-sampling asymptotics, cluster-robust inference, and the potential outcomes framework. They demonstrate that, even under interference, cluster-robust methods consistently estimate marginalized exposure effects. Moreover, when interference is absent and assignment is randomized, the framework naturally reduces to the standard individual-level randomized experiment, thereby preserving compatibility with classical individual-level inference.