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1. A Regularization-Based Adaptive Test for High-Dimensional Generalized Linear Models

   Wu, C; Xu, G; Shen, X. Pan (June 2020, Journal of machine learning research)

   In spite of its urgent importance in the era of big data, testing high-dimensional parameters in generalized linear models (GLMs) in the presence of high-dimensional nuisance parameters has been largely under-studied, especially with regard to constructing powerful tests for general (and unknown) alternatives. Most existing tests are powerful only against certain alternatives and may yield incorrect Type I error rates under high-dimensional nuisance parameter situations. In this paper, we propose the adaptive interaction sum of powered score (aiSPU) test in the framework of penalized regression with a non-convex penalty, called truncated Lasso penalty (TLP), which can maintain correct Type I error rates while yielding high statistical power across a wide range of alternatives. To calculate its p-values analytically, we derive its asymptotic null distribution. Via simulations, its superior finite-sample performance is demonstrated over several representative existing methods. In addition, we apply it and other representative tests to an Alzheimer’s Disease Neuroimaging Initiative (ADNI) data set, detecting possible gene-gender interactions for Alzheimer’s disease. We also put R package “aispu” implementing the proposed test on GitHub. 

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2. Boosting the power of kernel two-sample tests

   https://doi.org/10.1093/biomet/asae048  

   Chatterjee, A; Bhattacharya, B B (January 2025, Biometrika)

   The kernel two-sample test based on the maximum mean discrepancy is one of the most popular methods for detecting differences between two distributions over general metric spaces. In this paper we propose a method to boost the power of the kernel test by combining maximum mean discrepancy estimates over multiple kernels using their Mahalanobis distance. We derive the asymptotic null distribution of the proposed test statistic and use a multiplier bootstrap approach to efficiently compute the rejection region. The resulting test is universally consistent and, since it is obtained by aggregating over a collection of kernels/bandwidths, is more powerful in detecting a wide range of alternatives in finite samples. We also derive the distribution of the test statistic for both fixed and local contiguous alternatives. The latter, in particular, implies that the proposed test is statistically efficient, that is, it has nontrivial asymptotic (Pitman) efficiency. The consistency properties of the Mahalanobis and other natural aggregation methods are also explored when the number of kernels is allowed to grow with the sample size. Extensive numerical experiments are performed on both synthetic and real-world datasets to illustrate the efficacy of the proposed method over single-kernel tests. The computational complexity of the proposed method is also studied, both theoretically and in simulations. Our asymptotic results rely on deriving the joint distribution of the maximum mean discrepancy estimates using the framework of multiple stochastic integrals, which is more broadly useful, specifically, in understanding the efficiency properties of recently proposed adaptive maximum mean discrepancy tests based on kernel aggregation and also in developing more computationally efficient, linear-time tests that combine multiple kernels. We conclude with an application of the Mahalanobis aggregation method for kernels with diverging scaling parameters. 

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3. New multivariate tests for assessing covariate balance in matched observational studies

   https://doi.org/10.1111/biom.13395  

   Chen, Hao; Small, Dylan_S (November 2020, Biometrics)

   Abstract We propose new tests for assessing whether covariates in a treatment group and matched control group are balanced in observational studies. The tests exhibit high power under a wide range of multivariate alternatives, some of which existing tests have little power for. The asymptotic permutation null distributions of the proposed tests are studied and theP‐values calculated through the asymptotic results work well in simulation studies, facilitating the application of the test to large data sets. The tests are illustrated in a study of the effect of smoking on blood lead levels. The proposed tests are implemented in anRpackageBalanceCheck. 

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4. High-dimensional analysis of variance in multivariate linear regression

   https://doi.org/10.1093/biomet/asad001  

   Lou, Zhipeng; Zhang, Xianyang; Wu, Wei Biao (January 2023, Biometrika)

   Summary In this paper, we develop a systematic theory for high-dimensional analysis of variance in multivariate linear regression, where the dimension and the number of coefficients can both grow with the sample size. We propose a new U-type statistic to test linear hypotheses and establish a high-dimensional Gaussian approximation result under fairly mild moment assumptions. Our general framework and theory can be used to deal with the classical one-way multivariate analysis of variance, and the nonparametric one-way multivariate analysis of variance in high dimensions. To implement the test procedure, we introduce a sample-splitting-based estimator of the second moment of the error covariance and discuss its properties. A simulation study shows that our proposed test outperforms some existing tests in various settings. 

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5. Causal Discovery with General Non-Linear Relationships Using Non-Linear Independent Component Analysis,

   Ricardo Pio Monti, Kun Zhang (January 2019, Proceedings of Conference on Uncertainty in Artificial Intelligence (UAI) 2019)

   We consider the problem of inferring causal relationships between two or more passively ob-served variables. While the problem of such causal discovery has been extensively studied,especially in the bivariate setting, the majority of current methods assume a linear causal relationship, and the few methods which consider non-linear relations usually make the assumption of additive noise. Here, we propose a framework through which we can perform causal discovery in the presence of general nonlinear relationships. The proposed method is based on recent progress in non-linear in-dependent component analysis (ICA) and exploits the nonstationarity of observations in order to recover the underlying sources. We show rigorously that in the case of bivariate causal discovery, such non-linear ICA can be used to infer causal direction via a series of in-dependence tests. We further propose an alternative measure for inferring causal direction based on asymptotic approximations to the likelihood ratio, as well as an extension to multivariate causal discovery. We demonstrate the capabilities of the proposed method via a series of simulation studies and conclude with an application to neuroimaging data. 

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