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Register a new userNonpar MANOVA via Independence Testing
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The $k$-sample testing problem tests whether or not $k$ groups of data points are sampled from the same distribution. Multivariate analysis of variance (MANOVA) is currently the gold standard for $k$-sample testing but makes strong, often inappropriate, parametric assumptions. Moreover, independence testing and $k$-sample testing are tightly related, and there are many nonparametric multivariate independence tests with strong theoretical and empirical properties, including distance correlation (Dcorr) and Hilbert-Schmidt-Independence-Criterion (Hsic). We prove that universally consistent independence tests achieve universally consistent $k$-sample testing and that $k$-sample statistics like Energy and Maximum Mean Discrepancy (MMD) are exactly equivalent to Dcorr. Empirically evaluating these tests for $k$-sample scenarios demonstrates that these nonparametric independence tests typically outperform MANOVA, even for Gaussian distributed settings. Finally, we extend these non-parametric $k$-sample testing procedures to perform multiway and multilevel tests. Thus, we illustrate the existence of many theoretically motivated and empirically performant $k$-sample tests. A Python package with all independence and k-sample tests called hyppo is available from https://hyppo.neurodata.io/.
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Complex data structures such as time series are increasingly present in modern data science problems. A fundamental question is whether two such time-series are statistically dependent. Many current approaches make parametric assumptions on the random processes, only detect linear association, require multiple tests, or forfeit power in high-dimensional, nonlinear settings. Estimating the distribution of any test statistic under the null is non-trivial, as the permutation test is invalid. This work juxtaposes distance correlation (Dcorr) and multiscale graph correlation (MGC) from independence testing literature and block permutation from time series analysis to address these challenges. The proposed nonparametric procedure is valid and consistent, building upon prior work by characterizing the geometry of the relationship, estimating the time lag at which dependence is maximized, avoiding the need for multiple testing, and exhibiting superior power in high-dimensional, low sample size, nonlinear settings. Neural connectivity is analyzed via fMRI data, revealing linear dependence of signals within the visual network and default mode network, and nonlinear relationships in other networks. This work uncovers a first-resort data analysis tool with open-source code available, directly impacting a wide range of scientific disciplines.
We consider the hypothesis testing problem of detecting conditional dependence, with a focus on high-dimensional feature spaces. Our contribution is a new test statistic based on samples from a generative adversarial network designed to approximate directly a conditional distribution that encodes the null hypothesis, in a manner that maximizes power (the rate of true negatives). We show that such an approach requires only that density approximation be viable in order to ensure that we control type I error (the rate of false positives); in particular, no assumptions need to be made on the form of the distributions or feature dependencies. Using synthetic simulations with high-dimensional data we demonstrate significant gains in power over competing methods. In addition, we illustrate the use of our test to discover causal markers of disease in genetic data.
A number of universally consistent dependence measures have been recently proposed for testing independence, such as distance correlation, kernel correlation, multiscale graph correlation, etc. They provide a satisfactory solution for dependence testing in low-dimensions, but often exhibit decreasing power for high-dimensional data, a phenomenon that has been recognized but remains mostly unchartered. In this paper, we aim to better understand the high-dimensional testing scenarios and explore a procedure that is robust against increasing dimension. To that end, we propose the maximum marginal correlation method and characterize high-dimensional dependence structures via the notion of dependent dimensions. We prove that the maximum method can be valid and universally consistent for testing high-dimensional dependence under regularity conditions, and demonstrate when and how the maximum method may outperform other methods. The methodology can be implemented by most existing dependence measures, has a superior testing power in a variety of common high-dimensional settings, and is computationally efficient for big data analysis when using the distance correlation chi-square test.
In this article, we consider the problem of high-dimensional conditional independence testing, which is a key building block in statistics and machine learning. We propose a double generative adversarial networks (GANs)-based inference procedure. We first introduce a double GANs framework to learn two generators, and integrate the two generators to construct a doubly-robust test statistic. We next consider multiple generalized covariance measures, and take their maximum as our test statistic. Finally, we obtain the empirical distribution of our test statistic through multiplier bootstrap. We show that our test controls type-I error, while the power approaches one asymptotically. More importantly, these theoretical guarantees are obtained under much weaker and practically more feasible conditions compared to existing tests. We demonstrate the efficacy of our test through both synthetic and real datasets.
In this article, we propose a new hypothesis testing method for directed acyclic graph (DAG). While there is a rich class of DAG estimation methods, there is a relative paucity of DAG inference solutions. Moreover, the existing methods often impose some specific model structures such as linear models or additive models, and assume independent data observations. Our proposed test instead allows the associations among the random variables to be nonlinear and the data to be time-dependent. We build the test based on some highly flexible neural networks learners. We establish the asymptotic guarantees of the test, while allowing either the number of subjects or the number of time points for each subject to diverge to infinity. We demonstrate the efficacy of the test through simulations and a brain connectivity network analysis.
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