Multiscale Fisher's Independence Test for Multivariate Dependence
This work addresses the problem of scalable dependency testing for researchers and practitioners dealing with massive multivariate datasets, offering a computationally efficient alternative to existing methods.
The paper tackles the computational inefficiency of existing nonparametric independence tests for multivariate data, which scale poorly with sample size and require resampling, by introducing a scalable, resampling-free method that transforms the task into multiple univariate tests with almost linear complexity, achieving strong control of family-wise error rate and demonstrating substantial computational advantages and decent statistical power in simulations.
Identifying dependency in multivariate data is a common inference task that arises in numerous applications. However, existing nonparametric independence tests typically require computation that scales at least quadratically with the sample size, making it difficult to apply them to massive data. Moreover, resampling is usually necessary to evaluate the statistical significance of the resulting test statistics at finite sample sizes, further worsening the computational burden. We introduce a scalable, resampling-free approach to testing the independence between two random vectors by breaking down the task into simple univariate tests of independence on a collection of 2x2 contingency tables constructed through sequential coarse-to-fine discretization of the sample space, transforming the inference task into a multiple testing problem that can be completed with almost linear complexity with respect to the sample size. To address increasing dimensionality, we introduce a coarse-to-fine sequential adaptive procedure that exploits the spatial features of dependency structures to more effectively examine the sample space. We derive a finite-sample theory that guarantees the inferential validity of our adaptive procedure at any given sample size. In particular, we show that our approach can achieve strong control of the family-wise error rate without resampling or large-sample approximation. We demonstrate the substantial computational advantage of the procedure in comparison to existing approaches as well as its decent statistical power under various dependency scenarios through an extensive simulation study, and illustrate how the divide-and-conquer nature of the procedure can be exploited to not just test independence but to learn the nature of the underlying dependency. Finally, we demonstrate the use of our method through analyzing a large data set from a flow cytometry experiment.