Multi-scale Fisher’s independence test for multivariate dependence

Abstract

SUMMARY 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 in the presence of massive sample sizes. 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 2 × 2 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. We derive a finite-sample theory that guarantees the inferential validity of our adaptive procedure at any given sample size. We show that our approach can achieve strong control of the level of the testing procedure at any sample size without resampling or asymptotic approximation and establish its large-sample consistency. We demonstrate through an extensive simulation study its substantial computational advantage in comparison to existing approaches while achieving robust statistical power under various dependency scenarios, and illustrate how its divide-and-conquer nature 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 dataset from a flow cytometry experiment.

Towards a universal representation of statistical dependence

“…Remark 4.1. Such 'multi-scale' approach, through artificially created (2 × 2)-tables partitioning S XY , resonates with recent ideas of Ma and Mao (2019), Zhang (2019) or Gorsky and Ma (2022) for independence…”

AUGUST: An Interpretable, Resolution-based Two-sample Test

“…Computational burdens could be eased by performing inference across a carefully-selected range of binary depths. For example, as a multivariate test of dependence, the coarse-to-fine sequential adaptive method of [19] chooses a subset of available univariate tests at each resolution using spatial knowledge of dependency structures.…”

Discussion of ‘Multi-scale Fisher’s independence test for multivariate dependence’

“…We read with interest the work of Gorsky and Ma (2022) on statistical dependence testing using a Multiscale Fisher's Independence Test (MultiFIT). The procedure consists in first transforming the data to map to the unit ball, then performing univariate Fisher's exact tests of independence on a collection of 2 × 2 contingency tables, and finally correcting for the use of multiple testing.…”

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We discuss how MultiFIT, the Multiscale Fisher's Independence Test for Multivariate Dependence proposed by Gorsky and Ma (2022), compares to existing linear-time kernel tests based on the Hilbert-Schmidt independence criterion (HSIC). We highlight the fact that the levels of the kernel tests at any finite sample size can be controlled exactly, as it is the case with the level of MultiFIT.

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