The post-hoc test for local dependence

TL;DR

This paper introduces a novel copula-based method for testing both global and local independence in data, providing detailed insights into where dependence occurs and how strong it is, with enhanced interpretability.

Contribution

It proposes a new approach using critical surfaces and quantile dependence functions to identify and visualize local dependence while maintaining overall significance levels.

Findings

The method effectively detects local dependence discrepancies.

It maintains the overall significance level of the independence test.

Provides graphical representations for better interpretability.

Abstract

The concept of independence plays a crucial role in probability theory and has been the subject of extensive research in recent years. Numerous approaches have been proposed to test for independence; however, most of them address the problem only at a global level. From a practical perspective, it is important not only to determine whether the data are dependent but also to identify where this dependence occurs and how strong it is. The graphical presentation of results is another essential aspect that should not be neglected, as it considerably enhances interpretability. The main objective of this work is to propose a solution that considers these aspects simultaneously. Relying on copula-based results, we introduce a novel method for testing global and local statistical independence using the quantile dependence function. Rather than assessing whether the value of the test statistic exceeds a single critical threshold and subsequently deciding whether to reject the independence hypothesis, we introduce so-called critical surfaces that guaranty a locally equal probability of exceeding them under independence. This approach enables a detailed examination of local discrepancies and an assessment of their statistical significance while preserving the overall significance level of the test.