# Validation of Association

## TL;DR

This paper introduces a new quantile dependence function to measure and visualize complex dependence structures between variables, providing new tests for independence and insightful diagnostic plots.

## Contribution

It develops a novel function-valued measure of dependence, new estimators, and tests for independence, enhancing interpretability and detection of dependence in joint distributions.

## Key findings

- The new tests outperform existing independence tests in simulations.
- The dependence function reveals detailed dependence structures in real data.
- Graphical tools aid in interpreting complex dependence patterns.

## Abstract

Recognizing, quantifying and visualizing associations between two variables is increasingly important. This paper investigates how a new function-valued measure of dependence, the quantile dependence function, can be used to construct tests for independence and to provide an easily interpretable diagnostic plot of existing departures from the null model. The dependence function is designed to detect general dependence structure between variables in quantiles of the joint distribution. It gives an insight into how the dependence structures changes in different parts of the joint distribution. We define new estimators of the dependence function, discuss some of their properties, and apply them to construct new tests of independence. Numerical evidence is given on the test's benefits against three recognized independence tests introduced in the previous years. In real-data analysis, we illustrate the use of our tests and the graphical presentation of the underlying dependence structure.