Multi-Panel Kendall Plot in Light of an ROC Curve Analysis Applied to Measuring Dependence

TL;DR

This paper introduces a new dependence measure based on Kendall plots and ROC curve analysis, capable of capturing complex relationships between variables, with demonstrated effectiveness through simulations and real data examples.

Contribution

It proposes a novel dependence index derived from extended Kendall plots and ROC analysis, addressing limitations of existing measures like AUK.

Findings

The new index effectively detects various dependence structures.

Simulations confirm the measure's mathematical properties.

Real data examples demonstrate practical applicability.

Abstract

The Kendall plot ($\K$-plot) is a plot measuring dependence between the components of a bivariate random variable. The $\K$-plot graphs the Kendall distribution function against the distribution function of $VU$, where $V$ and $U$ are independent uniform $[0,1]$ random variables. We associate $\K$-plots with the receiver operating characteristic ($\ROC$) curve, a well-accepted graphical tool in biostatistics for evaluating the ability of a biomarker to discriminate between two populations. The most commonly used global index of diagnostic accuracy of biomarkers is the area under the $\ROC$ curve ($\AUC$). In parallel with the $\AUC$, we propose a novel strategy to measure the association between random variables from a continuous bivariate distribution. First, we discuss why the area under the conventional Kendall curve ($\AUK$) cannot be used as an index of dependence. We then suggest a simple and meaningful extension of the definition of the $\K$-plots and define an index of dependence that is based on $\AUK$. This measure characterizes a wide range of two-variable relationships, thereby completely detecting the underlying dependence structure. Properties of the proposed index satisfy the mathematical definition of a measure. Finally, simulations and real data examples illustrate the applicability of the proposed method.