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

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.