Multi-Panel Kendall plot in light of an ROC curve analysis applied to measuring dependence

ABSTRACT

The Kendall plot ($\mathrm{K}$-plot) is a plot measuring dependence between the components of a bivariate random variable. The $\mathrm{K}$-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 $\mathrm{K}$-plots with the receiver operating characteristic ($\mathrm{ROC}$) 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 $\mathrm{ROC}$ curve ($\mathrm{AUC}$). In parallel with the $\mathrm{AUC}$, we propose a novel strategy to measure association between random variables from a continuous bivariate distribution. First, we discuss why the area under the conventional Kendall curve ($\mathrm{AUK}$) cannot be used as an index of dependence. We then suggest a simple and meaningful extension of the definition of the $\mathrm{K}$-plots, and define an index of dependence that is based on $\mathrm{AUK}$. 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.

KEYWORDS:

• $\mathrm{AUC}$
• dependence measure
• Kendall distribution function
• Kendall's τ
• Kendall plot
• nonparametric association
• $\mathrm{ROC}$ curves

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Acknowledgments

The authors are grateful to the Editor, the Associate Editor and the referees for suggestions that led to a substantial improvement of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Dr. Vexler's effort was supported by the National Institutes of Health (NIH) grant 1G13LM012241-01. Dr. Markatou would like to thank the Jacobs School of Medicine and Biomedical Sciences for facilitating this work through institutional financial resources (to M. Markatou) that supported the work of the second author of this paper. She also acknowledges PCORI for partially supporting this work under award 1507-31640.