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D-plots: Visualizations for Analysis of Bivariate Dependence Between Continuous Random Variables

Author

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  • Arturo Erdely

    (Programa de Actuaría, FES Acatlán, Universidad Nacional Autónoma de México, Avenida Alcanfores y San Juan Totoltepec S/N, Santa Cruz Acatlán, Naucalpan de Juárez 53150, Mexico
    These authors contributed equally to this work.)

  • Manuel Rubio-Sánchez

    (Departamento de Informática y Estadística, Universidad Rey Juan Carlos, C/Tulipan s/n, Móstoles, 28933 Madrid, Spain
    These authors contributed equally to this work.)

Abstract

Scatter plots are widely recognized as fundamental tools for illustrating the relationship between two numerical variables. Despite this, based on solid theoretical foundations, scatter plots generated from pairs of continuous random variables may not serve as reliable tools for assessing dependence. Sklar’s theorem implies that scatter plots created from ranked data are preferable for such analysis, as they exclusively convey information pertinent to dependence. This is in stark contrast to conventional scatter plots, which also encapsulate information about the variables’ marginal distributions. Such additional information is extraneous to dependence analysis and can obscure the visual interpretation of the variables’ relationship. In this article, we delve into the theoretical underpinnings of these ranked data scatter plots, hereafter referred to as rank plots. We offer insights into interpreting the information they reveal and examine their connections with various association measures, including Pearson’s and Spearman’s correlation coefficients, as well as Schweizer–Wolff’s measure of dependence. Furthermore, we introduce a novel visualization ensemble, termed a d-plot , which integrates rank plots, empirical copula diagnostics, and traditional summaries to provide a comprehensive visual assessment of dependence between continuous variables. This ensemble facilitates the detection of subtle dependence structures, including non-quadrant dependencies, that might be overlooked by traditional visual tools.

Suggested Citation

  • Arturo Erdely & Manuel Rubio-Sánchez, 2025. "D-plots: Visualizations for Analysis of Bivariate Dependence Between Continuous Random Variables," Stats, MDPI, vol. 8(2), pages 1-23, May.
    • Handle: RePEc:gam:jstats:v:8:y:2025:i:2:p:43-:d:1663667
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    References listed on IDEAS

    as
    1. Friendly, Michael, 2006. "Data Ellipses, HE Plots and Reduced-Rank Displays for Multivariate Linear Models: SAS Software and Examples," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 17(i06).
    2. Dungang Liu & Shaobo Li & Yan Yu & Irini Moustaki, 2021. "Assessing Partial Association Between Ordinal Variables: Quantification, Visualization, and Hypothesis Testing," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(534), pages 955-968, April.
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