Visualizing Multiple Quantile Plots

Abstract

Multiple-quantile plots provide a powerful graphical method for comparing the distributions of two or more populations. This article develops a method of visualizing triple-quantile plots and their associated confidence tubes, thus extending the notion of a quantile–quantile (QQ) plot to three dimensions. More specifically, we consider three independent one-dimensional random samples with corresponding quantile functions Q₁, Q₂, and Q₃. The triple-quantile (QQQ) plot is then defined as the three-dimensional curve Q(p) = (Q₁(p), Q₂(p), Q₃(p)), where 0 < p < 1. The empirical likelihood method is used to derive simultaneous distribution-free confidence tubes for Q. We apply our method to an economic case study of strike durations and to an epidemiological study involving the comparison of cholesterol levels among three populations. These data as well as the Mathematica code for computation of the tubes are available in the online supplementary materials.

Key Words:

• Confidence region
• Empirical likelihood
• Three-sample comparison

ACKNOWLEDGMENTS

The authors are grateful to Jaap Abbring for pointing out the strike data and to Daniel McGee for providing the cholesterol dataset. Ian McKeague’s research was partially supported by the National Institutes of Health (NIH) grant R01GM095722-01.

Additional information

Notes on contributors

Marko A. A. Boon

Marko A. A. Boon, Assistant Professor, Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (E-mail: [email protected]). John H. J. Einmahl, Professor, Department of Econometrics & OR and CentER, Tilburg University, PO Box 90153, 5000 LE Tilburg, The Netherlands (E-mail: [email protected]). Ian W. McKeague, Professor, Department of Biostatistics, Columbia University, 722 West 168th Street, New York, NY 10032 (E-mail: [email protected]).

John H. J. Einmahl

Marko A. A. Boon, Assistant Professor, Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (E-mail: [email protected]). John H. J. Einmahl, Professor, Department of Econometrics & OR and CentER, Tilburg University, PO Box 90153, 5000 LE Tilburg, The Netherlands (E-mail: [email protected]). Ian W. McKeague, Professor, Department of Biostatistics, Columbia University, 722 West 168th Street, New York, NY 10032 (E-mail: [email protected]).

Ian W. McKeague

Marko A. A. Boon, Assistant Professor, Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (E-mail: [email protected]). John H. J. Einmahl, Professor, Department of Econometrics & OR and CentER, Tilburg University, PO Box 90153, 5000 LE Tilburg, The Netherlands (E-mail: [email protected]). Ian W. McKeague, Professor, Department of Biostatistics, Columbia University, 722 West 168th Street, New York, NY 10032 (E-mail: [email protected]).