The Annals of Statistics

Graph-Theoretic Measures of Multivariate Association and Prediction

Jerome H. Friedman and Lawrence C. Rafsky

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Interpoint-distance-based graphs can be used to define measures of association that extend Kendall's notion of a generalized correlation coefficient. We present particular statistics that provide distribution-free tests of independence sensitive to alternatives involving non-monotonic relationships. Moreover, since ordering plays no essential role, the ideas are fully applicable in a multivariate setting. We also define an asymmetric coefficient measuring the extent to which (a vector) $X$ can be used to make single-valued predictions of (a vector) $Y$. We discuss various techniques for proving that such statistics are asymptotically normal. As an example of the effectiveness of our approach, we present an application to the examination of residuals from multiple regression.

Article information

Ann. Statist., Volume 11, Number 2 (1983), 377-391.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62G10: Hypothesis testing
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.)

Multivariate association interpoint distances graph theory linear permutation statistics examination of residuals


Friedman, Jerome H.; Rafsky, Lawrence C. Graph-Theoretic Measures of Multivariate Association and Prediction. Ann. Statist. 11 (1983), no. 2, 377--391. doi:10.1214/aos/1176346148.

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