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Original Articles

Many Faces of the Correlation Coefficient

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Abstract

Some selected interpretations of Pearson's correlation coefficient are considered. Correlation may be interpreted as a measure of closeness to identity of the standardized variables. This interpretation has a psychological appeal in showing that perfect covariation means identity up to positive linearity. It is well known that |r| is the geometric mean of the two slopes of the regression lines. In the 2 × 2 case, each slope reduces to the difference between two conditional probabilities so that |r| equals the geometric mean of these two differences. For bivariate distributions with equal marginals, that satisfy some additional conditions, a nonnegative r conveys the probability that the paired values of the two variables are identical by descent. This interpretation is inspired by the rationale of the genetic coefficient of inbreeding.

Acknowledgments

This study was supported by the Sturman Center for Human Development, the Hebrew University, Jerusalem. We are grateful to Raphael Falk for his continuous help in all the stages of this study.

Notes

Note 1: Formulas tying r to various test statistics ‐‐ thus suggesting additional interpretations ‐‐ can be found, for example, in CitationCohen (1965), CitationFriedman (1968), CitationLevy (1967), CitationRodgers and Nicewander (1988), and CitationRosenthal and Rubin (1982). Geometric and trigonometric interpretations of r can be found, among other sources, in CitationCahan (1987), CitationGuilford (1954, pp. 482–483), and CitationRodgers and Nicewander (1988).

Note 2: Note that the formula for Spearman's rank-order coefficient, rS, when there are no ties,

where di denotes the difference between the ranks of the ith pair, is structured similarly to (3). Spearman's rS is thus a measure of closeness to identity of the matched sets of ranks (see CitationCohen and Cohen 1975, p. 38, and CitationSiegel and Castellan 1988, pp. 235–241).

Note 3: Recently, CitationRovine and von Eye (1997) showed that when k of the n standardized values of the variables X and Y are identical (i.e., there are k matches) and the other n – k values are unrelated, the (nonnegative) correlation coefficient between X and Y approximately equals the proportion of matches.

A postscript version of this article (falk.ps) is available.

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