Many Faces of the Correlation Coefficient

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.

• Association in 2 × 2 table
• Correlation as probability
• Inbreeding
• Regression slopes

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 Cohen (1965), Friedman (1968), Levy (1967), Rodgers and Nicewander (1988), and Rosenthal and Rubin (1982). Geometric and trigonometric interpretations of r can be found, among other sources, in Cahan (1987), Guilford (1954, pp. 482–483), and Rodgers and Nicewander (1988).

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

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

Note 3: Recently, Rovine 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.