Spearman Rank Correlation Coefficient

The Spearman rank correlation coefficient, also known as Spearman's rho, is a nonparametric (distribution-free) rank statistic proposed by Spearman in 1904 as a measure of the strength of the associations between two variables (Lehmann and D'Abrera 1998). The Spearman rank correlation coefficient can be used to give an R-estimate, and is a measure of monotone association that is used when the distribution of the data make Pearson's correlation coefficient undesirable or misleading.

The Spearman rank correlation coefficient is defined by

(1)

where  is the difference in statistical rank of corresponding variables, and is an approximation to the exact correlation coefficient

(2)

computed from the original data. Because it uses ranks, the Spearman rank correlation coefficient is much easier to compute.

The variance, kurtosis excess, and higher-order moments are

			(3)
			(4)
			(5)

Student was the first to obtain the variance.

REFERENCES:

Hogg, R. V. and Craig, A. T. Introduction to Mathematical Statistics, 5th ed. New York: Macmillan, pp. 338 and 400, 1995.

Lehmann, E. L. and D'Abrera, H. J. M. Nonparametrics: Statistical Methods Based on Ranks, rev. ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 292, 300, and 323, 1998.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:Cambridge University Press, pp. 634-637, 1992.