Spearman Rank Correlation Coefficient

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.

The Spearman rank correlation coefficient is defined by

$$r'\equiv 1-6\sum {d^2\over N(N^2-1)},$$ (1)

where $d$ is the difference in Rank of corresponding variables, and is an approximation to the exact Correlation Coefficient

$$r\equiv {\sum xy\over \sqrt{\sum x^2\sum y^2}}$$ (2)

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

The Variance, Kurtosis, and higher order Moments are

$$\sigma^2 = {1\over N-1}$$ (3)

$$\gamma_2 = -{114\over 25N}-{6\over 5N^2}-\ldots$$ (4)

$$\gamma_3 = \gamma_5=\ldots=0.$$ (5)

Student was the first to obtain the Variance.

See also Correlation Coefficient, Least Squares Fitting, Linear Regression, Rank (Statistics)

References

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.