rankcorrelation

Rank correlation is a statistical measure of the strength and direction of association between two variables when the data are considered as ranks rather than raw values. It is nonparametric and is especially appropriate for ordinal data or for detecting monotonic relationships that may be nonlinear.

The two most common rank correlation coefficients are Spearman's rho and Kendall's tau. Spearman's rho is defined as the Pearson correlation coefficient computed on the ranks of the data. When there are no ties, it can be expressed as rho = 1 - (6 * sum d_i^2) / (n(n^2 - 1)), where d_i is the difference between the ranks of the paired observations and n is the number of pairs. With ties, average ranks are used and corrections may be applied.

Kendall's tau is based on the relative ordering of pairs. It equals (C - D) / [n(n-1)/2], where C is the number of concordant pairs and D the number of discordant pairs. Extensions such as tau-b and tau-c adjust for ties to produce values in the range [-1, 1].

Interpretation of rank correlations is straightforward: values near 1 indicate a strong positive monotonic relationship, values near -1 a strong negative monotonic relationship, and values around 0 suggest little or no monotonic association. Compared with Pearson correlation, rank correlations are more robust to outliers and do not assume linearity, making them useful for ordinal data or when the relationship is monotonic but not linear. Limitations include the restriction to monotonic relationships and the loss of information about the exact form of the association; ties can complicate calculation and interpretation.