An algorithm and program for calculation of Kendall’s rank correlation coefficient

Abstract

Although less widely used than Spearman's rho, Ken-dall's (1938, 1975) rank correlation coefficient (tau) possesses advantages that may make it a preferred statistic: Its distribution under the null hypothesis is approximately normal even when the sample size (n) is fairly small'; it allows determination of confidence interval bounds; and it can be applied to partial correlation (Kendall, 1975). Nevertheless, many statistics texts do not discuss tau, and some that do offer only fragmentary information. A few texts (e.g., McCall, 1980; Walker & Lev, 1953) present rho in considerable detail, mention the advantages of tau, and then ironically say nothing more about the use or calculation of tau. Kendall (1975) described the calculation of tau as involving comparison of every pair of ranks within each of the two distributions being studied. For a given pair of ranks Xi and Xi> where i < j, a score of +1 is assigned if Xi < Xj; a score of-1 is assigned if Xi > Xj; and a score of 0 is assigned ifr, = x; The statistic S, which is linearly related to tau, is then obtained by summing the products of the resulting scores for each corresponding pair of ranks in the two distributions. After a little simplification, the calculations can be summarized by the equation n-[ n S = E E sgn [(xj-X,)(yj-Yi)], (1) i=lj=i+l where X and Yrepresent ranks in the first and second distributions , respectively. Note that X and Ycan represent raw scores other than ranks and that this procedure requires neither sorting of the data nor assignment of ranks. Equation 1 comprises the main part of an algorithm for calculation of tau. [Tau = SID, where D is the maximum possible value of S for a given n. When there are no tied ranks, D = n(n-1)12. Ties decrease D, and thus increase tau for a given value of S.] Kendall (1938, 1975) proposed shortcut methods for calculating S, one of which appears to be the most common current method. It requires the sorting into natural order of ranks in one distribution, together with ranks for corresponding subjects in the second distribution. A score of +1 or-1 is assigned to each pair of ranks in the second distribution, depending on whether the rank for the second member of the pair is greater than or less than