# Kendall Tau

Kendall tau is a nonparametric measure of correlation defined as:

T = (# agreements - # disagreements) / total number of pairs

For small n (n <10), the exact probability can be calculated. The tabulated values can be found in Siegel and Castellan. However, the exact sampling distribution of T approaches a normal distribution very quickly with increasing n size. For n = 10 or more, refer to the normal distribution (Hays, 1988).

Kendall tau is equivalent to the Spearman R statistic with regard to the underlying assumptions. It is also comparable in terms of its statistical power. However, Spearman R and Kendall tau are usually not identical in magnitude because their underlying logic, as well as their computational formulas are very different. Siegel and Castellan (1988) express the relationship of the two measures in terms of the inequality:

-1 < = 3 * Kendall tau - 2 * Spearman R < = 1

More importantly, Kendall tau and Spearman R imply different interpretations: While Spearman R can be thought of as the regular Pearson product-moment correlation coefficient as computed from ranks, Kendall tau rather represents a probability. Specifically, it is the difference between the probability that the observed data are in the same order for the two variables versus the probability that the observed data are in different orders for the two variables. Kendall (1948, 1975), Everitt (1977), and Siegel and Castellan (1988) discuss Kendall tau in greater detail. Two different variants of tau are computed, usually called taub and tauc. These measures differ only with regard as to how tied ranks are handled. In most cases these values will be fairly similar, and when discrepancies occur, it is probably always safest to interpret the lowest value.