Statistics Based on Ranks - Kendall's Tau

Kendall's tau-b is computed as:

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

To account for tied rankings, Siegel (1956) gives the computational formula:

where P and Q are the number of concordant pairs (# agreements) and discordant pairs (# disagreements), respectively.

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 tau approaches a normal distribution very quickly with increasing n size. For n = 10 or more, refer to the normal distribution (Hays, 1988).

Kendall's 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's 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's tau - 2 * Spearman R <= 1

More importantly, Kendall's 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's 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's tau in greater detail.

Stuart's tau-c. Two different variants of tau are computed, usually called tau-b,- and Stuart's tau-c:

where m = min(R,C).

Stuart's tau-c makes a correction for table size in addition to a correction for ties. In most cases these values will be fairly similar, and when discrepancies occur, it is probably always safest to interpret the lowest value.