KendallTau

Kendall's tau, or Kendall tau, is a nonparametric statistic that measures the strength and direction of association between two variables based on the ordering of data. It was introduced by Maurice Kendall in 1938 and relies on pairwise comparisons of observations.

For each pair of observations, Kendall’s method classifies the pair as concordant if the ranks of both variables agree (x_i > x_j and y_i > y_j, or x_i < x_j and y_i < y_j) and discordant if they disagree (x_i > x_j and y_i < y_j, or x_i < x_j and y_i > y_j). Pairs that tie on one or both variables are recorded separately. Kendall’s tau summarizes association as a function of these counts.

There are several variants to handle ties. Kendall’s tau-a uses only the difference between concordant and discordant pairs divided by the total number of pairs, ignoring ties: tau-a = (N_c − N_d) / N_0, where N_0 = n(n−1)/2. Kendall’s tau-b adjusts for ties: tau-b = (N_c − N_d) / sqrt((N_0 − N_x)(N_0 − N_y)), where N_x and N_y are the numbers of pairs tied on X and on Y, respectively. Kendall’s tau-c (Stuart–Maxwell form) is a variant for rectangular data grids and provides a normalization that yields a maximum of 1 regardless of marginal totals.

Interpretation: values range from −1 to 1. Positive values indicate a monotone positive association, negative values indicate a monotone negative association, and values near zero suggest little monotone relation. Significance can be tested with asymptotic or exact methods.

Similar to other rank-based measures, Kendall’s tau is robust to outliers and appropriate for ordinal data. It is commonly computed in statistical software packages such as R and SciPy.