Kendall Tau

A non-parametric measure of association between two ranked variables, based on the number of concordant and discordant pairs of observations.

Source: Kendall 1938

Kendall's tau measures the association between two ranked variables by comparing every pair of observations and counting how many are concordant, ordered the same way on both variables, against how many are discordant. The balance of concordant and discordant pairs gives a coefficient between negative one and positive one, capturing whether the two tend to rise together. As a non-parametric measure it makes no assumption about the shape of the data, relying only on rank order. Comparing the ordering of pairs is its method. Kirkwood and Sterne (2003) describe this measure.

Source: Kirkwood & Sterne 2003

Kendall's tau is calculated by examining all pairs of observations and counting how many are concordant, ordered the same way by both variables, and how many are discordant, ordered oppositely, then taking the difference between these counts and dividing by the total number of pairs, with adjustments for ties in some versions. So Kendall's tau is calculated from the balance of concordant and discordant pairs relative to all pairs, giving a value between negative one and positive one, where a preponderance of concordant pairs yields a positive tau and of discordant pairs a negative one, reflecting the direction and strength of the monotonic association between the ranks.

Source: Kendall 1938

Kendall's tau and Spearman's correlation are both rank-based measures of monotonic association, but they are computed differently: Kendall's tau is based on the counts of concordant and discordant pairs, while Spearman's is essentially the Pearson correlation of the ranks. They usually agree in direction but differ in magnitude, with Kendall's tau typically smaller. So the two differ in their construction and scale, with Kendall's tau interpretable in terms of pair concordance and Spearman's as a rank correlation, and although they generally give similar conclusions about association, Kendall's tau is often preferred for its interpretation and its behaviour with small samples and ties.

Source: Pearson 1896

Kendall's tau is used to measure association between two ordinal variables or when a rank-based, non-parametric measure is wanted that does not assume a linear relationship or a particular distribution, and it is robust to outliers. It is suitable for ordinal data and small samples. So Kendall's tau is used when the relationship between two variables is to be assessed through their ranks rather than their values, which is appropriate for ordinal data, non-linear monotonic relationships, or data with outliers, providing a measure of association that, like Spearman's, does not rely on the assumptions of Pearson's correlation.

Source: Kendall 1938

Kendall's tau is interpreted by its sign and magnitude: a positive value indicates that the variables tend to rank observations in the same order, a negative value in opposite orders, and a value near zero little monotonic association, with values near one or negative one indicating strong association. It can be read as relating to the probability of concordance versus discordance. So Kendall's tau is interpreted as the strength and direction of monotonic association based on pair concordance, with larger absolute values indicating more consistent ordering, providing an interpretable rank-based measure, though, like all measures of association, it does not imply causation.

Source: Kendall 1938