Kendall’s Tau: Understanding its Logic and Computation

When researchers need to measure the relationship between two sets of ranked data, most reach for Spearman’s rho out of habit. But there’s another rank correlation measure – Kendall’s tau (τ) – that approaches the same problem from a fundamentally different angle, and in many statistical contexts, it actually does the job better. Kendall’s tau is a nonparametric coefficient that quantifies the strength and direction of association between two ranked variables, not by comparing differences in ranks, but by examining pairs of observations and asking a simple question: do these two subjects rank in the same order on both variables, or in the opposite order? This pair-by-pair logic – rooted in the concepts of concordance and discordance – makes τ one of the most interpretable correlation measures available to psychology researchers.

The logic behind Kendall’s tau 🔗

Unlike Spearman’s rho, which applies Pearson’s product-moment formula to ranked data, Kendall’s tau takes a more direct route. It is a nonparametric measure of association based on counting concordant and discordant pairs among all possible pairings of observations. For any two observations – say, participant i and participant j – the pair is concordant if both variables are ordered in the same direction (i ranks higher than j on both variable X and variable Y). The pair is discordant if the ordering is reversed: higher on X but lower on Y, or vice versa.

Kendall’s tau was introduced by Maurice Kendall in the mid-20th century as an alternative to Spearman’s rank correlation, offering robustness in situations where ties are present in the data. The coefficient τ is essentially the net balance between these two types of pairs, scaled to fall between −1 and +1. A value of +1 means every single pair is concordant – perfect agreement in rankings. A value of −1 means every pair is discordant – rankings are completely reversed. A value near 0 indicates no consistent ordering relationship between the two variables.

This concordance-discordance logic also gives τ a clean probabilistic meaning: it represents the probability that a randomly selected pair of observations is concordant, minus the probability that it is discordant. This makes the interpretation of Kendall’s tau in terms of agreeable and non-agreeable pairs very direct, which is especially useful when communicating findings to audiences who are not statisticians.

How Kendall’s tau is computed 🔗

The basic formula for Kendall’s tau (often called tau-a, used when there are no ties) is:

τ = (C − D) / (C + D)

where C is the total count of concordant pairs and D is the total count of discordant pairs. Concordant and discordant pairs are determined from ranked values, not the raw data. For a dataset of n observations, the total number of unique pairings is n(n−1)/2. Each pairing is examined and classified as concordant, discordant, or tied.

A worked example 🔗

To make this concrete, consider a clinical example drawn from StatsDirect: a tutor ranks ten clinical psychology students on two dimensions – career suitability and psychology knowledge. The question is whether students ranked higher in knowledge are also ranked higher in career suitability. After listing all possible pairs of students and identifying which pairs are concordant and which are discordant, we get a Kendall’s tau of 0.51. A two-sided significance test yields p = .047, allowing us to reject the null hypothesis and conclude that the two rankings are not independent. Students the tutor rated as more knowledgeable were also rated as more career-suitable, and this association is statistically significant.

As a simpler illustration: suppose two doctors each rank six patients by physical health. We take one doctor’s ranking as the reference (sorted 1 to 6) and compare it with the other doctor’s ranking. For each rank, we look at all ranks below it and note whether they are larger (concordant) or smaller (discordant). If we count 11 concordant and 4 discordant pairs, tau = (11 − 4) / (11 + 4) = 0.47, indicating a moderate positive agreement between the two doctors’ assessments.

Adjusting for tied ranks 🔗

Real-world data – especially in psychology – frequently contains ties. Rating scales with a limited number of response options often cause multiple participants to receive identical scores, and when assigned the same rank, they cannot be classified as either concordant or discordant. Kendall’s tau addresses this through a variant called tau-b (τb).

Tau-b uses a correction in the denominator that accounts separately for ties on the X variable, ties on the Y variable, and ties on both simultaneously, ensuring the coefficient still ranges between −1 and +1. When there are no ties in the data, τb equals the simpler τa. Tau-b is the default used by most statistical software, including R’s cor() function, when ties are present.

A third variant, tau-c (also called Stuart-Kendall tau-c), is designed for situations where the two variables have different numbers of possible values – for instance, when one variable is continuous and the other is binary or has fewer ordinal levels. Tau-c can reach ±1 for non-square contingency tables, unlike tau-b, which is bounded below ±1 in such cases. For most psychological research involving rating scales and preference rankings, tau-b is the appropriate choice.

Testing for significance 🔗

Once τ is computed, the next step is to determine whether it differs significantly from zero – that is, whether there is a real association in the population, rather than a pattern that could have arisen by chance.

Small samples (n < 10) 🔗

When data contain no ties and the sample is small (fewer than 9 observations), the significance of tau is evaluated using a recurrence formula to derive exact probability values. This exact method is preferred over approximations when sample sizes are very small, because the normal approximation is less reliable in these conditions.

Larger samples 🔗

For larger samples, a z-statistic is computed and compared to the standard normal distribution. When reporting a Kendall’s tau result, researchers typically include the coefficient, the degrees of freedom (n − 2), and the p-value – for example: “A Kendall’s tau correlation indicated a moderate positive association, τ(4) = 0.47, p = .188.” When ties are present, inferences are drawn from a normal approximation using an extended variance formula.

Kendall’s tau vs. Spearman’s rho 🔗

Both Kendall’s τ and Spearman’s ρ are rank-based, nonparametric measures of monotonic association, and in most datasets they point to the same conclusion. But they differ in meaningful ways, and these differences matter in certain research situations.

How the coefficients compare in value 🔗

For the same dataset, Spearman’s rho typically yields a numerically larger value than Kendall’s tau. This is not because one is more “accurate” than the other – it reflects the different scales and mathematical structures of the two formulas. Despite the difference in magnitude, both coefficients typically lead to the same inferences. The key difference is in how they handle deviations: Spearman’s rho is more sensitive to errors and discrepancies in the data, while Kendall’s tau has better distributional properties.

When tau is preferred 🔗

Kendall’s tau has several specific advantages that make it the better choice in certain scenarios:

Small samples with tied ranks: Kendall’s tau should be preferred over Spearman’s correlation when the dataset is small and there are many tied ranks. Tied values are handled more precisely through tau-b’s correction factors.

Psychiatric and clinical data: A study examining correlation methods in psychiatric research found that Kendall’s tau-b maintained adequate control of type I error rates and provided less biased estimates than Spearman’s r when applied to psychiatric symptom rating data. This has clear implications for clinical psychology research, where symptom severity scales frequently produce skewed, tied data.

Probabilistic interpretability: In the textbook Statistical Methods for Psychology, Dave Howell concludes that Kendall’s tau is generally preferred over Spearman’s rho because it is a better estimate of the corresponding population parameter and its standard error is known. The probabilistic interpretation of tau – the net probability of concordance over discordance – is also more conceptually transparent than the rank-difference basis of rho.

Non-normal and ordinal data: Kendall’s tau is robust to outliers and works well with skewed or non-normally distributed data. This is particularly relevant in behavioral and social science research, where data rarely follows a perfect normal distribution.

When Spearman’s rho is still used 🔗

Despite tau’s statistical advantages, Spearman’s rho remains more widely reported in published research, largely due to convention and familiarity. In most situations, the two measures align closely and lead to the same inferences, so either is defensible. Spearman’s rho also has computational advantages for very large datasets, where tau’s pairwise comparison approach can be more demanding to compute.

Applications in psychology research 🔗

Kendall’s tau appears across a wide range of psychological research contexts where rank data is collected. In social science and psychology, survey results and behavioral studies often yield ordinal data – Likert-scale responses, ranked preferences, observer ratings – that do not meet the assumptions of parametric tests. In such cases, tau provides a reliable, assumption-light alternative.

In clinical settings, tau is used to examine associations between symptom severity rankings and treatment outcome rankings. In educational psychology, it quantifies whether students who rank highly on one assessment tend to rank highly on another. In personality research, it can capture monotonic relationships between trait rankings that aren’t necessarily linear in nature.

The measure’s direct grounding in the logic of agreement – do these two subjects maintain the same relative ordering across both variables? – makes it particularly well-suited to any research question where the meaningful unit is the comparison between individuals rather than their absolute scores.

What do you think? When a researcher finds the same significant result using both Spearman’s rho and Kendall’s tau, does it matter which one they report – and should journals have standardized guidelines for choosing between them? If you were designing a study comparing clinicians’ rankings of patient improvement, would you trust Kendall’s tau more than Spearman’s rho to detect a real association, and why?

References

1. https://www.statisticshowto.com/kendalls-tau/
2. https://online.stat.psu.edu/stat509/lesson/18/18.3
3. https://www.numberanalytics.com/blog/understanding-kendalls-tau-key-stats-correlation
4. https://www.statisticssolutions.com/free-resources/directory-of-statistical-analyses/kendalls-tau-and-spearmans-rank-correlation-coefficient/
5. https://library.virginia.edu/data/articles/correlation-pearson-spearman-and-kendalls-tau
6. https://www.statsdirect.com/help/nonparametric_methods/kendall_correlation.htm
7. https://numiqo.com/tutorial/kendalls-tau
8. https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient
9. https://pubmed.ncbi.nlm.nih.gov/10221741/
10. https://www.researchgate.net/post/Does-Spearmans-rho-have-any-advantage-over-Kendalls-tau
11. https://ishanjainoffical.medium.com/understanding-kendalls-tau-rank-correlation-c959a7daea56
12. https://statisticaloddsandends.wordpress.com/2019/07/08/spearmans-rho-and-kendalls-tau/