Kendall’s Tau

Rank correlation based on concordant vs discordant pairs; robust to ties and small samples

Introduction

Kendall’s \(\tau\) quantifies association via the proportion of pairs of observations that are concordant (both variables increase or both decrease) minus the proportion that are discordant. It is more robust than Spearman in small samples and under ties.

Prerequisites

Ranks, concordant / discordant pairs.

Theory

For \(n\) observations, there are \(\binom{n}{2}\) pairs. A pair \((i, j)\) is concordant if \(\mathrm{sign}(X_i - X_j) = \mathrm{sign}(Y_i - Y_j)\), discordant if the signs differ, and tied otherwise.

\[\tau = \frac{\text{concordant} - \text{discordant}}{\binom{n}{2}}.\]

Variants (for ties):

• tau-a: simple difference divided by all pairs (including ties).
• tau-b: corrects for tied pairs in both variables; most commonly used with tied data.
• tau-c (Stuart’s tau-c): for different-sized marginal distributions.

\(\tau \in [-1, 1]\); 0 indicates no association. The null distribution (no association) is asymptotically normal; exact distributions exist for small \(n\).

Assumptions

Same as Spearman: ordinal or continuous data, monotonic relationship, independent pairs.

R Implementation

library(DescTools)
set.seed(2026)

n <- 40
x <- rnorm(n)
y <- 0.6 * x + rnorm(n, sd = sqrt(1 - 0.6^2))

cor.test(x, y, method = "kendall")

# Spearman for comparison
cor.test(x, y, method = "spearman")

# Tau-b (explicit)
KendallTauB(x, y, conf.level = 0.95)

# With ties
x2 <- round(x, 1)
y2 <- round(y, 1)
cor.test(x2, y2, method = "kendall")

Output & Results

        Kendall's rank correlation tau

data:  x and y
z = 3.68, p-value = 0.0002
alternative hypothesis: true tau is not equal to 0
tau: 0.394

        Spearman's rho: 0.57

Both correlations are significantly positive; Kendall’s \(\tau\) is numerically smaller than Spearman (typical, as they measure different things).

Interpretation

Report: “Kendall’s \(\tau = 0.39\) (z = 3.68, p < 0.001), indicating a moderate positive monotonic association between \(x\) and \(y\).”

Practical Tips

• Kendall and Spearman measure different quantities; do not treat them as interchangeable.
• Kendall is more robust in small samples; Spearman is more common in reporting.
• For heavily tied data (e.g., Likert scales), prefer Kendall tau-b over Spearman.
• DescTools::KendallTauB() provides the tau-b with a CI, which base R’s cor.test does not.
• Kendall tau has a direct interpretation as (probability of concordance - probability of discordance), which Spearman does not.