Kendall’s Tau

kendall

correlation

ranks

concordance

non-parametric

Rank correlation based on concordant and discordant pairs; robust to small samples and tied ranks

Published

April 17, 2026

Research question

Kendall’s tau (\(\tau\)) is a rank-correlation coefficient based on the difference between concordant and discordant pairs. It is preferred over Spearman’s \(\rho\) in small samples (say \(n < 20\)) and when many ties are present. Biomedical example: in a small diagnostic pilot with 18 patients, is a radiology reader’s ordered confidence rating (1-5) associated with the final pathology grade (G1-G4)?

Assumptions

Assumption	How to verify in R
Both variables at least ordinal	scale
Independent pairs	design

Kendall makes no assumption of linearity or normality.

Hypotheses

\[H_0: \tau = 0 \qquad H_1: \tau \ne 0\]

R code

library(tidyverse); library(rstatix); library(DescTools)
set.seed(42)

# 18 patients: reader confidence (1-5) vs pathology grade (1-4)
pilot <- tibble(
  confidence = sample(1:5, 18, replace = TRUE),
  grade      = NA_integer_
) |>
  mutate(grade = pmin(4L, pmax(1L, confidence + sample(-1:1, 18, replace = TRUE))))

cor.test(pilot$confidence, pilot$grade, method = "kendall")

pilot |> cor_test(confidence, grade, method = "kendall")

# Kendall's tau-b (adjusts for ties)
DescTools::KendallTauB(pilot$confidence, pilot$grade)

Interpreting the output

Kendall’s \(\tau_b = 0.64\), \(z = 3.35\), \(p < .001\). The reader’s confidence rating is strongly associated with the pathological grade. The tau-b variant corrects for ties and is recommended when the data contain repeated values.

Effect size

Kendall’s tau is its own effect size. Cohen’s thresholds (adapted): small 0.10, medium 0.30, large 0.50. Tau values are systematically smaller than Spearman’s rho on the same data.

Reporting (APA 7)

Reader confidence was positively associated with the pathological grade (Kendall’s tau-b = .64, z = 3.35, p < .001).

Common pitfalls

Reporting Spearman’s rho and Kendall’s tau as interchangeable. Tau is smaller on average and has a different interpretation (probability of concordance minus probability of discordance).
Ignoring ties: Kendall’s tau-a treats each tie as a pair counted neither concordant nor discordant; tau-b corrects for this.
Using Kendall in large samples without reason; Pearson or Spearman are more common in reporting and equally valid under their own assumptions.

Parametric vs. non-parametric alternative

Linear, bivariate normal: Pearson correlation.
Monotonic, moderate sample: Spearman rank correlation.