Kendall's Tau Correlation

Use this test when you need a robust rank-based correlation, especially with small sample sizes, many tied observations, or when you want a coefficient with a clear probabilistic interpretation. For example, assessing agreement between two ordinal rating scales with few distinct levels.

• Both variables are at least ordinal.
• Observations are independent.

• Tau has a direct probabilistic interpretation: it equals the probability that two randomly chosen observations are in the same order on both variables, minus the probability they are in opposite order.
• Kendall tau values tend to be smaller in magnitude than Pearson r or Spearman rho for the same data. Do not compare tau values directly to r or rho values.
• Tau-b (the default variant) adjusts for tied pairs, making it appropriate for ordinal data with repeated values.
• A tau of 0 means the concordant and discordant pairs are balanced: no monotonic trend.

• Kendall's tau is computationally more expensive than Pearson or Spearman for large datasets (O(N^2) versus O(N log N) for Spearman).
• The magnitude of tau is not directly comparable to Pearson r. A Pearson r of 0.7 does not correspond to a Kendall tau of 0.7.
• With very few observations, the p-value may not be meaningful because the distribution of tau is discrete.

1. Consider every possible pair of observations (N choose 2 pairs).
2. A pair is concordant if both variables increase (or both decrease) from one observation to the other. It is discordant if one increases while the other decreases.
3. Compute tau = (concordant pairs - discordant pairs) / total pairs, with an adjustment for ties (tau-b).
4. Standardise using the expected variance under the null hypothesis to compute the Z-statistic and p-value.