Statistics Quest Part of Formula Quest

Rank correlation: Spearman and Kendall

Spearman's ρ and Kendall's τ-b measure monotone association using ranks, so they resist outliers and do not require a straight line.

r_s = \operatorname{Pearson}\big(\operatorname{rank}(x), \operatorname{rank}(y)\big)

Spearman's rank correlation (average ranks for ties)

$r_s$: Spearman's ρ: Pearson's r computed on the ranks
$\operatorname{rank}(x)$: ranks 1…n of x, tied values sharing their average rank

\tau_b = \frac{C - D}{\sqrt{(n_0-n_1)(n_0-n_2)}},\qquad n_0=\frac{n(n-1)}{2}

Kendall's τ-b

$C,\ D$: numbers of concordant and discordant pairs of observations
$n_0$: total number of pairs
$n_1,\ n_2$: pairs tied in x and pairs tied in y (both 0 without ties)

When to use

The relationship is monotone (always rising or always falling) but not necessarily linear.
Ordinal data (ratings, rankings) or data with outliers or strong skew.

Assumptions

Independent pairs. No distributional assumption for the coefficients themselves.
The engine tests Spearman's ρ with the t approximation $t = r_s\sqrt{(n-2)/(1-r_s^2)}$ (approximate for small n); Kendall's τ-b uses the exact null distribution when there are no ties and n is small, otherwise a tie-corrected normal approximation.

How to interpret

Both range from −1 to 1; ±1 means a perfectly monotone relationship.
Kendall's τ has a direct reading: $(C-D)/n_0$ (without ties) is P(a random pair is concordant) − P(discordant).
τ is usually smaller in size than ρ for the same data; do not compare one with the other.

Common mistakes

Treating rank correlation as a test of linearity; it ignores the spacing of values.
Comparing Spearman's ρ from one study with Pearson's r or Kendall's τ from another.
Forgetting that a monotone association is still not causation.

Example

Two judges rank five entries: x = 1, 2, 3, 4, 5 and y = 2, 1, 4, 3, 5.

Spearman: $r_s = 0.80$ (t = 2.31, 3 df, p = 0.104).
Kendall: 10 pairs, C = 8 concordant, D = 2 discordant, so $\tau_b = (8-2)/10 = 0.60$ , exact p = 0.233.
With only five entries, neither is clear evidence of agreement beyond chance.