Chapter 12 Kendall’s Tau

Kendall’s Tau \(\tau\) is a measure of association between X and Y based on concordance.

12.1 How it Works

Concordance suggests that observations are consistent with one another; that they are positively associated.

More formally, we say that a pair of points \(\left(X_{i}, Y_{i}\right)\) and \(\left(X_{j}, Y_{j}\right)\) are:

Concordant if \(X_{i}<X_{j} \Rightarrow Y_{i}<Y_{j}\), or \(\left(X_{i}-X_{j}\right)\left(Y_{i}-Y_{j}\right)>0\)
- If pairs are more likely to be concordant \(\implies\) positive association
Discordant if \(X_{i}<X_{j} \Rightarrow Y_{i}>Y_{j}\), or \(\left(X_{i}-X_{j}\right)\left(Y_{i}-Y_{j}\right)<0\)
- If pairs are more likely to be discordant \(\implies\) negative association

We define Kendall’s Tau as: \[ \begin{aligned} r_{\tau} &= \frac{(\text{# concordant pairs})-(\text{# discordant pairs})}{\binom{n}{2}} \end{aligned} \]

Why use \(\binom{n}{2}\) instead of \(n\)?

\[ \begin{array}{l} p_{\text {lower tail}}=\frac{\# r_{\tau}^{*} \leq r_{\tau, obs}}{n !} \\ p_{\text {upper tail}}=\frac{\# r_{\tau}^{*} \geq r_{\tau, obs}}{n !} \\ p_{\text {two sided}}=\frac{\# |r_{\tau}^{*}| \geq |r_{\tau, obs}|}{n !} \end{array} \]

12.2 Limitations

Both look for a monotonic trend only, neither can detect a parabolic trend (\(r_s=r_{\tau}=0\))
We must have independent paired observations (can’t use time series b/c dependence)