24.3 Kendall’s \(\tau\)

Depends on the order not the value of the data

Concordant:
When one pair dominates the other given \((x_1, y_1)\) and \((x_2, y_2)\)

• \(x_1 > x_2\) and \(y_1 > y_2\) or,

• \(x_2 > x_1\) and \(y_2 > y_1\)

Discordant:
When the pair is mixed

Kendall’s \(\tau\)

\[\begin{equation} \tau = \dfrac{C - D}{\text{# of pairs}} \tag{24.2} \end{equation}\]

• \(\tau = \dfrac{C - D}{C + D}\) if there are no ties

• \(\tau \in [-1, 1]\) with same interpretation as Pearson’s

Focus on the rank \(\Rightarrow\) changes in one extreme value won’t change the indication

Continuous Kendall’s \(\tau\)

\[\begin{equation} \tau = 4 \mathrm{E}[C(u,v)] - 1 \tag{24.3} \end{equation}\] \[\begin{equation} \tau = -1 + 4 \int_0^1 \int_0^1 C(u,v)c(u,v)dudv \tag{24.4} \end{equation}\]

Definition 24.1 (Correlation vs Dependency)

• Correlation = mathematical calculation, a statistic, and measure of dependency

• Dependency = interaction between random variables (broader term)

• e.g. correlation can be 0 while there are still dependency