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A measure of concordance between two random variables based on ranks.

Kendall's tau is a measure of concordance for two random variables. It is based on ranks, and has many properties in common with Spearman's rho.

We say that $(y_{i,1},y_{i,2})$ and $(y_{j,1},y_{j,2})$ are concordant if:

$$(y_{i,1} - y_{j,1}) \times (y_{i,2} - y_{j,2}) >0$$

And discordant if the product is $< 0$. For a given dataset, let $c$ = # of concordant observations and $d$ = # of discordant observations. Then:

$$\hat{\tau} = \frac{c-d}{n \choose 2}$$

If pairs are tied (i.e. $y_{i,1} = y_{j,1}$), then $\hat{\tau}$ is not bound by -1 and +1. There are different approaches to handling ties.