Kendall’s tau correlation is another non-parametric correlation coefficient which is defined as follows.
Let x1, …, xn be a sample for random variable x and let y1, …, yn be a sample for random variable y of the same size n. There are C(n, 2) possible ways of selecting distinct pairs (xi, yi) and (xj, yj). For any such assignment of pairs, define each pair as concordant, discordant or neither as follows:
- concordant if (xi > xj and yi > yj) or (xi < xj and yi < yj)
- discordant if (xi > xj and yi < yj) or (xi < xj and yi > yj)
- neither if xi = xj or yi = yj (i.e. ties are not counted).
Now let C = the number of concordant pairs and D = the number of discordant pairs. Then define tau as
We can use Kendall’s tau for hypothesis testing even when x and y are not binormally distributed and even when there are outliers. Click on the following topics for more details:
- Basic concepts
- Hypothesis testing using table of critical values
- Hypothesis testing using the normal approximation
- Hypothesis testing with ties