Kendall's Tau

What is Kendall's Tau?

Kendall's tau is a correlation coefficient and therefore measures the relationship between two variables.

In contrast to Pearson correlation, Kendall's rank correlation is a non-parametric test procedure. For Kendall's tau, the data do not need to be normally distributed and the two variables need only be ordinal.

The same is true for Spearman rank correlation. Kendall's tau is very similar to Spearman's rank correlation coefficient.

However, Kendall's tau should be preferred over Spearman's correlation when the dataset is small and there are many rank ties.

Calculate Kendall's Tau

We can calculate Kendall's Tau with this formula:

where C is the number of concordant pairs and D is the number of discordant pairs. But what are concordant and discordant pairs?

Kendall's tau example

Suppose two doctors rank 6 patients by physical health. We use the female doctor's ranking as the reference and sort the patients from 1 to 6.

Now we compare the reference ranks with the second doctor's ranks. For example, the patient who is ranked 3 by the female doctor is ranked 4 by the male doctor.

We want to know whether there is a correlation between the two assessments using Kendall's tau. To calculate it, we use the ranks on the right-hand side, i.e. the ones from the male doctor.

We now look at each rank and note whether the values below it are smaller or larger than it.

As shown in the figure above, we start with the first rank, corresponding to the value 3. 1 is smaller than 3, so it gets a minus, 4 is larger, so it gets a plus, 2 is smaller, so it gets a minus, 6 is larger, so it gets a plus, and 5 is also larger, so it also gets a plus.

We now do the same for the second rank, corresponding to the value 1. Of course, each subsequent rank has a greater value than 1, so we have a plus everywhere.

For the rank with value 4, 2 is smaller and 6 and 5 are larger. We proceed in the same way for the ranks with value 2 and 6.

We can now calculate the number of concordant and discordant pairs. We get the number of concordant pairs by counting all pluses. In our example, we have a total of 11.

We get the number of discordant pairs by counting all minuses. In our example, we have a total of 4.

C is 11 and D is 4, so Kendall's tau is (11 - 4) / (11 + 4), resulting in a value of 0.47.

An alternate formula for Kendall's tau uses S = C-D and n is the number of cases. Here, S is 7 and n is 6.

Substituting gives 7/15.

Kendall's tau significance

For Kendall's tau, the null and alternative hypotheses are:

• Null hypothesis: the correlation coefficient tau = 0 (there is no correlation).
• Alternative hypothesis: the correlation coefficient tau ≠ 0 (there is a correlation).

Now we want to know if the correlation coefficient is significantly different from zero. You can determine this either by hand or with software like numiqo.

For a hand calculation, we can use the z-distribution as an approximation. However, we should have more than 40 cases. The 6 cases from our example are too few. We get the z-value using this formula:

Calculate Kendall's tau with numiqo

If you want to calculate Kendall's tau online with numiqo, copy your own data into the table in the Kendall's tau calculator and click on correlation.

Then select the variables for which you want to calculate Kendall's tau. Now all you have to do is click on Kendall's tau and you're done.

If you are not sure how to interpret the results, click on Summary in words.