Statistics: Kendall’s Tau(Simply Explained!)
What is Kendall’s Tau? Kendall’s Tau is a correlation coefficient which means that it measures the relationship between two variables. But what is the difference between Pearson’s correlation and Kendall’s rank correlation?
Unlike Pearson’s, Kendall’s rank correlation is a non-parametric test procedure. Meaning that in order to calculate Kendall’s Tau, the data needs to be normally distributed and the two variables have to have ordinal scale levels. This is also true for the Spearmen rank correlation! Kendall’s Tau is very similar to Spearmen’s rank correlation coefficient but Kendall’s Tau should be preformed over Spearmen’s correlation if very few data with many rank ties is available.
But how is Kendall’s Tau calculated? Kendall’s Tau is calculated with the following formula:
In the formula above, “C” is the number of concordant pairs and “D” is the number of discordant pairs. But what are concordant and discordant pairs?
Lets say we have 2 doctors who we ask to rank 6 patients according to their physical health. One of the doctors is defined as a reference and the patients are sorted from one through 6. Now the sorted ranks are matches with the ranks of the other doctor. For example, who is in 3rd place with the reference doctor is in 4th place with the other doctor.
Now using Kendall’s Tau we want to know if there is a correlation between the two rankings.
For the correlation of Kendall's Tau we only need the other doctors ranks(the doctor to the right). We now have to look at each individual rank and note which values are smaller or greater than its self.
So lets start at the first rank 3 and 1 the number below it is smaller so it gets a minus. 4 is greater so it gets a plus. 2 is smaller so it gets a minus. 6 is greater so it gets a plus. 5 is also greater so it also gets a plus.
Now we do that same thing for 1(compare all the numbers to it). Here every subsequent rank is greater than 1 so we have a plus everywhere. At rank 4, 2 is smaller and 5 and 6 are greater. Now we do this for rank 2 and rank 6.
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Now we can easily calculate the number of concordant and discordant pairs. We get the number of concordant pairs by counting all the plus signs and the number of discordant pairs by counting all the minus signs. In our example we would have a total of 11 plus signs and 4 discordant pairs.
This means that C is 11 and D is 4. When we plug these numbers into the equation we get the Kendall’s Tau to be a 0.47.
Here is the alternative formula for Kendall’s Tau where S = C — D and n is the number of cases:
Using this formula we get 0.47 as well. Just like the Pearson’s correlation coefficient r, Kendall’s Tau also varies between -1 and 1. With the help of the coefficient we can determine two things.
- How strong the correlation is
- The direction of the correlation
If the coefficient is between -1 and 0 the correlation is negative otherwise if it is between 0 and 1 its positive. If the result is exactly 0 there is no correlation.
Many times when we start from a sample we want to test a hypothesis about a population. We calculated the correlation coefficient from the sample data. Now we can test if the correlation coefficient is significantly different from 0. The null then must be that the correlation coefficient Tau equals 0 and the alternative is that it doesn’t. Now we want to know if it is significantly different from 0.
For the calculation by hand we can use z-distribution as an approximation. However for this we should have at least 40 cases.
