Kendall’s Tau-a

Kendall’s Tau-a is a non-parametric correlation which is an alternative to the Spearman correlation. The advantage of Kendall’s tau-a is that it’s got a more straightforward interpretation (unlike Spearman’s) and what is more Kendall’s tau-a is a very simple correlation to calculate (as long as there aren’t any ties). It doesn’t take much maths, but it is a bit arduous.

We are only going to look at how to calculate the Kendall correlation when there are no ties, because, frankly, it makes our life far too complicated if there are ties.

The basic idea of the Kendall correlation is easy, although it is so arduous that we are going to do it on a cut-down version of Roney’s dataset.

The first stage is to sort the data into order, and then rank them, as before.

Table 1: Short version of Roney, et al, data, ranked and sorted.

If the relationship were perfect and positive, then we would expect that the person who had the lowest score for change in testosterone would also have the lowest score for display. For each person who had a display score that was lower than that person’s change score, the worse the correlation would be. The same is true for the second person, and the third person, and so on, for every possible pair of people. To calculate the Kendall tau-a correlation, all we do is count those which are concordant with the theory, and those which are discordant with the theory.

So, let’s do it.

We have already put the variables in order, in Table 1. This isn’t strictly necessary, but it makes our life easier.

Taking the first person, who is ranked 1 for change in testosterone, how many people are ranked above that person for display? These are concordant – and the answer is 7 people, so C = 7. The number of discordant people, who are ranked above, is zero, so D = 0.

Take the second person. 6 people are ranked above that person, and they are concordant, so C = 6, and 1 person (the person ranked 4^th in display) is equal, so they are discordant, D = 1.

We keep doing this for each person, but we can make our lives easier by putting this into a table, which is shown in Table 2. For each pair of people, we say whether the scores are concordant, in which case we give them a C, or discordant, in which case we give them a D.

Table 2: Calculating the Kendall tau-a Coefficient

We count the total number of Cs, and find there are 21. We count the number of Ds, and find there are 7. Kendall’s tau-a is given by the following formula:

Where C and D represent the number of Cs and Ds.

In our data:

It is possible to calculate the probability value associated with Kendall’s tau-a, but it’s rather tricky, and it is especially tricky when there are ties. However, the probability value will be the same as the probability for the Spearman correlation. Calculation of the confidence intervals is also possible, but not easy. If you really want to know, it’s covered in .

The Kendall tau-a correlation has a simpler interpretation than the Spearman correlation. The Kendall tau-a correlation represents the difference between two probabilities – let’s say that you and I are raters, who are rating some object on some characteristic. I say that A has a higher score than B. The tau-a correlation is the probability that you will say that they are in the same order minus the probability that you will say that they are in the opposite order. If we are in complete agreement, this will be 1 – 0 = 1. If we are in complete disagreement, this will be 0 – 1 = -1, and if we are both random, this will be 0.5 – 0.5 = 0.

Despite its simpler interpretation, the Kendall tau-a correlation tends to be used much less than the Spearman correlation. We think there are two reasons for this. First, the correlation is much harder to calculate. Nowadays, almost all statistical analysis is done with the help of a computer, so we don’t really care about this. (Although it causes a historical legacy – people see Spearman’s correlation used, so they use it.) Second, and perhaps more importantly, although the p-values are the same, the correlations are lower. In our example dataset, the Kendall correlation was 0.50, the Spearman correlation was 0.64. If you want to get your research published (and if you don’t get it published, there wasn’t much point doing it) a higher correlation looks better, and researchers might perceive has a better chance of getting into print.