# Difference between Spearman and Kendall-Tau correlation test

My question is not about the definition of the two rank correlation methods, but it is a more practical question: I have two variables, X and Y, and I calculate the rank correlation coefficient with the two approaches. With the Kendall-tau-b (which accounts for ties) I get tau = 0 and p-value = 1; with Spearman I get rho = -0.13 and p-value = 0.44.

Why does Kendall tell me there is exactly no correlation, while Spearman does see a nonzero correlation?

• Spearman's rho and Kendall's tau are calculated in different ways. Why do you believe that something is going wrong with the Kendall tau calculation (and not the Spearman one)? Oct 25, 2017 at 17:09
• Because I get a P-value of 1... Is not that suspicious? Oct 25, 2017 at 18:42
• $p=1$ is a direct consequence of $\tau_b=0$, so the question is why $\tau_b=0$ and $\rho= -.13$ is a problem for you. Oct 25, 2017 at 18:46
• Then, I re-formulate my question: why according the Kendall-tau there is an exact no-correlation, while Spearman is less strict on that? Oct 25, 2017 at 19:52
• Thank you. That clarification makes for a much better question. I took the liberty of editing it into your post (and upvoting) and answering. Hope my answer is helpful. Oct 25, 2017 at 20:43

Spearman's $\rho$ and Kendall's $\tau$ are calculated differently. That is, they have different notions of "correlation". (As does Pearson's $r$.) Thus, they will output different correlation coefficients.

I don't see anything surprising about having one type of correlation zero and one nonzero.

Here is a fun little dataset with $\rho<0$, $r=0$ and $\tau>0$:

> xx <- 1:6
> yy <- c(-0.2,2.5,1,2,3.5,-1)
> plot(xx,yy,pch=19)
> cor(xx,yy,method="spearman")
 -0.02857143
> cor(xx,yy,method="kendall")
 0.06666667
> cor(xx,yy,method="pearson")
 1.748437e-18 So different measures of correlation will output different results. Your next question probably is which coefficient you should use. Happily enough, we already have a question on this: Kendall Tau or Spearman's rho?