Which correlation should be used for non-normal data: Spearman's rho versus Kendall's tau versus Kendal's tau-b?

I'm trying to see if there is a correlation between the height of grass and the height under branches available for grass to grow. I have 227 paired observations:

GrassHeight HeightUnderDebris
0            0
0            0
0            0
8            16
0            0
0            0
0            0
2            2
6            6
0            0
0            0
1            1
0            0
0            0
0            0
8            15
0            0
7            7
15           15


My data is not normally distributed and it fails at the assumption of bivariate normality:

result<-hzTest(data,cov = TRUE,qqplot = FALSE)
result<-mardiaTest(data,cov = TRUE,qqplot = FALSE)
result<-roystonTest(data,qqplot = FALSE)


Therefore, I need to use a Spearman's rho or Kendall's tau. Firstly, Spearman's rho results in a warning message:

cor.test(GrassHeight, HeightUnderDebris, method="spearman")

Spearman's rank correlation rho

data:  GrassHeight and HeightUnderDebris
S = 123090, p-value < 2.2e-16
alternative hypothesis: true rho is not equal to 0
sample estimates:
rho
0.9368622

Warning message:
In cor.test.default(GrassHeight, HeightUnderDebris, method = "spearman") :
Cannot compute exact p-value with ties


So I then decided to use Kendall's tau as it can deal with ties:

cor.test(GrassHeight, HeightUnderDebris, method="kendall")

Kendall's rank correlation tau

data:  GrassHeight and HeightUnderDebris
z = 17.202, p-value < 2.2e-16
alternative hypothesis: true tau is not equal to 0
sample estimates:
tau
0.858494


Firstly, should I be concerned that my data has many zeros? They are important as they reflect that if there is no space under branches, then there is no space for grass growth hence why the grass height is 0.

Secondly, how would you interpret Kendall's results? Is it right that the two variables are uncorrelated at 0.05 significance level if their correlation coefficient is zero? In this case, tau is 0.858. That is not zero and will be rounded up to 1. Can I say that the two variables are correlated based on this?

Should I rather look at rpudplus and the function rpucor, which now uses Kendall’s tau-b to compute the correlation coefficient?

What post-hoc test can I do to find out the nature of the correlation, i.e: as the height between the ground and branch increases, grass height increases?

Pearson's correlation doesn't assume normality, so you should use it. You really don't need Kendall's tau in your example.

In your analysis, you should start off with a simple plot. Like this:

grass  <- c(0,0,0,8,0,0,0,2,6,0,0,1,0,0,0,8,0,7,15)
height <- c(0,0,0,16,0,0,0,2,6,0,0,1,0,0,0,15,0,7,15)
plot(grass, height)


This is clearly linear and monotonic positive, and so it was pointless for you to do a normality test.

Both Pearson and Spearman give similar results:

cor(grass, height) # 0.9300721
cor(grass, height, method='spearman') # 0.9947245


In this example, it's not really not that important to do a correlation test because you know the p-value will be very small and your result will be significant. But let's do it anyway:

cor(grass, height) # 8.227e-09
cor.test(grass, height, method='spearman', exact=FALSE) # 2.2e-16


In the first line, we did a test for Pearson's correlation. Your very small p-value give you confidence that your correlation is not zero (not very surprising).

In the second line, we have a test for Spearman's correlation. Note that your zeros made the default exact test impossible, so you'll need to set exact to FALSE to approximate your test statistic with t approximation. Again, this is very significant.

• what should have normal dist are the residues Oct 19, 2017 at 18:49

Spearman’s rho and Kendall’s tau can both be used for non-parametric data, as they are both measures of rank correlation. One key difference: Spearman’s rho can be used for both continuous AND discrete variables, while Kendall’s tau should only be used for discrete variables. Spearman’s rho takes into account how far the ranks are from each other, while Kendall’s tau only considers whether they are equal or not.

In your case, your data are discrete and either is thus appropriate. However, if you were to measure the heights as continuous variables (e.g. with decimal points) then Spearman’s rho would be the preferred measure.

I would not be concerned about the zeroes. As you say, they reflect a real consideration for your study.

Interpretation of Spearman’s rho and Kendall’s tau are similar. They can range from -1 to +1, with -1 reflecting a perfect negative correlation (as one variable increases, the other decreases), 0 reflecting no relationship between the variables and +1 reflecting a perfect positive correlation (as one variable increases, the other will also increase). In both instances, your values are close to 1 and it is thus safe to say that the data are strongly positively correlated, i.e. as the available height under branches increases, the height of the grass can be expected to increase. This is easily interpreted from the p values. In both cases, p < 0.05 and the relationship is thus significant. Also in both cases, the rho/tau values are positive, hence a significant, positive correlation is evident. This is usually the case, i.e. rho and tau generally lead to similar inferences.

In this case, I don’t think it would be necessary to use tau-b for the calculation. You are unlikely to find manifest differences that would influence your interpretation.

There is no need for a post-hoc test to determine the nature of the correlation. As stated above, the rho or tau value provides you with this information. Closer to -1 indicates strongly negative correlation, closer to 0 indicates weak/no correlation, closer to +1 indicates strongly positive correlation.