Non-parametric methods for the study of the correlation: Spearman's rank correlation coefficient and Kendall tau rank correlation coefficient

We saw in the previous post, how to study the correlation between variables that follow a Gaussian distribution with the Pearson product-moment correlation coefficient. If it is not possible to assume that the values follow gaussian distributions, we have two non-parametric methods: the Spearman's rho test and Kendall's tau test.

For example, you want to study the productivity of various types of machinery and the satisfaction of operators in their use (as with a number from 1 to 10). These are the values:

Productivity: 5, 7, 9, 9, 8, 6, 4, 8, 7, 7
Satisfaction: 6, 7, 4, 4, 8, 7, 3, 9, 5, 8

Begin to use first the Spearman's rank correlation coefficient:

a <- c(5, 7, 9, 9, 8, 6, 4, 8, 7, 7)
b <- c(6, 7, 4, 4, 8, 7, 3, 9, 5, 8)

cor.test(a, b, method="spearman")

        Spearman's rank correlation rho

data:  a and b 
S = 145.9805, p-value = 0.7512
alternative hypothesis: true rho is not equal to 0 
sample estimates:
      rho 
0.1152698 

The statistical test gives us as a result rho = 0.115, which indicates a low correlation (not parametric) between the two sets of values.
The p-value > 0.05 allows us to accept the value of rho calculated, being statistically significant.

Now we check the same data with the Kendall tau rank correlation coefficient:

a <- c(5, 7, 9, 9, 8, 6, 4, 8, 7, 7)
b <- c(6, 7, 4, 4, 8, 7, 3, 9, 5, 8)
 
cor.test(a, b, method="kendall")

        Kendall's rank correlation tau

data:  a and b 
z = 0.5555, p-value = 0.5786
alternative hypothesis: true tau is not equal to 0 
sample estimates:
     tau 
0.146385

Even with the Kendall test, the correlation is very low (tau = 0.146), and significant (p-value > 0.05).