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*).

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