Correlation/association hypothesis test

A hypothesis test formally tests if there is correlation/association between two variables in a population.

The hypotheses to test depends on the type of association:

• For a product-moment correlation, the null hypothesis states that the population correlation coefficient is equal to a hypothesized value (usually 0 indicating no linear correlation), against the alternative hypothesis that it is not equal (or less than, or greater than) the hypothesized value.
• For rank correlation methods, the hypotheses are restricted and more general. The null hypothesis states the variables are independent, against the alternative hypothesis that there is an association, such as a monotonic function.

When the test p-value is small, you can reject the null hypothesis and conclude that the population correlation coefficient is not equal to the hypothesized value, or for rank correlation that the variables are not independent. It is important to remember that a statistically significant test may not have any practical importance if the correlation coefficient is very small.

Pearson's and Kendall's tests are preferred as both have associated estimators of the population correlation coefficient (rho and tau respectively). Although the Spearman test is popular due to the ease of computation, the Spearman correlation coefficient is a measure of the linear association between the ranks of the variables and not the measure of association linked with the Spearman test.