Nonparametric Hypothesis Testing: Rank and Permutation Methods with Applications in R

6 Testing for correlation and concordance

6.1 Introduction

In this chapter, we study correlation and concordance. This section introduces the problems to be addressed and the method presented in the successive sections.

In Section 6.2 we are interested in the statistical relationship between the two components X and Y of the bivariate variable underlying the population of interest. The objective is to understand whether or not these variables are independent and in case they are not independent to assess the degree of dependency among them. The Pearson product moment correlation coefficient, that is the most familiar measure of correlation, is introduced. Its distribution under the null hypothesis that X and Y are independent depends on the distribution of the bivariate variable (X, Y). For this reason it is not suitable as the test statistic within a nonparametric framework for testing the hypothesis that X and Y are independent.

In Section 6.3 we consider nonparametric tests for independence. More precisely, in Section 6.3.1 we consider the nonparametric test based on the Spearman correlation coefficient and in Section 6.3.2 we consider the nonparametric test based on the Kendall correlation coefficient. Both tests are based on ranks and the null distributions of the corresponding test statistics do not depend on the distribution of (X, Y): the tests are distribution free. The only assumption is that X and Y are continuous. Applications to real life problems in finance, management ...