If you want a summary statistic giving you some information about the dependency between the two data sets then Pearson's linear correlation coefficient or a measure of rank correlation meet that requirement.
The former only quantifies a particular form of dependency. namely linear dependency. A value of zero only implies independent if there is no non-linear dependency present. This would be true if the two are related by an elliptical dependency structure (e.g. Gaussian/Normal).
So, Pearson's correlation coefficient is limited in its application.
Rank correlation coefficients measure broader forms of dependency (i.e. non-linear). They are also a single summary statistic quantifying the degree of dependency - they give no information about the structure of that relationship.
If you want more information about the dependency structure then one way of describing that structure is by using a copula.
So, maybe we should be careful about using the word correlation? It is generally assumed that it means Pearson's linear correlation! It is perhaps safer to use the term dependency when meaning broader forms of inter-relationship.
Last edited: Aug 31, 2012
