Correlation is frequently applied, but in fact often the interest of the experimentalist is not so much to find a significant correlation as a significant dependence of the variables, and clearly independence is not the same as lack of correlation.
To give a particular example, consider two genes and their expression across a number of individuals. If genes are co-regulated (and therefore possibly functionally related), their expression values will not be independent. Whether they will be correlated, however, is another matter.
Now, there are tests out there that can be seen as an alternative to a test for a significant correlation -- test of variable independence for continuous variables, such as the Longest Increasing Subsequence Independence Test (R package LIStest) or a non-parametric test described here.
I have very little background and no experience with these tests. My question is: are these tests comparable in power to correlation tests? Do you have any practical experience? What would you recommend?