Correlation vs. independence: statistical power, available tests 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?
 A: If you expect a linear relationship use a linear correlation. If you do not then do not use one. 
Independence tests are for any kind of relationship. If you have no idea what to expect those are OK. You might also look into GAM (generalized additive modelling) or polynomial regression.
You may anticipate a specific nonlinear relationship where the variables could be converted so that it is then linear (e.g. exponential decay). In that case model the relationship (better) or correlated the converted data.
Among these, the most powerful test will be the most appropriate test. A linear correlation is not going to be more powerful than an independence test when you have a curvilinear relationship. A specific model that accurately represents the data is going to be more powerful than a method that can model any relationship. That said, it is possible that a very flexible modelling method can be equally as powerful as a constrained one, but it won't be appreciably greater. And it would need to be greater to justify the increased complexity.
