I have a health outcome (measured as a rate of cases per 10,000 people in an administrative zone) that I'd like to associate with 15 independent variables (social, economic, and environmental measures of those same administrative zones) through some kind of model (I'm thinking a Poisson GLM or negative binomial if there's overdispersion). From scatterplot investigation I know there is multicollinearity among the variables that I need to investigate and deal with. I am not against removing variables, and within any problematic combinations can justify choosing one variable over the other(s) for reasons of interest or cost/ease of collection.
In the past I used the correlation matrix to detect multicollinearity, but I've been reading around on this site and discovered VIF and the condition index/number, which seem to be generally accepted as better options. My question is how any of these measures work with non-linear correlations, which is what I have between nearly all of my variables (determined graphically). What are my options for evaluating multicollinearity besides Spearman correlations?
Aside: I realize this is probably a separate question, but in case it's relevant here, I also have a lot of non-monotonic correlations among my variables that I don't know what to do with...
Thanks for any help!