Spearman's rho or no? is one of my datasets non-monotonic?

#1
Hi, I'm dyslexic with limited stat education (and the internet is not helping thus far) and I'm having horrible trouble understanding what tests I can and cannot use for potentially non-monotonic data (brain, meet brick wall...). This is for my dissertation (social sciences), and I'm questioning a Spearman's rho I already ran in SPSS, following the example from a published paper which used the exact same variables for the same purpose. (Three sets of variables are universal constants, and this paper and I are comparing these constants to our own observed data). This other paper does not produce their data in a scatterplot nor do they confirm that it is all monotonic, which, as far as I'm aware, is an assumption of the Spearman's rho. (What I'm not sure about is, if assumptions are also rules, meaning results cannot be trusted unless assumptions are true).

The observed data is interval, but the constants can be considered ordinal, as they are ranked, standardized constants.

Problem
My data, I've now realized, is only partially monotonic... I think. What I'm doing is comparing constants A, B, and C to my observed data, D, and within D there are four sets of observed data. Each comparison has four scatterplots representing four sets of observed data (total of 12 plots). In scatterplots, A-D and B-D show monotonic relationships for all four sets of observed data (no, I have not tested this in any way - just visual plots). Unfortunately, scatterplots for relationship C-D show three out of the four plots visually looking non-monotonic, like this.



Worryingly, several of the C-D relationships from the SPSS Spearman's rho report significant correlations (as do some other comparisons between datasets). Now I'm not sure that these results are valid.

What I need to know is this - if some datasets are visually monotonic, and some are not, how does this affect the Spearman's rho as a whole? Of course, constant C is removable from the data table without any effect to the other constants/results. It's not necessary to test C in order to test A and B. For the purpose of my research, however, it's fairly important to somehow make assumptions based on some sort of observable evidence (Is there a relationship between constant C and my observed data?)

So, if I can't use Spearman's rho, what is left?
I'm fairly certain that I could run individual nonlinear correlation tests, but is there a simpler way?

Thanks so much. I hope this is at least somewhat understandable.
 
#2
You cannot use Spearman's rho for non-monotonic relationships. To capture the wiggly curve that you have illustrated you have to

1) build a parametric model (using polynomials or splines, for example)

or

2) exploit a non-parametric estimator (like a kernel smoother).