# Methods For Measuring Non-Linear Correlation?

I have been learning about standard methods in Statistics such as the Pearson's Correlation Coefficient, Spearman's Correlation and Kendall's Tau.

My understanding of this so far is that:

• Pearson Correlation Coefficient measures the linear correlation between two sets of data

• Spearman's Correlation measures the "monocity" between two sets of data (e.g. do they both increase and decrease at the same time?)

• Kendall's Tau measures the ordinal association between two sets of data - supposedly Kendall's Tau is similar to the Spearman Correlation, but Kendall's Tau has a more logical confidence intervals.

I had the following question - can any of these methods be used for measuring a specific form of "Non Linear Correlation" between two sets of data?

For example - suppose I want to see how strongly two sets of data are correlated relative to a "second order curve" :

Is there something that could measure the "curved correlation"?

The two ideas I came up with:

• Try to use some data transformations (e.g. Log) to transform one of the variables into a more linear pattern that will make it suitable for one of the above measures

• Fit a polynomial regression model (of order 2) to this data and measure the MSE

But I am not sure if either of these approaches are suitable.

• Interesting question. Some of the trouble of defining a curved correlation will be deciding on what kind of curvature you want to measure. After all, a logarithm-type of graph has different curvature than a quadratic. Further, determining the sign will be challenging, since many curves (such as quadratics) allow for increasing and decreasing sections. I’ve wondered if the concavity of a parabola (up-opening vs down-opening) could be used for this, but parabolas are just one type of curve. (Maybe you can do this if you restrict to convex or concave functions.)
– Dave
Nov 9, 2022 at 6:59
• (1) What do you mean by "measuring"? If you want a measure of the "strength" of such a correlation, then you could indeed run a polynomial regression and report the MSE. Possibly cross-validated, otherwise if you re-ran this for higher order polynomials, you would "find" that the "second-order correlation" is smaller than the "third-order correlation" and so on. Conversely, if you want to do statistical inference, the null and alternative hypotheses will need some thinking about - are $x$ and $x^3$ for $-1<x<1$ "significantly second order correlated"? ... Nov 9, 2022 at 7:41
• ... (2) Especially for inference, the question comes up whether you want to test a specific polynomial correlation, or a general second-order polynomial, or a general polynomial of up to second order. Perhaps you could explain what you want to do with such a nonlinear correlation? Nov 9, 2022 at 7:42
• Another way to consider Stephan's comments is that every regression you could estimate for the two variables in your plot is, in a sense, a correlation measurement. Testing and comparing arbitrarily many regressions has problems with false discovery and statistical validity, so "just try stuff" isn't a great way to go about it: you need to be specific about what questions you want to ask your data & how you want to ask it. The plot you show is roughly monotonic & Spearman's correlation would characterize the extent. Lots of nonlinear functions are monotonic, so Spearman's is an answer.
– Sycorax
Nov 10, 2022 at 3:28