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Let’s say I want to measure the correlation between two variables, both measured on a ratio scale. Let’s say however that the relationship between the variables isn’t linear (so I can’t calculate Pearson’s r because one of the assumptions is linearity). Let’s also say that the relationship between the variables isn’t monotonic (so I can’t calculate Spearman’s rho because one of the assumptions is linearity). The only other option I know of is Kendall’s tau… but it seems to say online that the variables having a monotonic relationship is an assumption… so it seems I can’t run this test either. So is there some other option? Thanks, FBH

FastBallooningHead
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    Other option for what, exactly? You give us a clear enough account of your problem to demonstrate you're not asking about a correlation in the usual sense (of measuring degree of a linear association). However, a lack of monotonic relationship could be just about anything. Please describe for us what aspect of that relationship you do want to measure or describe. – whuber Feb 18 '23 at 23:41
  • I want to know what the association is between two variables. So let's say between anxiety levels and cortisol levels as an example. And so I could calculate Pearson's r... but when I look at a scatterplot let's say the relationship doesn't look linear... And so I see if I can calculate Spearman's... but the scatterplot shows the relationship isn't monotonic.... So what other test is available to run? – FastBallooningHead Feb 18 '23 at 23:47
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    If it's not monotonic, is it an "association" at all? We need to know what you mean by that. But perhaps you're overthinking this. After all, Spearman's correlation is $\pm 1$ for purely monotonic relationships and otherwise lies between those two extremes, giving some sense of the "lack of monotonicity." Maybe this already does what you want. – whuber Feb 18 '23 at 23:49
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    Aaaah, that makes sense, right. So I could still run Spearman's, and if Spearman's rho is close to 0 then there's no monotonic (or linear) relationship. Thanks! – FastBallooningHead Feb 18 '23 at 23:51
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    There are some attempts at defining "measures of association" here: https://stats.stackexchange.com/questions/534454/a-formal-definition-of-a-measure-of-association – Galen Feb 19 '23 at 00:07
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    Can you show us a plot? – kjetil b halvorsen Feb 19 '23 at 01:29
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    Note that if you choose any test based on looking at the data so that the test makes significant exactly the pattern you see, the test is invalid anyway, because the theory behind such tests assumes that they are chosen before seeing the data, i.e., not conditionally on the data. – Christian Hennig Feb 19 '23 at 11:17
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    There is in fact nonmonotonic association possible between variables that is not measured by Pearson, Spearman, or Kendall (for example if $y=x^2+e$ with $x$ symmetric about 0). Arguably this should not be called "correlation" though (although some people do that). – Christian Hennig Feb 19 '23 at 11:19
  • Serious, FBH, go learn about nonparametric regression Buja, A., Hastie, T., & Tibshirani, R. (1989). Linear Smoothers and Additive Models. The Annals of Statistics, 17(2), 453–510. – Alexis Feb 20 '23 at 19:00

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