real world examples of causality without correlation

Question

There are several hypothetical instances of causality despite the lack of correlation between two variables, some of them have been mentioned in this forum - Does no correlation imply no causality?

Would anyone be able to share examples of real-world published cases in which there was clear causality without (linear) correlation?

Heteroscedastic residuals often provide examples where there is a non-linear causal relationship but linear regression has removed any correlation between the residuals and values of the causal variable — Henry, Nov 18 '22 at 19:15
These cases are common in self-regulating processes in biology - see my answer to a related question. — Lizzie Silver, Nov 20 '22 at 22:37

score 3 · Answer 1 · edited Nov 19 '22 at 13:41

Yes, of course, we can have examples of real-world cases where we have a causal relationship without a strong correlation; an obvious place to check is causal time-series models. Assume for example we have the structure:

It is easy for to say that $Y_t\perp \!\!\!\perp X_{t-1}|Y_{t-1}$ based on $d$-separation, i.e. we have conditional independence between $X_{t-1}$ and $Y_t$ given $Y_{t-1}$. Here, only the value of $X_t$ would be assistive to predict $Y_t$ and thus Granger causality (or some other "basic correlation" method) would not detect the influence of $X$ on $Y$ because the past of $X$ influences $Y_t$ only via the past of $Y$. As a result, the causal relation of $X$ on $Y$ is completely missed based on correlation measurements; I "borrowed" this example from Ch. 10 in "Elements of Causal Inference" (2017) by Peters et al., check it out for more formal treatment. Now for a published result: "A study of problems encountered in Granger causality analysis from a neuroscience perspective" (2017) by Stokes and Purdon is a quick one; they use a VAR model instead of a simple ARX as above but the logic is the same - a vector auto-regressive (VAR) model generalising a single-variable autoregressive (AR) model by allowing for multivariate time series. In general, search the terms "Granger Causality" and "Common-cause fallacy" alongside each other and a few references pop up; "A review of the Granger-causality fallacy" (2015) by Maziarz is a good place to start too.

Agreed, I meant to focus on the AR class of models generally, thank you, amended. — usεr11852, Nov 19 '22 at 12:49
Thank for your thorough answer (+1). Are you aware of a more down-to-Earth example? I read somewhere else that the tobacco industry messed up the data on the causal relation between smoking and cancer with the intent to show they are not correlated. However, if true, this is a case of scientific fraud. I wonder if there is some other example like this that does not involve data fraud. — Ben, Nov 19 '22 at 19:59
I am glad I could help. I haven't come across something like that for the smoking industry, sorry. I think their main line of defence was the absence of RCTs as the correlation was always there; what killed their argument was the use of instrumental variables related to genetic markers (which is something different to "no correlation, but with causation"). — usεr11852, Nov 20 '22 at 01:39

real world examples of causality without correlation

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