<p>In psychology, we often investigate whether two variables are related. We often investigate this with Pearson's, Kendall's or Spearman's correlation. The obvious drawback of those methods is that they are not able to detect all relationships. Pearson's correlation might miss nonlinear and the other two nonmonotonic relationships.</p>
<p>I am wondering whether anybody is aware of an example where this is actually the case. That is, I am looking for a <strong>real-data example</strong> where it is obvious that two variables are related, but neither of the traditional correlation methods detects it; let's say, defined as all having <span class="math-container">$p$</span> value &gt; .05. I have already searched through the literature and was not able to find a convincing example. An example where only Pearson's correlation fails to detect the relationship would already be nice.</p>

CC BY-SA 4.0
2022-11-14T15:06:29.287Z
Usually the simple stats class example is a parabolic relationship, something like y = x^2. It doesn't make sense to define this using p-values, though; if your model is that wrong, your p-value is likely meaningless. Additionally, even a method with very low power to detect a given relationship will give a significant p-value with a large enough sample size. Don't overuse p-values for anything besides their intended purpose!
Bryan Krause
CC BY-SA 4.0
2022-11-14T15:15:28.823Z
Thanks so much for your comments and warnings. I realized that I crucially forgot to add that I am looking for a real-data example.
Julian Karch
CC BY-SA 4.0
2022-11-14T15:18:13.607Z
https://scholar.google.com/scholar?hl=en&as_sdt=0%2C50&q=U-shaped+relationship+psychology&btnG=
Bryan Krause
CC BY-SA 4.0
2022-11-14T15:31:34.037Z
Thanks for the suggestion. I went through much of that list already. For all examples I investigated, either 1) one of the traditional methods also detects the relationship, or 2) the relationship is so weak that it is questionable whether it's real or just a false positive.
Julian Karch
CC BY-SA 4.0
2022-11-14T15:34:02.110Z
"Detects the relationship" the way you've defined it is not typically considered good enough by statisticians (this is part of the point I was trying to make in the first comment). What are you trying to do?
Bryan Krause
CC BY-SA 4.0
2022-11-14T15:39:02.873Z
I am indeed (some kind of) statistician. I agree with all your arguments, but am just trying to find an example for educational purposes.
Julian Karch
CC BY-SA 4.0
2022-11-14T15:42:46.310Z
I'd either stick with theoretical examples, or if you want to talk about practical ones then use a practical outcome, like power. And please, please, don't teach your students that their model is good if it gives them a significant p-value.
Bryan Krause
<correlation>
Example Nonlinear Relationship Not Detected by Pearson, Kendall, Spearman Correlation
Julian Karch
# Example Nonlinear Relationship Not Detected by Pearson, Kendall, Spearman Correlation

(Asked by: username_0 on 2022-11-14)

In psychology, we often investigate whether two variables are related. We often investigate this with Pearson's, Kendall's or Spearman's correlation. The obvious drawback of those methods is that they are not able to detect all relationships. Pearson's correlation might miss nonlinear and the other two nonmonotonic relationships.

I am wondering whether anybody is aware of an example where this is actually the case. That is, I am looking for a **real-data example** where it is obvious that two variables are related, but neither of the traditional correlation methods detects it; let's say, defined as all having $p$ value > .05. I have already searched through the literature and was not able to find a convincing example. An example where only Pearson's correlation fails to detect the relationship would already be nice.

**Tags:** correlation

> **Comments:**
> **username_1**: Usually the simple stats class example is a parabolic relationship, something like y = x^2. It doesn't make sense to define this using p-values, though; if your model is that wrong, your p-value is likely meaningless. Additionally, even a method with very low power to detect a given relationship will give a significant p-value with a large enough sample size. Don't overuse p-values for anything besides their intended purpose!  
> **username_0**: Thanks so much for your comments and warnings. I realized that I crucially forgot to add that I am looking for a real-data example.  
> **username_1**: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C50&q=U-shaped+relationship+psychology&btnG=  
> **username_0**: Thanks for the suggestion. I went through much of that list already. For all examples I investigated, either 1) one of the traditional methods also detects the relationship, or 2) the relationship is so weak that it is questionable whether it's real or just a false positive.  
> **username_1**: "Detects the relationship" the way you've defined it is not typically considered good enough by statisticians (this is part of the point I was trying to make in the first comment). What are you trying to do?  
> **username_0**: I am indeed (some kind of) statistician. I agree with all your arguments, but am just trying to find an example for educational purposes.  
> **username_1**: I'd either stick with theoretical examples, or if you want to talk about practical ones then use a practical outcome, like power. And please, please, don't teach your students that their model is good if it gives them a significant p-value.  

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