Good Examples of Causation does not Imply Correlation

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WWGD
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Ok, so if the causality relation between A,B is not linear, then it will go unnoticed by correlation, i.e., we may have A causing B but Corr(A, B)=0. I am trying to find good examples to illustrate this but not coming up with much. I can think of Hooke's law, where data pairs (x, kx^2) would have zero correlation. Is this an " effective" way of illustrating the point that causation does not imply ( nonzero) correlation? Any other examples?
 

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  • #2
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If you apply a voltage across a resistor it causes power to dissipate in the resistor. The power is quadratic in the voltage so the linear correlation coefficient is zero.
 
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  • #3
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Why would (x,x^2) not have a high correlation for positive x?
 
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Something like (x,xsin(x)) would have little correlation
 
  • #5
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Why would (x,x^2) not have a high correlation for positive x?
I haven't double-checked the actual values of the correlation ( difficult to do on the phone) but because points in a parabola do not closely resemble/fit points in a line.
 
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Something like (x,xsin(x)) would have little correlation
Thanks. Can you find a causal relation des cribed by such pairs?
 
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Correlation(x,x^2)~0.97 for x=1:100
 
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Why would (x,x^2) not have a high correlation for positive x?
I wasn’t limiting it to positive x. The correlation is 0 for a balanced positive and negative sample
 
  • #9
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I guess the fact that it's quadratic isn't interesting here, (x,|x|) would have similarly small correlation. Basically anytime you have a signed input, and an unsigned output whose magnitude depends on the magnitude of the input.

The correlation of the charge on an ion and the angle of curvature when it passes through a magnetic field? Actually constructing these examples is annoying.

What about something like the correlation between day of the year and temperature. Days 1 and 365 are both cold (at least in the northern hemisphere), the middle days are warm, so correlation is zero.
 
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I guess the fact that it's quadratic isn't interesting here, (x,|x|) would have similarly small correlation. Basically anytime you have a signed input, and an unsigned output whose magnitude depends on the magnitude of the input.

The correlation of the charge on an ion and the angle of curvature when it passes through a magnetic field? Actually constructing these examples is annoying.

What about something like the correlation between day of the year and temperature. Days 1 and 365 are both cold (at least in the northern hemisphere), the middle days are warm, so correlation is zero.
Thanks, but it is not just any dataset, or, like you said, it is relatively-straightforward. I am looking for one describing a causal relation.
 
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I guess the fact that it's quadratic isn't interesting here, (x,|x|) would have similarly small correlation. Basically anytime you have a signed input, and an unsigned output whose magnitude depends on the magnitude of the input.

The correlation of the charge on an ion and the angle of curvature when it passes through a magnetic field? Actually constructing these examples is annoying.

What about something like the correlation between day of the year and temperature. Days 1 and 365 are both cold (at least in the northern hemisphere), the middle days are warm, so correlation is zero.
Oops! Realized I forgot to shift the ## y=kx^2 ## to avoid symmetry. Consider, e.g., ## y=k(x-1)^2##. That should do it.
 
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Using the word correlation to imply linear correlation is a little uncomfortable to me when used in the phrase, "Causation does not Imply Correlation". I always interpret "correlation" as general correlation in the converse.
 
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  • #13
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I think the examples given here all have zero general correlation.
 
  • #14
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ultimately if measured properly, causation should result in linear correlation, some adjustment of variables will result in linear correlation in the examples above. In the quadratic example centered at the origin, for instance, a simple look at the data will reveal the relationship and all one has to do is take the absolute value of the input.
 
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I think the examples given here all have zero general correlation.
I think zero correlation means knowing the value of one would give you absolutely no information that is useful to predict the value of the other.
 
  • #16
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For context, I may be teaching a small online class that includes this general area and was looking for examples that are " natural". I am thinking too of including Anscombe's quartet somehow. More interesting to me, but beyond the scope, is having different RVs with the same distribution: like the RVs counting heads or tails in a binomial with p=0.5.
 
  • #17
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The other situation is a missing variable, where A impacts B, but does not show up statistically because the impact of C is not accounted for
 
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The other situation is a missing variable, where A impacts B, but does not show up statistically because the impact of C is not accounted for
I'm not sure, but encryption might be a good example.
 
  • #19
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The other situation is a missing variable, where A impacts B, but does not show up statistically because the impact of C is not accounted for
You mean lurking variables?
 
  • #20
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Using the word correlation to imply linear correlation is a little uncomfortable to me when used in the phrase, "Causation does not Imply Correlation". I always interpret "correlation" as general correlation in the converse.
Since this thread is in the statistics section I assumed that standard statistical correlation was implied, but you do make a good point. That isn’t the only meaning to the term.
 
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  • #21
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Since this thread is in the statistics section I assumed that standard statistical correlation was implied, but you do make a good point. That isn’t the only meaning to the term.
I assume everything outside of General Discussion to be interpreted technically. " Big Picture" questions, no less important/interesting than the latter, I assume belong in GD.
 
  • #22
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Since this thread is in the statistics section I assumed that standard statistical correlation was implied, but you do make a good point. That isn’t the only meaning to the term.
I assume everything outside of General Discussion to be interpreted technically. " Big Picture" questions, no less important/interesting than the latter, I assume belong in GD. Edit: Unless explicitly stated otherwise. The linked content below makes me think this is the way PF is organized.
 
  • #23
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Causation implies you can make a prediction about the value is basically a tautology, and doesn't really help much. How do I figure out if there exists an arbitrarily shaped function which results in at least ##\epsilon## predictive power?

One method of testing this is by measuring the correlation. If it exists, then a predictive function exists (even if the relationship is not casual). The fact that correlation can be zero and you can still have perfect predictive power is an interesting result in my opinion.
 
  • #24
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While we're talking about correlation. Anyone know if we can consider Spearman Rho for more than 2 datasets? Edit: I know we can use Kruskal -Wallis one-way Anova for simiular but just curious as to Spearman Rho.
 
  • #25
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The spearman coefficient is the Pearson coefficient of pairs of integers describing the relative values of the data sets, so if there's a Pearson coefficient there is a spearman one.

I don't know of anything specific, but as far as the Pearson coefficient measures how good a line fits the data, you can certainly measure e.g. the variance explained by the first pca factor and take the square root of that. I think that won't give pearson's coefficient since the line returned by pca on two variables is not the best fit line, but I might be wrong about that.
 

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