A Good Examples of Causation does not Imply Correlation

[WWGD]

If you are going to teach an introductory class, I think you should be careful about these terms. Saying that A and B are uncorrelated implies that A and B are random variables. Saying that A causes B implies that A and B are events. The two implications are conflicting. It would be better to talk about random variables X and Y being correlated and about the event $X \in A$ implying (not causing) the event $Y \in B$. (You could talk about events A and B being independent, but not uncorrelated).
Also, you should be careful to indicate that "causation" is a logic problem that depends on subject knowledge, not a statistical problem.

Well, this is part of the problem of trying to popularize not-so-simple topics. I have to do enough handwaving to start a tornado.

 

[WWGD]

This is just a Stats 101 for incoming freshmen , with little to no background in Math/Philosophy. I was just asked to incorporate this topic to the existing curriculum. At any rate, there is still a lot of handwaving when introducing the CLT, Hypothesis Testing, etc. I will include the caveat of the necessary oversimplification and just direct the curious ones to more advanced sources.

 

[Stephen Tashi]

Consider Hooke's law and the causal relation ( Causality is still somrwhat of a philosophical term at this stage, so I am settling for accepted Physical laws as describing /defining causality) with a shift, to$y=k(x-1)^2$ . Then the samples at opposite sides of the above equalities described by it, as well as by other Physical laws may give rise to uncorrelated data sets. I cannot afford to enter or present serious background on causality in an intro-level class.

For the purposes of statistics, the main point is not the philsopical definition of causality, but rather the mathematical point that the correlation between A and B is ( as @FactChecker says) only defined for random varables A and B. If A and B are physical measurements, they are only random variables if some scheme is specified for taking random samples of the measurements. So a "Hookes Law" relation between A and B does not define A and B as random variables. To suggest to an introductory class that the names of two measurements (e.g. length, force or height, weight) implies the concept of a correlation or a lack of correlation between the measurements is incorrect. A fundamental problem in applications of statistics is how to design sampling methods. You cannot provide a coherent example of "measurement A is not correlated with measurement B" without including the sampling method.

 

[FactChecker]

You do not need to dwell in class on the technicalities, but you can arm yourself with a few simple, intuitive, examples. I think that is what you want from this thread.
If you randomly select a person and measure the lengths of their arms, those lengths are random variables. The event that the selected right arm is more than 2 feet long is an event. The right arm lengths are highly correlated with left arm lengths, but the right arm being over 2 feet long does not cause the left arm to be over two feet long -- it just implies, it does not cause.

 

[WWGD]

For the purposes of statistics, the main point is not the philsopical definition of causality, but rather the mathematical point that the correlation between A and B is ( as @FactChecker says) only defined for random varables A and B. If A and B are physical measurements, they are only random variables if some scheme is specified for taking random samples of the measurements. So a "Hookes Law" relation between A and B does not define A and B as random variables. To suggest to an introductory class that the names of two measurements (e.g. length, force or height, weight) implies the concept of a correlation or a lack of correlation between the measurements is incorrect. A fundamental problem in applications of statistics is how to design sampling methods. You cannot provide a coherent example of "measurement A is not correlated with measurement B" without including the sampling method.

My actual take on causation of B by A would be that in several independent experiments, variable A was controlled for, ( instances of it were) selected randomly so that the major non-error variation in B is explained through variation in A. But there is little room to delve into this, to specify how/where random variables or events come into place in this setting. And, yes, I was assuming a scheme to take random samples from each has been defined. Students I have had have trouble understanding what a probability distribution is, so delving into events and random variables is overkill, as I am only allotted a single class to go into this topic.

 

[Stephen Tashi]

But there is little room to delve into this, to specify how/where random variables or events come into place in this setting.

I don't understand how an introductory course in statistics can have little room for discussing random variables!

 

[WWGD]

I don't understand how an introductory course in statistics can have little room for discussing random variables!

Nursing and other humanities students , high school with hardly any/ very poor Math or Science background, I guess. Not a comfortable position for me to be in, for sure.

 

[Dr. Courtney]

Consider Hooke's law and the causal relation ( Causality is still somrwhat of a philosophical term at this stage, so I am settling for accepted Physical laws as describing /defining causality)

But even in Hooke's law, does the displacement cause the force, or does the force cause the displacement?

Yes!

 

[Ygggdrasil]

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?

Here's a nice figure with some examples illustrating your point:

https://janhove.github.io/teaching/2016/11/21/what-correlations-look-like