If we have two non-zero-correlated random variables, then why do we say that "correlation does not imply causation"?

Question

If we have two non-zero correlated random variables then they are dependent. Why then do we have the saying "Correlation does not imply Causation". A change in one variable may not cause exactly the same change in another but there is at least some 'causal' link.

Answer 1

Things can be correlated by having a third common cause while not having any causal influence on each other. For example, a fire siren going off and flashing warning lights may be near 100% correlated but neither causes the other. Instead, there is a third causal influence that is the common cause of both, namely pulling the fire alarm.

I recommend learning about the work primarily developed by Judea Pearl on causal inference. An assumption in causal inference, called Reichenbach's common-cause assumption and sloganized as "no correlation without causation", states that two variables are only dependent if one is the cause of the other or there is a third variable causing both. So you definitely aren't completely off in your intuition.

Answer 2

There are three different concepts in play, two from probability theory and one from philosophy.

The two mathematical concepts are (in)dependence and correlation. Independence is a general property of events and random variables. Correlation is a property typically associated with random variables that are square integrable. If two square integrable variables are independent they they have zero correlation but not vice versa.

Causation is a philosophical concept that implies comparing the observable world with alternative worlds (that could have happened but did not). It is so dangerous that only lawyers and judges use it. The closest approximation that scientists use is causality, and it is mainly a negative axiom: an event cannot be the cause of another event unless it precedes the other event in time. There is a slightly adapted version in relativity theory due to the fact that time is not an absolute observable.

The statement 'correlation does not imply causation' refers to the fact that correlation can be inferred statistically, whereas causal connections imply much more a priori choices of what worlds are possible.

As an example, it is relatively easy to establish a positive correlation, and therefore also a dependence, between smoking and lung cancer. To prove that smoking causes lung cancer, however, you need to set up an experiment that effectively manages several parallel worlds, one in which a person smokes and another where they don't smoke with all other things being equal whatever that means.

As a good surrogate, scientists will approximate cause-effect relationships by multivariate correlation analysis. The best thing we can then come up with is statements like 'smoking is positively correlated to lung cancer even after factoring out annual income and level of education'. That is how research is reported in professional journals. Unfortunately the word cause will often creep back into the press release.

Answer 3

If we have two non-zero correlated random variables then they are dependent.

Yes, this is true. Non-zero correlation implies dependence.

Why then do we have the saying "Correlation does not imply Causation"

Because this is also true.

A change in one variable may not cause exactly the same change in another but there is at least some 'causal' link

There is not necessarily any causal link. If two things change together, it doesn't mean that one is the cause of the other. For example, in the Summer, ice cream sales increase and, apparently, the crime also increases, but one is not the cause of the other. There is one or more hidden variables which makes these variables to change together. In general, we often think that if two things change together, then they have a causal relationship, but this is actually rarely the case.