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Correlation does not imply causation. Causation does imply correlation but not necessarily linear correlation. ... So does correlation imply high-order causation? If A and B are correlated, is it always possible to find K={K1 K2 ... Kn} variables such that 

A~K1, K1~K2, ...,Kn-1~Kn and Kn~B

where ~ denotes causality.

In other words, if there is no causality in a correlation, does a common-causal variable(s) always exist?

mribeirodantas
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Behzad Momahed Heravi
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  • "Correlation does not imply causation" trumps any other other statements of the form "correlation implies [any modifying word(s)] causation". – Thomas Aug 01 '13 at 13:34
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    Your `K` etc. statement is certainly possible, but I note that `B` does not occur in it. Would you mind clarifying what you are getting at there? – gung - Reinstate Monica Aug 01 '13 at 13:39
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    Definitions are over-rated, but I don't think you can make real progress here without some sharpening of terms. Notation doesn't help if it's not explained. If correlation is more general than linear relationship, then you seem to be saying that relationships between variables must be explicable somehow. Fine, but that's too weak a statement to be useful. There are shelves of books and papers on this. Judea Pearl. 2009. _Causality_ Cambridge University Press is more aware of discussions across several disciplines than works that are discipline-bound (e.g. in philosophy, physics, economics). – Nick Cox Aug 01 '13 at 13:41
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    you may read [this](http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/?utm_source=buffer&utm_campaign=Buffer&utm_content=buffer2f355&utm_medium=twitter). – SAAN Aug 01 '13 at 13:49
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    Are you looking for a term that is more general than correlation. If so, do you mean [dependence](http://www.collinsdictionary.com/dictionary/english/statistical-dependence) rather than correlation? (In statistics it's possible to have variables that depend on one another without there being a correlation, or at least without there being a linear correlation.) – TooTone Aug 01 '13 at 14:53
  • @gung Kn~B should be added – Behzad Momahed Heravi Aug 01 '13 at 15:34
  • @TooTone maybe a more general term for causality. if a correlation is evident between two variables and one is not caused by another then is there always a hidden variable (or a chain of variables) that links them together? – Behzad Momahed Heravi Aug 01 '13 at 15:48
  • Causation does not imply correlation, not linear correlation and not non-linear correlation either. You can check more details about this in another question clicking [here](https://stats.stackexchange.com/questions/26300/does-causation-imply-correlation). – mribeirodantas Dec 12 '21 at 00:00

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You can simply have an omitted variable which has a causal impact on A and a causal impact on B. In that case, you will have a correlation between A and B but no "high-order causation". This issue is known in econometrics as the omitted variable bias. It is discussed in most econometrics textbooks. See for instance Cameron and Trivedi Microeconometrics. For a more advanced discussion see Judea Pearl's book on causality.

PAC
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  • So any two correlated variables can be linked with an omitted variable? Doesn't seem true to me. Maybe a chain of causes or a parallel combination of causes can link the two variables. – Behzad Momahed Heravi Aug 01 '13 at 15:43
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    I don't get your point. I say that you may have an omitted variable, not that you always have one. – PAC Aug 01 '13 at 16:04
  • Do you mean that common-causal variable(s) or omitted variable(s) is NOT inevitable? – Behzad Momahed Heravi Aug 01 '13 at 16:07
  • I agree with this answer, and to add an example: being tall makes one heavier and more probable to have bigger shoes, but gaining weight does not increase shoe size (although maybe width (: ) – Mike Flynn Aug 01 '13 at 17:12
  • Just for the joke, look at xkcd : http://xkcd.com/552/ – PAC Aug 01 '13 at 17:49
  • Seconding the point about Pearl's excellent text on Causality. We would infer that `A !~ B` if `Pr(B | A, K1, K2, ... Kn) = Pr(B | K1, K2, ... Kn)` or that `B` is conditionally independent of `A`. This would nullify any causal relationship between `A` and `B`. – AdamO Aug 01 '13 at 19:16
  • @AdamO I don't think that statement "nullifies any causal relationship between and B". After all, you are conditioning on $K_i$, which can d-separate $A$ and $B$. If one of the $K_i$ mediated the causal relationship between $A$ and $B$, then not conditioning on that $K_i$ would leave $A$ as an indirect cause of $B$. – Neil G Dec 12 '21 at 03:45
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No.

For example, we can find a correlation between global average temperature and the world population of pirates ("Arrrgh!"), but nobody would suggest there is any sort of causality involved:

enter image description here

Edit:

Okay, the pirates vs global temperatures example is not a very good one here. Because (a) the x-axis of the chart is highly distorted, and (b) we could probably actually stretch to find an omitted variable (like some measure of industrialization).

A better example was given in answer to a previous question on CV:

enter image description here

Community
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The Photon
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  • This is a good example. Is there a causal structure that can explain this correlation? – Behzad Momahed Heravi Aug 01 '13 at 16:20
  • Yes and no. The key point is simply that the correlation here has to be explained indirectly. Sure, the hypothesis that pirates cause climate change (or whatever) is wrong according to current science, and it may not be of interest to investigate the indirect relationship, i.e. work out what the other variables are, but absence of a causal relationship between two particular variables does not refute a principle of causality in general. (Again, there have to be some definitions.) – Nick Cox Aug 01 '13 at 16:23
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    Here's an even better example, in which the statistician used superior data visualization methods: http://stats.stackexchange.com/questions/36/examples-to-teach-correlation-does-not-mean-causation/618#618 – The Photon Aug 01 '13 at 16:23
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    @NickCox, okay, for the pirates example we could talk about industrialization possibly having a causal effect on both variables. The lemons example linked in my last comment is a better one. – The Photon Aug 01 '13 at 16:25
  • For the Lemon example: Decline in highway fatality caused by increased road safety caused by government policy towards better safety caused by people demanding better health and lifestyle. The latter is a common cause for increasing demand in fresh fruits which causes increase in Fresh Lemon Imports. – Behzad Momahed Heravi Aug 01 '13 at 16:44
  • @BehzadMomahedHeravi, only if Mexico is the only country in the world that produces lemons. – The Photon Aug 01 '13 at 17:16
  • This isn't a good example because there is a causal relationship—just not a direct one. Industrialization can be seen, for example, as a common cause of the reduction of piracy and the increasing world temperatures. Except for spurious correlation, there is some causal chain. – Neil G Dec 12 '21 at 03:34
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If two events $A$ and $B$ are correlated given some observations $Z$ and interventions $Y$ in every context, then there must some causal mechanism mediating that correlation, which formally means that $A$ and $B$ are not d-separated by $Y$ and $Z$.

However, it's not clear what your notation means. Examples of non-d-separation include

  • $A$ causes $B$ (directly or indirectly),
  • they may have a common cause, and
  • they may both be causes of $Z$ or its ancestors.
Neil G
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