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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?

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  • $\begingroup$ "Correlation does not imply causation" trumps any other other statements of the form "correlation implies [any modifying word(s)] causation". $\endgroup$
    – Thomas
    Commented Aug 1, 2013 at 13:34
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    $\begingroup$ 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? $\endgroup$
    – gung - Reinstate Monica
    Commented Aug 1, 2013 at 13:39
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    $\begingroup$ 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). $\endgroup$
    – Nick Cox
    Commented Aug 1, 2013 at 13:41
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    $\begingroup$ you may read this. $\endgroup$
    – SAAN
    Commented Aug 1, 2013 at 13:49
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    $\begingroup$ Are you looking for a term that is more general than correlation. If so, do you mean 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.) $\endgroup$
    – TooTone
    Commented Aug 1, 2013 at 14:53

3 Answers 3

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Behzad Momahed Heravi
    Commented Aug 1, 2013 at 15:43
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    $\begingroup$ I don't get your point. I say that you may have an omitted variable, not that you always have one. $\endgroup$
    – PAC
    Commented Aug 1, 2013 at 16:04
  • $\begingroup$ Do you mean that common-causal variable(s) or omitted variable(s) is NOT inevitable? $\endgroup$
    – Behzad Momahed Heravi
    Commented Aug 1, 2013 at 16:07
  • $\begingroup$ 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 (: ) $\endgroup$
    – Mike Flynn
    Commented Aug 1, 2013 at 17:12
  • $\begingroup$ Just for the joke, look at xkcd : xkcd.com/552 $\endgroup$
    – PAC
    Commented Aug 1, 2013 at 17:49
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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

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  • $\begingroup$ This is a good example. Is there a causal structure that can explain this correlation? $\endgroup$
    – Behzad Momahed Heravi
    Commented Aug 1, 2013 at 16:20
  • $\begingroup$ 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.) $\endgroup$
    – Nick Cox
    Commented Aug 1, 2013 at 16:23
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    $\begingroup$ Here's an even better example, in which the statistician used superior data visualization methods: stats.stackexchange.com/questions/36/… $\endgroup$
    – The Photon
    Commented Aug 1, 2013 at 16:23
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    $\begingroup$ @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. $\endgroup$
    – The Photon
    Commented Aug 1, 2013 at 16:25
  • $\begingroup$ 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. $\endgroup$
    – Behzad Momahed Heravi
    Commented Aug 1, 2013 at 16:44
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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.
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