Under what conditions does correlation imply proximity to causation?

Question

I've recently had an experience with the whole "correlation does not imply causation", which is certainly true as far as a true/false proposition is concerned, but which also seems to be used too often to dismiss. What I wanted to ask refers to correlation as evidence, that is how reliable is correlation as a measure of proximity to a causal source. I've read the interesting related CrossValidated questions which I'll tease out the related snippets below...

1. Under what conditions does correlation imply causation?
2. Statistics and causal inference?
3. If 'correlation doesn't imply causation', then if I find a statistically significant correlation, how can I prove the causality?
4. Online resources for philosophy of causation for causal inference

Wikipedia Article: Correlation does not imply causation: Use of correlation as scientific evidence

However, sometimes people commit the opposite fallacy – dismissing correlation entirely, as if it does not suggest causation at all... This would dismiss a large swath of important scientific evidence

Answer to #3 by Peter

A very likely reason for 2 variables being correlated is that their changes are linked to a third variable. Other likely reasons are chance (if you test enough non-correlated variables for correlation, some will show correlation), or very complex mechanisms that involve multiple steps.

Answer to #1 by Yaroslav Bulatov

under what conditions can you reliably extract causal relations from data? ... infer the direction of causality from observations of pairs of variables where one variable was known to cause another

Answer to #2 by Gaetan Lion

It just confirms that certain events occur before others and that those events appear to have a consistent directional relationship. This seems to entail true causality but it is not always the case.

Answer to #4 by Graeme Walsh quoting Russell

that inductions do not make their conclusions probable unless certain conditions are fulfilled The main logical function that the assumptions have to fulfill is that of conferring a high probability on the conclusions and inductions that satisfy certain conditions.

All of these questions happen to fall upon the actual conclusion of some causal source from some correlated information, but I'm interested in the intermediate state where discovery is still taking place and conclusions are not yet drawn. Perhaps even to simplify the situation a bit let's say a valid statistically significant correlation is discovered but there is no direct causal link between the two correlates.

Please excuse any ambiguity and lack of articulateness in the following question since this is difficult to articulate what exactly I mean by "close", and "proximate". Basically if one happens to have discovered a correlation like mentioned above what is the likelihood that it occurs from absolute unrelated pure chance? Can two randomly chosen independent variables randomly correlate without there being an intermediary albeit unknown variable that influences each? It's hard for me to imagine that a true and persistent correlation would not at the least imply that an investigator is "close" to the source of causality. Put another way given the two variables that correlate there must at least be a third (or a chain to a third) nearby that I can at least be confident that I'm looking in the right direction as far as measuring the "right" things.

This question may be hard to apply in practical situations due to the fact that investigators don't really focus upon random facts or measurements but are entering and exploring a body of knowledge and phenomena that already binds relations to things they happen to be manipulating. Although I read somewhere in the above links that data miners are doing just such blind correlation discoveries, probably in the hopes of being proximate to a cause where proximate is good enough. In a nutshell should correlation be viewed more optimistically as hope for discovery of a cause than to pessimistically assume it demonstrates a suspicious relation or even invalid evidence.

Answer 1

You write, "if one happens to have discovered a correlation like mentioned above what is the likelihood that it occurs from absolute unrelated pure chance?" There are (at least) two senses here:

The first is that in the population, the two variables in question may be uncorrelated as well as having no causal connection. This is a type I error. If you were to look at a particular pairing out of the blue, the type I error rate is alpha. On the other hand, as you look at more and more possible correlations (which is what some data mining does), the probability of a type I error increases until you are almost guaranteed to find something. For more on this, it may help you to read this excellent CV thread: Look and you shall find (a correlation).

The second sense is what if the true correlation is non-zero (and hence not a type I error), but due to a confound / omitted variable? This probability can also be determined, but only if you can specify an accurate base rate. That is, you can use Bayes formula if you can come up with a valid prior somehow.

Regarding the question of whether something can be "close" to a causal relation, we would have to specify what that could possibly mean. It seems to me that there is either a causal connection or there isn't, in which case it cannot meaningfully be said that something can be "closer" to causal and still not quite be actually causal. It is certainly plausible that a causal force (whatever kind of thing that is exactly) can be weaker or stronger, which presumably means it is easier to detect stronger causal relations, but this would not help determine if a found correlation represents true causality. Instead, it seems to be related to the probability that an apparently uncorrelated pair might have a true causal relation nonetheless.

Answer 2

EDIT: I now realise that my answer is slightly off-mark. In my answer I address the conditions for identifiability (measurability) of causal effects, meaning the conditions under which I can correctly measure the effect of A over B. But the OP's question is about the conditions for identifiability of the existence of a causal effect between A and B - which is one of the main questions in causal discovery. Although the conditions are somewhat similar, I think that it's best to separate clearly between them. I will post a separate answer, and leave this one up for those interested in this one for other reasons.

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Such an interesting question. There is quite a bit of literature about causal inference and causal discovery, disciplines that deal precisely with these questions.

In my understanding (coming from a very narrow part of this literature which uses Structural Causal Models), the conditions that allow you to say "an (unconditional) observational correlation between A and B is equal to the causal effect of A over B" are

1. there is no confounding bias (in this case it would be fulfilled if A and B have no common causes)
2. there is no selection bias
3. there is no measurement bias (4. the association is not just random)

Note that these conditions are untestable assumed properties of the data generating process. Any causal statement supported by observational evidence is strongly dependent on the assumptions you make about the data generating process.

There are more general conditions that allow you to draw conclusions in more complex situations. One (very simplistic) example could be that A and B do share a common causes. In this case, under which circumstances can you measure the effect of A over B by looking at observational data alone? The circumstance is what Pearl calls the "backdoor criterion": you can measure the effect of A over B if all backdoor paths between A and B are "blocked". What are backdoor paths? They are causal paths in the data generating process that connect A to B from "behind" A, which result in spurious correlation if not "blocked". You can "block" a path by adjusting for a certain variable that "opens" a path. A path is opened either if

1. there in an unmeasured confounder
2. there is a inverted fork (aka collider) and the middle node is adjusted on

For example the path A <- U -> B is open if U is not measured. In this case we say that the causal effect between A and B is not identifiable. Another example involving colliders is the following: A <- X -> Y <- Z -> B. This path is unconditionally closed, meaning that if you just measure the unconditional association between A and B you will measure the true causal effect. But if you adjust by Y, then you will find a spurious correlation.

I warmly suggest reading Hernan's Causal Inference book (in particular chapters 6-9 where he defines these biases) and following his course on Coursera, where these concepts are explained much more precisely. Link to book: https://www.hsph.harvard.edu/miguel-hernan/causal-inference-book/