http://mathoverflow.net/questions/22490/correlation-and-causation-when-can-we-believe-correlation-reasonably-at-least

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I am voting to close, even though I am not a statistician and the two answers so far seem to indicate that it is possible to say something of a mathematical nature on the subject. However, in the end and answer to the question as stated cannot be mathematical in nature. As always, if you disagree, explain why here, so closure can perhaps be staved off (and if the discussion turns long, move it to meta). – Harald Hanche-Olsen

I vote against closing, even though the question could have been phrased somewhat more precisely. The importance of our understanding how to perform causal inference cannot be underestimated. Any improvement in our current understanding of this matter could be used to combat many kinds of ills in the world, such as diseases and crime. A very important general question is how to begin with a large amount of observational data about, say, how disease X is caused -- say 10,000 people's health status and auxiliary variables -- and extract the most likely guesses as to the etiology of X. – Daniel Asimov

As my comment below suggests, I too "vote against closing", as it were -- Shiva Kaul's answer suggests that something useful can be said (beyond the standard Stats 99 responses, which is about all I'd be able to contribute myself). – Yemon Choi

I'm abstaining from voting on this question because of the poster's reaction to my comment, but I will share my opinion here.

I think this is a good example of a slightly misdirected question. I think everyone would be happy if the title was "How can you measure causality?" and the question started with "We all know correlation does not imply causation, but..."

See 22635 for an example of a well directed question on a similarly often misunderstood fact of life.

Harald wrote:

"I am voting to close, even though I am not a statistician and the two answers so far seem to indicate that it is possible to say something of a mathematical nature on the subject."

Harald, you said it yourself that you're not a statistician. If you really think the question has no merit, please give the statisticians and probabilists in the community a little more time to contribute before you make the move to close it. It's better to err by leaving a bad question open then to close a potentially good one.

Sorry, I am traveling and don't have time to participate in this discussion right now. But to comment quickly: I may not be a statistician, but I have been hanging out in the skeptic movement long enough to know very well that no amount of correlation proves causation in the real world. That some of the rather interesting answers show that correlation does indeed imply correlation under narrowly defined circumstances, does not change this fact. So in a sense, I don't think those interesting answers did answer the question as it was asked. It was the presence of those answers that made me doubt my decision to vote close sufficiently to write the comment I did. But as asked, the question seemed to call for endless arguments about what causation is and how we can tell it's there.

A different issue is that I think questions should be closed relatively quickly or not at all. Maybe votes to close should go stale after a while, to diminish the ratcheting effect a little. In particular, this question has been around far too long that it should be closed now.

Disclaimer: The above is written by a sleep-addled brain with a caffeine deficiency. Handle with caution. Void where prohibited.

I agree with those who think this question has just enough content to survive.

A related phenomenon which I have noticed is that there seem to be relatively few active statisticians on MO, so that some simple stats questions which should be quickly closed slip through the net:

http://mathoverflow.net/questions/21813/which-statistical-test-is-significant-here
http://mathoverflow.net/questions/21821/how-to-test-whether-a-data-is-or-not-t4-distribution

Tom--I've just voted to close both of these.

@Harry: Although that may be a contributing factor, I think fgdorais is closer to the mark. Statistics is a distinct field and culture from mathematics, and I doubt many statisticians (as opposed to students taking statistics classes, or people from other fields who aren't aware of the difference) have ever come here in the first place.

Have just cast closing votes on both questions mentioned by Tom. I remember noticing both at the time, but deciding to wait in case someone could turn up in comments with specific suggestions of where else to look. (I didn't want to leave a comment which gave an impression of "we look down on statistics" as opposed to "most of us here don't really do statistics at any practising level". Thankfully, both of Tom's comments seemed to me to avert that particular misunderstanding.

I have a suggestion:

If someone is an expert in an underrepresented area on MO and doesn't have enough points to vote to close, all that person should need to do is leave a comment explaining to us that he or she is an expert in the subject and why he or she thinks that the question should be closed (or maybe post it on meta like Tom Smith just did).

A lot of us can't tell the difference between a good or bad questions in fields in which we have no experience, so we rely on these sorts of comments to avoid closing questions that might be good. (Obviously this isn't necessary for something like algebraic geometry, algebraic topology, commutative algebra, etc., for obvious reasons).

What do you guys think?

@fgdorais: I think that the poster's reaction to your comment stems from a misreading of the fourth word in your comment. The font makes it hard to see exactly what that word is so it looks a bit like "LITTLE" instead of the "TITLE" that it actually is (I've left a comment to this effect).

I've also voted to close. My reason is my standard one: great answers do not make great questions.

@Andrew: Thanks! I was wondering where that "little question" was coming from.

I think both fgdorais and Andrew nailed it in their comments. If anything, the (excellent) answers to the question only confirm my gut feeling that the question is not a good one: If I read them correctly, they all agree that correlation on its own never implies causation. To show causation, you need not only statistics but knowledge (expertise, even) within the relevant domain, so that you can, for instance, with a reasonable degree of confidence verify the conditions of Shiva Kaul's answer. To top it off, the “necessary conditions” of the question seem wrong-headed to me. As an example, we now know that smoking causes cancer. This conclusion is of course based on a lot of data which has been subjected to a lot of statistical analysis. But it is also based on detailed knowledge of the underlying mechanism, plus of course looking for and accounting for the effect of every conceivable confounding variable. (Besides, the cancer comes a long time after the smoking, which effectively demolishes the third “necessary condition” of the question.)

If the question had not already gotten some (good) answers by the time I saw it, I would just have voted to close and moved on. It was the presense of those answers that made me uncomfortable enough with my decision to vote for closure to express those doubts in a comment. I still feel that it was the right decision, but in deference to Tom LaGatta's comment above, I think I will abstain from voting to close any question on statistics from now on, regardless. We can hope there are enough 3000+ rep statisticians here so they can deal with the really bad questions themselves.

@gilkalai:

I find it hard to understad why people regard the question as borderline or even want to close it. But I would be happy to understand this line of thought.

And I find it difficult to understand that this is so hard to understand. At least I, from my side of the discussion, can see why others disagree. If I can't even explain myself well enough that you can understand my point of view, then either my powers of explanations are feeble indeed, or else our respective points of view are separated by a chasm too deep and wide to be easily bridged. Maybe some of the comments added after you wrote the above will help explain it; if not, then I think the only way forward would be to spend an evening over a beer or three hashing it out. In other words, MO needs an attached bar. Until then, I'll stick with my resolution not to vote on statistics questions.

But in all seriousness, is it kosher to say that MO is not really a place for statisticians? I mean, this would be a lot easier if we could just close all statistics questions as off-topic and lead them to greener pastures. I think it's reasonable, and I think that's something that the administration should seriously consider.

That seems a bit random. I don't think statistics questions are being enough of a nuisance that this makes sense. We'd just end up with some good, interesting probability questions getting closed (and causing a big stink) because a handful of 3k+ rep users heard that "it's MO policy to close statistics questions". There are a whole bunch of questions tagged [st.statistics]. Most of them look fine to me (based on a poor sample I just made), and lots of them have accepted answers. I don't think these questions are bothering anybody.

@fgdorais:

I think I will abstain from voting to close any question on statistics from now on, regardless.

I think this is a bad decision.

Maybe, but it's not cast in stone. I may change my mind about it eventually, but for the time being, it stands.

I agree with Tom that the remarks by Harald, Francios, Harry and others are very useful and very much welcome.

In thie case, I found the title and the formulation of the problem
"We always hear , when reading on correlation, that 'Correlation does not imply causation.'
Still, I have never seen any source that tries to answer the question of when can we reasonably conclude a causal relation between variables X,Y,from a correlation.",
very reasonable and also the three preliminary thoughts added to the question rather nice.

The problem is of importance in various areas of applied mathematice and theoretical computer science, and also in probability theory and mathematical physics in addition to its importance in statistics. Also there were many papers written about it so answers pointing to various good sources will be quite useful.

I can see that "overlly important" questions with some philosophical aspects are, perhaps, not the most useful MO questions, but still there are many of those.

Part of Francois's and Harald's objection is that they simply suggest the answer "no". Francois wrote "The answer to your title question is never" and many people felt that this is an excellent response. Harald expressed a similar view and I can certainly identify with Harald's points regarding the importance of understanding specific mechanism in every particular case which is usually not a mathematical task. (And the reference to the skeptic movement in this context is interesting.) Let me mention that there are people that will not only agree that correlation does not imply causality but will question if "causality" is a meaningfull notion at all.

But I do not agree that the only contribution mathematics can make to the question if and when we can reasonably conclude causal relation from correlation is the answers "no" and "never". (Certainly there were attempts to say more which are worth looking at.)

@gilkalai: Count me among the causality-questioners :)

(I think "causality" is of a similar status to "rationality" - or, perhaps even more controversially, "free will" - but I digress).

My only objection is to the form of the question, not the content. I'm afraid the question cannot have lasting value in its current state.

In that case, I think the question should be edited. Could you either edit the question or indicate what about the form of the question needs to change? As I understand it, the objection is to the focus on the misguided question "when does correlation imply causation?" Instead, it should be pushing for something like "what, in addition to correlation, is needed to determine causation?" What exactly is the better question to ask?

Guys, as i said, in my opinion the title of the question "Correlation and causation: When can we believe correlation (reasonably, at least) imply causation"
is very nice (appart, perhaps from the English). I do not think the question "when does correlation imply causation?" is misguided. It is a great question. When you say "when" you usually mean in mathematics "under which additional conditions" and this is precisely what the question is about.

(I think that my citation index for that particular comment is higher than for any of my papers! Given that the probability of having a theorem or "mathematical thingy" named after me is fairly small, I'm quite chuffed at having a "Standard Objection" in the meantime.)

Dear Gil,

Correlation can be used as evidence of a causal relation, but it is not proof of such a relation nor is it a consequence of such a relation. In fact, the relation between the two is very limited as far as I know.

"A causal relation often leads to statistical correlation." Only when the causal relation leads to a linear dependency and then correlation only attempts to measure how linear this dependency is. Correlation can be used to check whether X, Y follow a given relation Y = f(X) by correlating Y and f(X) but you need to know f in advance. I have used correlation to support a relation that I derived by other means, but I never used correlation to derive a relationship. I think this is one of the principal ways that correlation is used in experimental sciences.

"[...] faced with significant statistical correlation it is often important to understand the causal structure leading to it." Maybe, but you could be in for a surprise! It is prudent to analyze other aspects of the data before embarking on such a quest. See Anscombe's Quartet to see how it is important to look at the data closely before relying on correlation and other basic statistics. It's also sometimes misleading to think that way. For example, road traffic density is highly correlated with certain times of day but neither is a direct cause of the other. In fact, the "causal structure" between the two is such a complex web that I pity the (presumably alien) scientist who unknowingly embarks on a quest to untangle this.

@Ben:

Can you clarify what you mean by "this"? There are certainly some very important related scientific/ethical questions regarding the use of correlation, but if "this" means the original question as posed then no.

Another important question with more mathematical content is how to measure causation. As we saw in the answers, this is an important research area in statistics.

And BTW, if we're talking about causality in (say) economics etc. then I'd just argue that economics consists of effective(?!) theories that are presumed to have some basis in progressively more exact sciences, and ultimately in physics.

Ah! I missed the part where we stopped talking about the actual question at hand. I think we all agree that there is a better question to ask. Judging from the answers that this question got, I'm sure that the better question will get some excellent answers. I think that it would be more productive for the interested parties to go and ask the better question on MO, so we can refocus this meta thread on whether or not question 22490 is appropriate.

That's a great find, Harald. Thanks!