(We have remnants of this type of reasoning in old-school “Correlation does not imply causation”, without the now-standard appendix, “But it sure is a hint”.)
Given the reasoning in this post and this post I think you can also infer that this old “Correlation does not imply causation” statement is not only flawed, but it’s also outright wrong
And should instead just be “Correlation does imply causation, but doesn’t tell which kind”
“imply” in the traditional phrase is used in the strong sense. You can have a correlation between 2 factors without there necessarily being a causal relationship between them.
If you can exclude coincidence, which is a question of confidence and what kind of data the correlation is based on, then you can say that the correlation does necessarily involve a causal relationship.
Well that’s just what I think. If you can show me how that’s wrong, then please do. Except I don’t think you can.
I agree. But it’s still inaccurate to say it does not imply causation.
correlation between A and B is explained by either 1. A->B 2. B->A 3. X->A & X->B or 4. By chance , or any combination of the aforementioned and which of 4. is usually confidently eliminated by anything that is statistically significant.
Point being there’s usually a causal relationship behind the correlation, even if it involves more factors than the ones that are being studied. Therefore that old phrase is misleading and—in my opinion—wrong.
As Peter noted, the meaning of “correlation does not imply causation” is “it is false that, for every X and Y, if positively-correlated(X,Y) then either causes(X,Y) or causes(Y,X).” Interpreted in this way, the principle is completely unimpeachable. If you object to it, you must be taking the principle to imply something much more general, like “it is false that for every X and Y, if positively-correlated(X,Y), then there is some Z that is in some way causally relevant to partly explaining this fact.” The latter version of the principle is much easier to deny.
But your own argument doesn’t quite get us to being able to deny either principle yet. For instance: What is meant by “X->A & X->B”? If this means direct causation, then it is surely false. But if it allows for transitive causal chains leading back to some X, then the principle risks triviality, since it is plausible that all events share at least some cause in common, if you go back far enough. A second problem: How can we rigorously unpack the meaning of “by-chance” correlations? And third: How do you know that statistically significant correlations are usually not “by chance” in your sense?
But if it allows for transitive causal chains leading back to some X, then the principle risks triviality, since it is plausible that all events share at least some cause in common, if you go back far enough.
However, wouldn’t that be extremely unlikely? And wouldn’t the likelihood be related to the amount of correlation?
A second problem: How can we rigorously unpack the meaning of “by-chance” correlations?
I’m not sure because I lack the skill in mathematics to answer this question the proper way.
And third: How do you know that statistically significant correlations are usually not “by chance” in your sense?
I’m not sure if there is a mathematical formalism for this, pretty much for the same reasons as for problem two: I don’t have the mathematical abilities required. However, I do know what they’re about, and I’m rather confident that you and I both can tell apart results that could be explained by mere chance and those that could not—it would be rather surprising if it was not achievable by means of math if you can achieve that by mere fallible intuition?
Well I apologize if I’m mistaken here, but I’m still trying to be reasonable.. Hmm.
Let’s create an examples to illustrate a point:
Students at some school take a special test each schoolyear and their tests results are compared with something fairly trivial. Let’s say the number of pencils the students bring to the tests.
Then by means of correlation it is found that the number of pencils brought by the students to the school has been increasing in a way that is correlated to prowess in the tests by the students.
In this case it’s not sufficient to say that the correlation implies that the number of pencils is causing the increasing prowess in the tests, nor that the prowess in the tests is causing the increased number of pencils. Which is what the phrase traditionally stands for.
But there still can be a causal relationship, for an example the school’s funding has been increasing and they’ve been giving more free material to students, and if increased material is correlated with increased prowess and increased number of pencils, or increasing economy.. and so forth, that’s causality, but not of the same kind.
However we can also say that this is just a coincidence, particularly if there has been only a couple of events. Or by some trivial causal chain like then you mentioned, but....
… you can also see how these results could be of a nature where casuality is actually required. If we look at a single testing event and notice that for the 500 students of the school there’s a strong correlation between number of pencils and test prowess, we’re starting to talk about extremely small probabilities that the results are by coincidence, are we not? Even if the pencils are not the cause , we can still deduce that there is a cause at high likelihood?
Well anyway maybe I’m just making excuses, at least it’s important to consider that at this point, and I see your point anyway, and I think I was wrong. Oops, sorry.
But not exactly. Because I think there’s something to this. And I think you should know what I mean. Maybe it’s important to start asking what this coincidence actually means? Isn’t this actually something about Markov Blanket ? (or something similar, sorry if I misused the term)
You can measure the likelihood that the profile the datasets are similar by chance. For an example simple increasing tendency—correlation that is—that can be explained by coincidentially similar increasing tendency, but if there’s an complex profile to correlation, you can measure what the likelihood for a coincidence is? Even further, the more complex the profiles are, the less likely a coincidence becomes? ( if they match )
it works like this: If people in general are erring on the side of over-associating correlation with causation rather than under-associating, then “correlation is not causation” is the better rule-of-thumb.
Agreed. I’m sorry for for commenting about this before thinking things really through, that was very lazy and thoughtless.
However in the course of you people being nice and pointing out how foolish I was not only the obvious error was corrected, but it appears that I also gained an insight(finding out something I didn’t know personally that is) into the matter. That being: In some cases you can estimate the probability of a correlation being merely coincidential versus it being the result of an actual causal relationship. Although since I’m not a mathematician I don’t actually know how do that, except by looking at graphs and letting the brain do all the work. It does though sound a little silly.
Does someone know how to do that mathematically? Estimate the probability of a correlation being coincidential versus due to a causal relationship of an unknown type?
Given the reasoning in this post and this post I think you can also infer that this old “Correlation does not imply causation” statement is not only flawed, but it’s also outright wrong
And should instead just be “Correlation does imply causation, but doesn’t tell which kind”
“imply” in the traditional phrase is used in the strong sense. You can have a correlation between 2 factors without there necessarily being a causal relationship between them.
If you can exclude coincidence, which is a question of confidence and what kind of data the correlation is based on, then you can say that the correlation does necessarily involve a causal relationship.
Well that’s just what I think. If you can show me how that’s wrong, then please do. Except I don’t think you can.
That’s begging the question, if by “coincidence” you just mean those cases where there is a correlation which does not involve a causal relationship.
I think that tradiotional wisdom is fairly accurate, bearing in mind that correlation between A and B doens’t imply causaiont between A and B.
I agree. But it’s still inaccurate to say it does not imply causation.
correlation between A and B is explained by either 1. A->B 2. B->A 3. X->A & X->B or 4. By chance , or any combination of the aforementioned and which of 4. is usually confidently eliminated by anything that is statistically significant.
Point being there’s usually a causal relationship behind the correlation, even if it involves more factors than the ones that are being studied. Therefore that old phrase is misleading and—in my opinion—wrong.
As Peter noted, the meaning of “correlation does not imply causation” is “it is false that, for every X and Y, if positively-correlated(X,Y) then either causes(X,Y) or causes(Y,X).” Interpreted in this way, the principle is completely unimpeachable. If you object to it, you must be taking the principle to imply something much more general, like “it is false that for every X and Y, if positively-correlated(X,Y), then there is some Z that is in some way causally relevant to partly explaining this fact.” The latter version of the principle is much easier to deny.
But your own argument doesn’t quite get us to being able to deny either principle yet. For instance: What is meant by “X->A & X->B”? If this means direct causation, then it is surely false. But if it allows for transitive causal chains leading back to some X, then the principle risks triviality, since it is plausible that all events share at least some cause in common, if you go back far enough. A second problem: How can we rigorously unpack the meaning of “by-chance” correlations? And third: How do you know that statistically significant correlations are usually not “by chance” in your sense?
However, wouldn’t that be extremely unlikely? And wouldn’t the likelihood be related to the amount of correlation?
I’m not sure because I lack the skill in mathematics to answer this question the proper way.
I’m not sure if there is a mathematical formalism for this, pretty much for the same reasons as for problem two: I don’t have the mathematical abilities required. However, I do know what they’re about, and I’m rather confident that you and I both can tell apart results that could be explained by mere chance and those that could not—it would be rather surprising if it was not achievable by means of math if you can achieve that by mere fallible intuition?
Well I apologize if I’m mistaken here, but I’m still trying to be reasonable.. Hmm.
Let’s create an examples to illustrate a point:
Students at some school take a special test each schoolyear and their tests results are compared with something fairly trivial. Let’s say the number of pencils the students bring to the tests.
Then by means of correlation it is found that the number of pencils brought by the students to the school has been increasing in a way that is correlated to prowess in the tests by the students.
In this case it’s not sufficient to say that the correlation implies that the number of pencils is causing the increasing prowess in the tests, nor that the prowess in the tests is causing the increased number of pencils. Which is what the phrase traditionally stands for.
But there still can be a causal relationship, for an example the school’s funding has been increasing and they’ve been giving more free material to students, and if increased material is correlated with increased prowess and increased number of pencils, or increasing economy.. and so forth, that’s causality, but not of the same kind.
However we can also say that this is just a coincidence, particularly if there has been only a couple of events. Or by some trivial causal chain like then you mentioned, but....
… you can also see how these results could be of a nature where casuality is actually required. If we look at a single testing event and notice that for the 500 students of the school there’s a strong correlation between number of pencils and test prowess, we’re starting to talk about extremely small probabilities that the results are by coincidence, are we not? Even if the pencils are not the cause , we can still deduce that there is a cause at high likelihood?
Well anyway maybe I’m just making excuses, at least it’s important to consider that at this point, and I see your point anyway, and I think I was wrong. Oops, sorry.
But not exactly. Because I think there’s something to this. And I think you should know what I mean. Maybe it’s important to start asking what this coincidence actually means? Isn’t this actually something about Markov Blanket ? (or something similar, sorry if I misused the term)
Oh well I think I can answer the question:
You can measure the likelihood that the profile the datasets are similar by chance. For an example simple increasing tendency—correlation that is—that can be explained by coincidentially similar increasing tendency, but if there’s an complex profile to correlation, you can measure what the likelihood for a coincidence is? Even further, the more complex the profiles are, the less likely a coincidence becomes? ( if they match )
So you don’t notice a lot of correlation-causation errors? I see them everywere. Practically every science story in the press.
How’d you get that from what I just said? Someone else making errors is not an excuse for you to do that too.
it works like this: If people in general are erring on the side of over-associating correlation with causation rather than under-associating, then “correlation is not causation” is the better rule-of-thumb.
Agreed. I’m sorry for for commenting about this before thinking things really through, that was very lazy and thoughtless.
However in the course of you people being nice and pointing out how foolish I was not only the obvious error was corrected, but it appears that I also gained an insight(finding out something I didn’t know personally that is) into the matter. That being: In some cases you can estimate the probability of a correlation being merely coincidential versus it being the result of an actual causal relationship. Although since I’m not a mathematician I don’t actually know how do that, except by looking at graphs and letting the brain do all the work. It does though sound a little silly.
Does someone know how to do that mathematically? Estimate the probability of a correlation being coincidential versus due to a causal relationship of an unknown type?