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Isn't this statement taking it too far? As far as I understand, the word "imply X" means that there is a chance >0% that X might be true. Therefore "not imply X

by erikb 12y ago

Isn't this statement taking it too far? As far as I understand, the word "imply X" means that there is a chance >0% that X might be true. Therefore "not imply X" would mean that the chance is <=0%. But when you have an unknown dataset showing X (with X being "A correlates with B", for instance), then the chance for X being true is in fact bigger than 0%. It might be dismissably more likely than 0%, so little more that you might conclude it is basically 0%, but it still is >0% in the mathematical sense. Therefore seeing correlation should imply correlation.

j2kun 12y ago
"Imply X" means "one can logically prove X follows as a consequence." When you say correlation does not imply causation you're saying "there are examples of things which correlate but have no casual relationship." Proving such a claim (by finding an example) is a logical disproof of the claim that correlation implies causation. Logical implication has nothing to do with probabilities in this sense. The point of the article is that correlation (of observed data) does not logically imply correlation (of the underlying phenomena, usually in a more general setting than the data allows).
- erikb 12y ago
  I see, then I misunderstood the definition of "imply X". Thanks :)
mcintyre1994 12y ago
I guess if you want to be formal you need to actually define imply like you have but the author hasn't. I'd go with A implies B means (not A) or B, ie binary implication. That'd seem to capture the spirit of "correlation does not imply causation" since we can have correlation (A) and not causing (not B), ie not((not A) or B), ie no implication. So I guess we can use it for this too as at least a possible definition. If we say A is correlation in the sample set, B is correlation in the population, I think we can replicate the above argument and say we can have correlation in the sample set (A) and not correlation in the population (not B), ie not implication. An obvious example would be if we have 2 variables and plotting them gave a simple quadratic curve (say, y=x^2 across x=[-10,10]) - we can find a sample set with strong positive correlation and another with strong negative correlation - but neither really describes the population as a whole. So being mathematical and using the boolean logic implication definition I think it's a valid statement - but defining implication in context is pretty important.
- afafsd 12y ago
  So the statement "correlation does not imply causation" says nothing more than the old "correlation does not equal causation", but is easier to misinterpret (accidentally or wilfully) due to the multiple definitions of the word "implies"?
  - erikb 12y ago
    I'd trust the definition of the other people more than my own, since English is only my second language. Also I am not a scientist. It's totally possible that my interpretation of the word "imply" is wrong, which is also what all the other other comments suggest.
deleted 12y ago
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msellout 12y ago
That is not a very useful definition of the word imply, since all events have a non-zero probability in a trivial, quantum mechanical anything-can-happen sense. Gelman likely uses the word imply in the more common meaning of indirect logical consequence.
mattfenwick 12y ago
> It might be dismissably more likely than 0%, so little more that you might conclude it is basically 0%, but it still is >0% in the mathematical sense. Therefore seeing correlation should imply correlation. I don't agree in the general case. A counter-example is that a purely random data set will have correlations, which by definition imply absolutely nothing about anything other than the random data set. This is addressed in the last sentence of the OP: > That is, correlation in the data you happen to have (even if it happens to be “statistically significant”) does not necessarily imply correlation in the population of interest.

by erikb 12y ago

j2kun 12y ago

"Imply X" means "one can logically prove X follows as a consequence." When you say correlation does not imply causation you're saying "there are examples of things which correlate but have no casual relationship." Proving such a claim (by finding an example) is a logical disproof of the claim that correlation implies causation. Logical implication has nothing to do with probabilities in this sense. The point of the article is that correlation (of observed data) does not logically imply correlation (of the underlying phenomena, usually in a more general setting than the data allows).

erikb 12y ago
I see, then I misunderstood the definition of "imply X". Thanks :)

mcintyre1994 12y ago

I guess if you want to be formal you need to actually define imply like you have but the author hasn't. I'd go with A implies B means (not A) or B, ie binary implication. That'd seem to capture the spirit of "correlation does not imply causation" since we can have correlation (A) and not causing (not B), ie not((not A) or B), ie no implication. So I guess we can use it for this too as at least a possible definition. If we say A is correlation in the sample set, B is correlation in the population, I think we can replicate the above argument and say we can have correlation in the sample set (A) and not correlation in the population (not B), ie not implication. An obvious example would be if we have 2 variables and plotting them gave a simple quadratic curve (say, y=x^2 across x=[-10,10]) - we can find a sample set with strong positive correlation and another with strong negative correlation - but neither really describes the population as a whole. So being mathematical and using the boolean logic implication definition I think it's a valid statement - but defining implication in context is pretty important.

afafsd 12y ago
So the statement "correlation does not imply causation" says nothing more than the old "correlation does not equal causation", but is easier to misinterpret (accidentally or wilfully) due to the multiple definitions of the word "implies"?
- erikb 12y ago
  I'd trust the definition of the other people more than my own, since English is only my second language. Also I am not a scientist. It's totally possible that my interpretation of the word "imply" is wrong, which is also what all the other other comments suggest.

deleted 12y ago

[deleted]

msellout 12y ago

That is not a very useful definition of the word imply, since all events have a non-zero probability in a trivial, quantum mechanical anything-can-happen sense. Gelman likely uses the word imply in the more common meaning of indirect logical consequence.

mattfenwick 12y ago

> It might be dismissably more likely than 0%, so little more that you might conclude it is basically 0%, but it still is >0% in the mathematical sense. Therefore seeing correlation should imply correlation. I don't agree in the general case. A counter-example is that a purely random data set will have correlations, which by definition imply absolutely nothing about anything other than the random data set. This is addressed in the last sentence of the OP: > That is, correlation in the data you happen to have (even if it happens to be “statistically significant”) does not necessarily imply correlation in the population of interest.