Causation without Correlation is Possible(theincidentaleconomist.com)
theincidentaleconomist.com
Causation without Correlation is Possible
http://theincidentaleconomist.com/causation-without-correlation-is-possible/
5 comments
No statistical method can prove causality. Also, there's no scientific method to confirm any one of causal relationships, either. Causality exists only in human mind. There's no methodic way to discern semipermanent coincidence (or identical equivalence) from causality.
>Causation without correlation. ...Suppose the value of y is known to be caused by x. The true relationship between x and y is mediated by another factor, call it A, that takes values of +1 or -1 with equal probability. The true process relating x to y is y = Ax.
>It is a simple matter to show that the correlation between x and y is zero. Perhaps the most intuitive way is to imagine many samples (observations) of x, y pairs. Over the sub-sample for which the pairs have the same sign (i.e. for which A happened to be +1) y=x and the correlation is 1. Over the sub-sample for which the pairs have the opposite signs (i.e. for which A happened to be -1) y=-x and the correlation is -1. Since A is +1 and -1 with equal probability, the contributions to the total correlation from the two sub-samples cancel, giving a total correlation of zero.
It seems to me that this doesn't quite make sense. Sure, the correlation of the average of the numbers is 0, but notice that |x - y| <= |2x|, or that |y| = |x|. That seems like a rather large correlation to me, even though half the time, x and y are positively correlated, and the other half, they're negatively correlated.
>It is a simple matter to show that the correlation between x and y is zero. Perhaps the most intuitive way is to imagine many samples (observations) of x, y pairs. Over the sub-sample for which the pairs have the same sign (i.e. for which A happened to be +1) y=x and the correlation is 1. Over the sub-sample for which the pairs have the opposite signs (i.e. for which A happened to be -1) y=-x and the correlation is -1. Since A is +1 and -1 with equal probability, the contributions to the total correlation from the two sub-samples cancel, giving a total correlation of zero.
It seems to me that this doesn't quite make sense. Sure, the correlation of the average of the numbers is 0, but notice that |x - y| <= |2x|, or that |y| = |x|. That seems like a rather large correlation to me, even though half the time, x and y are positively correlated, and the other half, they're negatively correlated.
The submitted article links out to a more scholarly article
http://www.springerlink.com/content/l787673gxg8425g6/fulltex...
with diagrams about issues to consider in observational studies.
I always like to recommend Peter Norvig's article on interpreting research studies
http://norvig.com/experiment-design.html
and in the medical context can also recommend Harriet Hall's lecture notes
http://www.skepticstoolbox.org/hall/
as examples of popular writings on research study interpretation that give vivid examples and bring up important issues.
http://www.springerlink.com/content/l787673gxg8425g6/fulltex...
with diagrams about issues to consider in observational studies.
I always like to recommend Peter Norvig's article on interpreting research studies
http://norvig.com/experiment-design.html
and in the medical context can also recommend Harriet Hall's lecture notes
http://www.skepticstoolbox.org/hall/
as examples of popular writings on research study interpretation that give vivid examples and bring up important issues.
It would be wonderful if the first line were true.
"It is well known that correlation does not prove causation."
Causality itself is hypothetical, an artifact of perception.
Causality itself is hypothetical, an artifact of perception.
That's the subject of the next post on this topic, forthcoming.
Long story short: causation without correlation is possible if you are unable to adjust for confounding.
The post suggests more, though doesn't illustrate it all (because I couldn't think of a simple way to do so). The key is that x and y can be statistically related yet still have zero correlation (because that is just one specific measure of relatedness). Interpretations of causation are not data driven. They come from theory.
The role of confounded factors came up in the examples because that is just an easy way to illustrate the points.
The role of confounded factors came up in the examples because that is just an easy way to illustrate the points.