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A more general lesson is that correlation and causation are unrelated: the former doesnt imply the latter, and the latter does not imply the former. Just becaus

A more general lesson is that correlation and causation are unrelated: the former doesnt imply the latter, and the latter does not imply the former. Just because one thing causes another does not mean it will be correlated with it. There is no contradiction in subsets having different correlations that the parent set. The apparent "paradox" arises from reading the data causally. The purpose of this lesson is to expose these assumptions in interpretation of data. Few seem to get the message though.

weinzierl 2y ago
If anyone has doubts about the second claim, think about a hash function. The input certainly causes the output, but they are not correlated in a statistical sense.
- mjburgess 2y ago
  Consider a medicine which kills everything with kidneys. It perfectly correlates with killing everything with a liver. Consider another medicine which kills everything with kidneys, unless they have a liver (eg., which filters it). Now there is no correlation at all with an effect on the kidneys, nor will there ever be (since all animals with one have the other) unless someone deliberately impairs a liver.
  - Anotheroneagain 2y ago
    Consider a medicine that completely cures Alzheimer's. It also necessarily increases the incidence of other causes of death, as those who won't die of Alzheimer's will die of something else instead.
kqr 2y ago
> A more general lesson is that correlation and causation are unrelated This is a bit extreme. The tongue-in-cheek variant I like (which I first read about in the book referenced by TFA) is "no correlation without causation". In order for two things to truly co-vary (and not just by accident, or as a consequence of poor data collection/manipulation), there needs to be some causal connection between the two – although it can be quite distant.
- mjburgess 2y ago
  > there needs to be some causal connection between the two Umm.. No, there doesn't... This idea features in the earlist 20th C. writings on statistics, but it's pseudoscience. If one carves up the whole history of the entire universe into all possible events, then there's likely to be a (near) infinite number of pairs of events which "perfectly" co-vary without any causal connection whatsoever. Indeed, one could find two galaxies that are necessarily causally isolated and find correlated events. This is, in part, because the properties of two casually independent systems can have indistinguishable distributions -- just by the nature of what a distribution is. It's this sort of thinking that I was aiming to rule out: really they have nothing to do with each other. It's early 20th C. frequentist pseudoscience that has given birth to this supposed connection, and it should be thrown out all together. Causation is a property of natural systems. Correlation is a property of two distributions. These have nothing to do with each other. If you want to "test" for causation, you need to have a causal theory and a causal analysis in which "correlation" shouldn't feature. If you induce correlation by causal intervention, i'd prefer we gave that a different name ("induced correlation") which is relevant to causation -- and it's mostly this confusion which those early eugenticists that created statistics were talking about.
  - Izkata 2y ago
    Examples: https://www.tylervigen.com/spurious-correlations https://www.tylervigen.com/spurious-correlations
  - kqr 2y ago
    > If one carves up the whole history of the entire universe into all possible events, then there's likely to be a (near) infinite number of pairs of events which "perfectly" co-vary without any causal connection whatsoever. But if they are not linked by a stable causal connection, wouldn't they eventually diverge, if we observe long enough?
    - godelski 2y ago
      There's subtleties in what you two are saying that I think are leading to miscommunication. I think it is better to think about this through mutual information rather than "correlation"[0], adding DAGs (directed acrylic graphs) also helps but are hard to draw here. If causation exists between A and B, the two must also have mutual information. This is more akin to the vernacular form of "correlation" which is how I believe you are using it. But statisticians are annoying and restrict "correlation" to be linear. In that case, no, causation does not necessitate nor imply linear correlation (/association). For mjburgess's universe example, I think it may depend on a matter of interpretation as to what is being considered causal here. A trivial rejection is that causation is through physics (they both follow the same physics) so that's probably not what was meant. I also don't really like the example because there's a lot of potential complexity that can lead to confusion[1], but let's think about the DAG. Certainly tracing causality back both galaxies converge to a single node (at worst, the Big Bang), right? They all follow physics. So both have mutual information to that node. *BUT* this does not mean that there is an arrow pointing from one branch to the other branch. Meaning that they do not influence one another and are thus not causally related (despite having shared causal "history", if you will). Maybe let's think of a different bland example. Suppose we have a function f(x) which outputs a truly random discrete outputs that are either 0 or 1 (no bias). Now we consider all possible inputs. Does there exist an f(a) = f(b) where a ≠ b? I think with this example we can see believe this is true but you can prove it if you wish. We can even believe that there is a stronger condition of a having no mutual information between a and b. In the same way here, if we tracked the "origin" of f(a) and f(b) we would have to come through f (f "causes" f(a) and f(b)), but a and b do not need to be constructed in any way that relates to one another. We can even complexity this example further by considering a different arbitrary function g which has a discrete output of [-1,0,1], or some other arbitrary (even same) output, and follow the same process. When doing that, we see no "choke point" and we could even pull a and b from two unrelated sets. So everything is entirely disjoint. Try other variations to add more clarity. [0] I also corrected mjburgess through this too because a subtle misunderstanding led to a stronger statement which was erroneous https://news.ycombinator.com/item?id=41228512 https://news.ycombinator.com/item?id=41228512 [1] Not only the physics part but we now have to also consider light cones and what physicists mean by causation
godelski 2y ago
> correlation and causation are unrelated This is incorrect (but what followed is correct). You have extended the meaning of the phrase "correlation does not imply causation" to a stronger case[0]. The correct way to say this is that "correlation are not necessarily related." The other way you might determine this was wrong is that ,,association''[2] always occurs when there is causation. So we have the classic A ⇒ B ⇏ B ⇒ A (A implies B does not imply B implies A), where ordering matters. Last, we should reference Judea Pearl's Ladder of Causality[1]. [0] Another similar example was given to us by Rumsfield with respect to the Iraq WMD search. Where the error was changing "the absence of proof is not proof of absence" to the much stronger "the absence of evidence is not evidence of absence". It also illustrates why we might want to "nitpick" here https://archive.is/20140823194745/http://logbase2.blogspot.com/2009/01/absence-of-evidence.html https://archive.is/20140823194745/http://logbase2.blogspot.c... [1] https://web.cs.ucla.edu/~kaoru/3-layer-causal-hierarchy.pdf https://web.cs.ucla.edu/~kaoru/3-layer-causal-hierarchy.pdf [2] Edit for clarity: The reason I (and Pearl) use the word "association" rather than "correlation" is because in statistics "correlation" often refers to linear relationship. So association clarifies that there is mutual information. There might be masked relationships, so non-linear. But if we are to use the standard vernacular of "correlation" (what most people think) then we could correctly say "causation implies correlation" (or more accurately, "causation implies correlation, but not necessarily linear correlation."). And of course, causation implies high mutual information, but high mutual information does not imply causation :) https://stats.stackexchange.com/questions/26300/does-causation-imply-correlation https://stats.stackexchange.com/questions/26300/does-causati...
- mjburgess 2y ago
  > ,association''[2] always occurs when there is causation This is incorrect. See Perl's work itself. Association does not occur when there is a collider. https://en.wikipedia.org/wiki/Collider_(statistics) https://en.wikipedia.org/wiki/Collider_(statistics) Since almost all variables we are measuring are on uncontrolled environments, in almost all cases, there is an opportunity to observe no association with causation. I give an example of this above: > Consider another medicine which kills everything with kidneys, unless they have a liver (eg., which filters it). Now there is no correlation at all with an effect on the kidneys, nor will there ever be (since all animals with one have the other) unless someone deliberately impairs a liver.

weinzierl 2y ago

If anyone has doubts about the second claim, think about a hash function. The input certainly causes the output, but they are not correlated in a statistical sense.

mjburgess 2y ago
Consider a medicine which kills everything with kidneys. It perfectly correlates with killing everything with a liver. Consider another medicine which kills everything with kidneys, unless they have a liver (eg., which filters it). Now there is no correlation at all with an effect on the kidneys, nor will there ever be (since all animals with one have the other) unless someone deliberately impairs a liver.
- Anotheroneagain 2y ago
  Consider a medicine that completely cures Alzheimer's. It also necessarily increases the incidence of other causes of death, as those who won't die of Alzheimer's will die of something else instead.

kqr 2y ago

> A more general lesson is that correlation and causation are unrelated This is a bit extreme. The tongue-in-cheek variant I like (which I first read about in the book referenced by TFA) is "no correlation without causation". In order for two things to truly co-vary (and not just by accident, or as a consequence of poor data collection/manipulation), there needs to be some causal connection between the two – although it can be quite distant.

mjburgess 2y ago
> there needs to be some causal connection between the two Umm.. No, there doesn't... This idea features in the earlist 20th C. writings on statistics, but it's pseudoscience. If one carves up the whole history of the entire universe into all possible events, then there's likely to be a (near) infinite number of pairs of events which "perfectly" co-vary without any causal connection whatsoever. Indeed, one could find two galaxies that are necessarily causally isolated and find correlated events. This is, in part, because the properties of two casually independent systems can have indistinguishable distributions -- just by the nature of what a distribution is. It's this sort of thinking that I was aiming to rule out: really they have nothing to do with each other. It's early 20th C. frequentist pseudoscience that has given birth to this supposed connection, and it should be thrown out all together. Causation is a property of natural systems. Correlation is a property of two distributions. These have nothing to do with each other. If you want to "test" for causation, you need to have a causal theory and a causal analysis in which "correlation" shouldn't feature. If you induce correlation by causal intervention, i'd prefer we gave that a different name ("induced correlation") which is relevant to causation -- and it's mostly this confusion which those early eugenticists that created statistics were talking about.
- Izkata 2y ago
  Examples: https://www.tylervigen.com/spurious-correlations https://www.tylervigen.com/spurious-correlations
- kqr 2y ago
  > If one carves up the whole history of the entire universe into all possible events, then there's likely to be a (near) infinite number of pairs of events which "perfectly" co-vary without any causal connection whatsoever. But if they are not linked by a stable causal connection, wouldn't they eventually diverge, if we observe long enough?
  - godelski 2y ago
    There's subtleties in what you two are saying that I think are leading to miscommunication. I think it is better to think about this through mutual information rather than "correlation"[0], adding DAGs (directed acrylic graphs) also helps but are hard to draw here. If causation exists between A and B, the two must also have mutual information. This is more akin to the vernacular form of "correlation" which is how I believe you are using it. But statisticians are annoying and restrict "correlation" to be linear. In that case, no, causation does not necessitate nor imply linear correlation (/association). For mjburgess's universe example, I think it may depend on a matter of interpretation as to what is being considered causal here. A trivial rejection is that causation is through physics (they both follow the same physics) so that's probably not what was meant. I also don't really like the example because there's a lot of potential complexity that can lead to confusion[1], but let's think about the DAG. Certainly tracing causality back both galaxies converge to a single node (at worst, the Big Bang), right? They all follow physics. So both have mutual information to that node. *BUT* this does not mean that there is an arrow pointing from one branch to the other branch. Meaning that they do not influence one another and are thus not causally related (despite having shared causal "history", if you will). Maybe let's think of a different bland example. Suppose we have a function f(x) which outputs a truly random discrete outputs that are either 0 or 1 (no bias). Now we consider all possible inputs. Does there exist an f(a) = f(b) where a ≠ b? I think with this example we can see believe this is true but you can prove it if you wish. We can even believe that there is a stronger condition of a having no mutual information between a and b. In the same way here, if we tracked the "origin" of f(a) and f(b) we would have to come through f (f "causes" f(a) and f(b)), but a and b do not need to be constructed in any way that relates to one another. We can even complexity this example further by considering a different arbitrary function g which has a discrete output of [-1,0,1], or some other arbitrary (even same) output, and follow the same process. When doing that, we see no "choke point" and we could even pull a and b from two unrelated sets. So everything is entirely disjoint. Try other variations to add more clarity. [0] I also corrected mjburgess through this too because a subtle misunderstanding led to a stronger statement which was erroneous https://news.ycombinator.com/item?id=41228512 https://news.ycombinator.com/item?id=41228512 [1] Not only the physics part but we now have to also consider light cones and what physicists mean by causation

godelski 2y ago

> correlation and causation are unrelated This is incorrect (but what followed is correct). You have extended the meaning of the phrase "correlation does not imply causation" to a stronger case[0]. The correct way to say this is that "correlation are not necessarily related." The other way you might determine this was wrong is that ,,association''[2] always occurs when there is causation. So we have the classic A ⇒ B ⇏ B ⇒ A (A implies B does not imply B implies A), where ordering matters. Last, we should reference Judea Pearl's Ladder of Causality[1]. [0] Another similar example was given to us by Rumsfield with respect to the Iraq WMD search. Where the error was changing "the absence of proof is not proof of absence" to the much stronger "the absence of evidence is not evidence of absence". It also illustrates why we might want to "nitpick" here https://archive.is/20140823194745/http://logbase2.blogspot.com/2009/01/absence-of-evidence.html https://archive.is/20140823194745/http://logbase2.blogspot.c... [1] https://web.cs.ucla.edu/~kaoru/3-layer-causal-hierarchy.pdf https://web.cs.ucla.edu/~kaoru/3-layer-causal-hierarchy.pdf [2] Edit for clarity: The reason I (and Pearl) use the word "association" rather than "correlation" is because in statistics "correlation" often refers to linear relationship. So association clarifies that there is mutual information. There might be masked relationships, so non-linear. But if we are to use the standard vernacular of "correlation" (what most people think) then we could correctly say "causation implies correlation" (or more accurately, "causation implies correlation, but not necessarily linear correlation."). And of course, causation implies high mutual information, but high mutual information does not imply causation :) https://stats.stackexchange.com/questions/26300/does-causation-imply-correlation https://stats.stackexchange.com/questions/26300/does-causati...

mjburgess 2y ago
> ,association''[2] always occurs when there is causation This is incorrect. See Perl's work itself. Association does not occur when there is a collider. https://en.wikipedia.org/wiki/Collider_(statistics) https://en.wikipedia.org/wiki/Collider_(statistics) Since almost all variables we are measuring are on uncontrolled environments, in almost all cases, there is an opportunity to observe no association with causation. I give an example of this above: > Consider another medicine which kills everything with kidneys, unless they have a liver (eg., which filters it). Now there is no correlation at all with an effect on the kidneys, nor will there ever be (since all animals with one have the other) unless someone deliberately impairs a liver.