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From the Wikipedia page titled correlation does not imply causality,

For any two correlated events, A and B, the different possible relationships include:

  1. A causes B (direct causation);
  2. B causes A (reverse causation);
  3. A and B are consequences of a common cause, but do not cause each other;
  4. A and B both causes C, which is (explicitly or implicitly) conditioned on.;
  5. A causes B and B causes A (bidirectional or cyclic causation);
  6. A causes C which causes B (indirect causation);
  7. There is no connection between A and B; the correlation is a coincidence.

What does the fourth point mean. A and B both causes C, which is (explicitly or implicitly) conditioned on. If A and B cause C, why do A and B have to be correlated.

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"Conditioning" is a word from probability theory : https://en.wikipedia.org/wiki/Conditional_probability

Conditioning on C means that we are only looking at cases where C is true. "Implicitly" means that we may not be making this restriction explicit, sometimes not even aware of doing it.

The point means that, when A and B both cause C, observing a correlation between A and B in cases where C is true, does not mean there is a real relationship between A and B. It's just conditioning on C (maybe unwillingly) that creates an artificial correlation.

Let's take an example.

In a country there exists exactly two sorts of diseases, perfectly independent. Call A : "person has first disease", B : "person has second disease". Assume $P(A)=0.1$, $P(B)=0.1$.

Now any person who has one of these diseases goes to see the doctor and only then. Call C : "person goes to see the doctor". We have $C=A \text{ or } B$.

Now let's calculate a few probabilities :

  • $P(C)=0.19$
  • $P(A|C)=P(B|C)=\frac{0.1}{0.19}\approx 0.53$
  • $P(A \text{ and } B|C)=\frac{0.01}{0.19}\approx 0.053$
  • $P(A|C)P(B|C)\approx 0.28$

Clearly, when conditioned on C, $A$ and $B$ are very far from being independent. Actually, conditioned on C, $not A$ seems to "cause" $B$.

If you use the list of persons who where recorded by their doctor(s) as a data source for an analysis, then there seems to be a strong correlation between diseases $A$ and $B$. You may not be aware of the fact that your data source is actually a conditioning. This is also called a "selection bias".

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The fourth point is an example of Berkson's paradox, also known as conditioning on a collider, also known as the explaining-away phenomenon.

As an example, consider a young woman who is frequently asked on dates by young men, and she must decide whether to accept or reject each date proposal. The young men vary in how attractive and charming they are, and let's suppose that these two traits are independent in the population of date-proposing men. Naturally the young woman is more inclined to accept a date proposal the more attractive or charming the man is. So a causal model for this situation may look like: $$ Attractive \rightarrow Accept \leftarrow Charming $$ That is, $Attractive$ and $Charming$ both cause $Accept$, which takes on values of 0 or 1 if the woman rejects or accepts the date proposal, respectively.

We supposed above that $Attractive$ and $Charming$ are independent in the population of date-proposing men. But are they still independent if we consider only the men whose proposals the woman accepted? In other words, we condition on $Accept=1$. Now suppose I tell you about a man who the woman agreed to date, and I tell you that he is (in the woman's opinion) not attractive at all. Well, we know that the woman agreed to date him anyway, so we would reasonably infer that he must be quite charming indeed. Conversely, if we learn about a man whose date proposal was accepted and who is not charming, we would reasonably infer that he must be quite attractive.

Do you see what's happened here? By conditioning on $Accept=1$, we've induced a negative correlation between $Attractive$ and $Charming$, even though these two traits are (by assumption) marginally independent. From the perspective of the woman, the attractive men she dates tend to be less charming, and the charming men she dates tend to be less attractive. But this is because, by thinking only of the men she has dated, she is implicitly conditioning on $Accept$. If she would instead consider all the men who have proposed dates, regardless of whether she accepted the proposal, she would see that there is no statistical association between the two traits.

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Simpson's paradox and Berkson's paradox can each give examples of "A and B both cause C, which is (explicitly or implicitly) conditioned on"

As an example suppose I have $1000$ stamps in my collection of which $100$ are rare ($10\%$) and $200$ are pretty ($20\%$). If there is no intrinsic relationship between rarity and prettiness, it might turn out $20$ of my stamps are both pretty and rare.

If I now display my $280$ interesting stamps, i.e. those which are rare or pretty or both, there will be an apparent negative correlation between rarity and prettiness ($20\%$ of displayed rare stamps are pretty while $100\%$ of displayed common stamps are pretty) due entirely to conditioning on being interesting.

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  • $\begingroup$ This is an example Berkson's paradox, not Simpson's paradox (see my answer). $\endgroup$ – Jake Westfall Jun 2 '17 at 15:55
  • $\begingroup$ @JakeWestfall You are probably right - I knew I had written the stamps example before somewhere but forgotten where and it turns out to be the Wikipedia page for Berkson's paradox $\endgroup$ – Henry Jun 2 '17 at 21:25
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The paragraph starts with "For any two correlated events, A and B,...", so my guess is that correlation is assumed at the beginning. In other words, they need not be correlated to simultaneously cause C, but if they were correlated and they did both cause C, it does not imply that there exists a causal relationship between them.

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