Q&A: Correlation and Causality
Correlation and Causality
Question
I know, of course, that causality cannot be inferred from correlation. My question is whether causality can be ruled out—not at the level of certainty, but at the level of probability—on the basis of a correlation that is not as high as expected. I’ll use an example to explain, one that came up on the site in the past.
There are those who claim there is a correlation between education and heresy / disbelief in God. If I were to show, for example, that the correlation between high socio-economic status and disbelief is much stronger, and there is also a correlation between education and socio-economic status, then it would be more likely that the causal factor is socio-economic status, and the correlation with education would be explained by the fact that socio-economic status is related to both.
That is—because we found that the correlation is weaker than what we see in other correlations, and it can be explained without causality—we can say that it is unlikely that there is causality here.
B. I’m asking this because as part of my research I examine correlations between the expression of different genes in different cells, and when you examine tens of thousands of genes there are thousands of correlations, and you have to separate the wheat from the chaff. The question is whether there are existing models that you know of which do manage to predict which correlations may represent causality with high probability and which are less likely to, because they can be explained on the basis of other, stronger correlations. I’d be happy if you could point me to further information on the subject.
(By the way, thanks for the recent columns on causality.)
Answer
I don’t think the strength of the correlation is a good measure of causality. For example, in Pearl’s example with the two light bulbs (column 461), the strength of the correlation is identical (=1) in both cases. And even if there is a low correlation, that doesn’t mean there is no causality. For example, clouds are linked to rain with a correlation that is not especially high, but it is clear that the clouds are the cause of the rain, whereas the correlation between the switch and the bulb in the second case is not causal.
The correlation between smoking and cancer is fairly low, but smoking is still the cause of the cancer. By contrast, the correlation between cancer and chemotherapy can be high, and yet the treatment is not the cause of the cancer (though here the cancer is the cause of the treatment).
In the next column I will explain that a causal relation is a sufficient but not a necessary condition, and that means that although when there is a cause there is an effect, the effect can occur without the cause. And the meaning of this is that the correlation in a causal relation can be as low as you like. That is also the meaning of the dependence on circumstances that I noted in the column here.
Discussion on Answer
It’s hard to answer questions about suspicions. Clearly there is no necessity here. I’m not at all sure that it’s even more likely, so long as both correlations are statistically significant (outside the margin of error), of course.
In an explosion with a spherical shell expanding outward from the center as a result of the explosion, you can find all the correlations.
In other words, causality has no meaning apart from the laws of physics. And if you do find meaning, it’s because you assumed what had to be proved.
In absolute terms, I completely agree that you can’t infer causality from the strength of a correlation. You can have 1 without causality, and almost 0 with it.
What I’m asking is in relative terms: if we continue with the issue of smoking and cancer, then in that context they also found a correlation between black coffee and cancer. Let’s assume we don’t know whether smoking causes cancer or black coffee does, but we do know that a cigarette and a cup of coffee go very well together.
In that case, can’t we say that we should suspect the causal factor to be the one with the stronger correlation to cancer (the working assumption being that one should not assume causality unnecessarily, and since the weaker correlation can also be explained without causality, there is no reason to suspect causality)?
Thanks