Friday, November 8, 2013

If correlation isn't causation, then what is?

I've started to get a little tired of writing entirely about media shenanigans, and in all likelihood so, dear readers, have you tired of reading about them.  So today I'm going to provide one of the 'educational pieces' alluded to in this blog's description; specifically I'm going to start talking about causal inference, which is the driving force behind the research I'm lucky enough to be paid to do.

We hear a lot in statistics classes (if you're inclined to go to them) and places like this blog, that "correlation is not causation."  Packaged within this pithy but stern warning is some very sound advice: just because you observe a relationship between two things doesn't tell you anything about the mechanism (if any) which creates that relationship.

For example, you might not be surprised to learn that Guardian readers are, on average, better educated than the general UK population.  You are unlikely to conclude from this that reading the Guardian makes you better educated, because this is absurd (unless you have a rather narrow view of what 'educated' means).  Perhaps instead you imagine that better educated people are (generally) more interested in the things the Guardian has to say, or perhaps that people from a middle-class background are both more likely to be well educated and to read the Guardian.

We often perform this sort of inference unconsciously; in the real world we are confronted with patterns and associations all the time and, usually being unable to experiment, we must infer the mechanism just from observing.  In some respects we are remarkably good at this, but this is often because we have learned a lot about the world around us through experience.

It can also trip us up: for centuries in the Western world it was believed that bad smells caused disease (think of malaria's etymology).  This does not seem unreasonable, since there is a strong association between foul smelling material and illness, but the conclusion is quite incorrect.  In particular, it suggests that the appropriate remedy is to cover up the bad smell with a good one, rather than washing one's hands more often.  It wasn't until the late 19th century when Louis Pasteur and Robert Koch showed that the bad smell and the diseases were both caused by bacteria, that our modern understanding developed.

To help us understand both the importance and the difficulty of causal inference, I will present a little piece on the history of understanding the relationship between smoking and lung cancer.

Fisher's Cancer Gene

Sir Ronald Aylmer Fisher was a highly distinguished and brilliant statistician, and is responsible for the foundation of many modern statistical methods; if you've studied statistics you'll already know this, because he has quite a few important things named after him.  He was also a difficult and stubborn man who entertained feuds with other statisticians, and who rejected one of the greatest triumphs of modern epidemiology: proving that smoking causes lung cancer.

Fisher was himself a heavy smoker, and this may have influenced his views.  By the 1950s, large observational data sets were available to show that people who smoked had a much higher risk of developing lung cancer.  Given what we now know, and the seemingly obvious potential problems caused by putting foreign substances into the body, it seems difficult to understand how such seemingly strong evidence could be ignored (on the other hand, people used to sell radioactive toothpaste).

We will represent the idea that some quantity $X$ influences another $Y$ in a causal way using an arrow: $X \rightarrow Y$ (I will try to define this more precisely in another post, though doing so is not straightforward: just ask a philosopher).  The modern consensus view, then, is represented by this diagram:

$$\text{Smoking} \longrightarrow \text{Cancer}$$
However, Fisher argued that it was not possible to rule out other explanations for the association between smoking and lung cancer.  Was it not possible, for example, that people who are already in the process of developing lung cancer (though unaware of it) might be more likely to use cigarettes as a palliative treatment, in order to relieve some related irritation of the lungs?  In such a case the proper conclusion would be that the cancer causes the smoking.
$$\text{Smoking} \longleftarrow \text{Cancer}$$
Or perhaps there is some other factor which we have not considered, and which both makes smoking and lung cancer more likely.  Social factors are quite plausible in this context: perhaps people who live in the city are both more likely to smoke and also to inhale factory soot.  However, such hypotheses can be checked by gathering additional data.  Fisher himself proposed a genetic explanation: perhaps there is a gene which provides its carriers with a strong craving to smoke, and which also increases their risk of developing cancer.  That is to say:
$$\text{Smoking} \longleftarrow \text{Gene} \longrightarrow \text{Cancer}$$
The key difference between the three hypotheses is this: if we forced or convinced everyone to stop smoking, in the first example it would reduce the number of deaths from lung cancer, whereas in the other two it would make no difference whatever because the cause is something else.  (This gives a hint of how one might try to define causality more formally.)

Logically Fisher is correct, an observed association can be explained in any of these ways.  In order to be certain that smoking causes lung cancer, we would need to do a randomised controlled trial: simply take 1000 people, divide them into two groups at random.  Force one group to do 40-a-day, ban the rest from smoking at all, and wait 20 years to see what happens.  Since any other mechanism which might cause participants to smoke (or not) has been broken, any remaining association must be causal.  Sadly though, this experiment didn't make it past the ethics committee.

So how was Fisher proved wrong?  Ultimately it came down to the sheer weight of evidence: Fisher's genetic explanation is highly implausible, because it would require two genetic effects (one on cancer and one on smoking) to be absurdly strong; genetic effects can also be controlled for (to some extent) by studying people who are related.  The ethics committees of the past did allow animal experiments, which demonstrated that smoking in (for example) beagles certainly does cause lung cancer.  And external circumstances may be used as a proxy for randomised trials: a higher cigarette tax reduces the amount by which people smoke, but shouldn't have any other effect on lung cancer (this is called an 'instrument', beloved of economists in search of causal conclusions).

The last great hope for causality

Causal inference is difficult, but it is not impossible (even without randomised trials).  It is certainly try that any causal conclusions drawn from observational studies should be treated with great caution (here's a particularly terrible example from our old friend the DM, I couldn't resist).  But the defeatist (or rather obstructive) attitude of Fisher is not the answer.  Causal inference is a fast developing field which shows that it is possible to obtain evidence of causal effects even without ideal study designs, and there is a huge potential benefit for many fields in discovering how.

I hope to share some of those methods with you in future posts.


Stolley, P.D. - When Genius Errs: R. A. Fisher and the Lung Cancer Controversy, Am. J. Epidemiol. 133 (5): 416-425, 1991.


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