When correlation can be used to infer causation

It is not uncommon to find correlations in the behavior of two or more phenomena and such correlations are sometimes used to imply causation. One of the most common objections posed to such arguments is that ‘correlation does not imply causation’, and is one of the first things that people learn about statistics. Even if they have not studied the subject, many people know enough to able to bring up this objection. But people may be sometimes too quick to pull that trigger.

The objection is based on fact that A correlates with B does not mean that A causes B. It could also be that B causes A or that both A and B are caused by some third factor. What an observed correlation often does is trigger further experiments to tease out which of the three options is operational. But we often use correlations to imply causation anyway. For example, a few years ago, I noticed that eating broccoli was followed shortly after by my having heartburn. I inferred that the broccoli was the cause of the heartburn and eliminated it from my diet and the heartburn went away. My inference is definitely not conclusive. The heartburn could have been due to something else that also stopped around the same time I stopped eating broccoli, but this correlation was sufficient for me to stop altogether because, what the hell, it was just broccoli and giving it up was easy. What persuaded me was the temporal sequence. I got heartburn after eating the broccoli and that occurred enough times to be suggestive.

While that example was fairly clear cut in that I could conduct informal causal experiments by eating and not eating broccoli and seeing if the heartburn came or not, other cases are not as simple, especially when the relationship is more subtle. If the question is important enough and the implications serious enough, a real test of causality involving large scale, randomized, double-blind tests would need to be done. But sometimes, due to ethical and practical reasons, one cannot conduct such experiments and correlations may be all that one has.

In his highly cited 1965 presidential address The Environment and Disease: Association or Causation delivered to the Section of Occupational Medicine, Austin Bradford Hill, Professor Emeritus of Medical Statistics at the University of London, listed seven factors that can be used to infer causation when controlled experiments cannot be done. He was specifically looking at the important but difficult topic of the role of the environment in disease. I have listed the key parts of each indicator he cites but his paper goes into more detail of each.

(1) Strength: First upon my list I would put the strength of the association. To take a very old example, by comparing the occupations of patients with scrotal cancer with the occupations of patients presenting with other diseases, Percival Pott could reach a correct conclusion because of the enormous increase of scrotal cancer in the chimney sweeps. ‘Even as late as the second decade of the twentieth century’, writes Richard Doll (1964), ‘the mortality of chimney sweeps from scrotal cancer was some 200 times that of workers who were not specially exposed to tar or mineral oils and in the eighteenth century the relative difference is likely to have been much greater.’

(2) Consistency: Next on my list of features to be specially considered I would place the consistency of the observed association. Has it been repeatedly observed by different persons, in different places, circumstances and times?

(3) Specificity: One reason, needless to say, is the specificity of the association, the third characteristic which invariably we must consider. If as here, the association is limited to specific workers and to particular sites and types of disease and there is no association between the work and other modes of dying, then clearly that is a strong argument in favor of causation.

(4) Temporality: My fourth characteristic is the temporal relationship of the association – which is the cart and which is the horse? This is a question which might be particularly relevant with diseases of slow development. Does a particular diet lead to disease or do the early stages of the disease lead to those particular dietetic habits? Does a particular occupation or occupational environment promote infection by the tubercle bacillus or are the men and women who select that kind of work more liable to contract tuberculosis whatever the environment – or, indeed, have they already contracted it? This temporal problem may not arise often, but it certainly needs to be remembered, particularly with selective factors at work in the industry.

(5) Biological gradient: Fifthly, if the association is one which can reveal a biological gradient, or dose-response curve, then we should look most carefully for such evidence. For instance, the fact that the death rate from cancer of the lung rises linearly with the number of cigarettes smoked daily, adds a very great deal to the simpler evidence that cigarette smokers have a higher death rate than non-smokers. The comparison would be weakened, though not necessarily destroyed, if it depended upon, say, a much heavier death rate in light smokers and a lower rate in heavier smokers. We should then need to envisage some much more complex relationship to satisfy the cause and effect hypothesis. The clear dose-response curve admits of a simple explanation and obviously puts the case in a clearer light.

(6) Plausibility: It will be helpful if the causation we suspect is biologically plausible. But this is a feature I am convinced we cannot demand. What is biologically plausible depends upon the biological knowledge of the day.

To quote again from my Alfred Watson Memorial Lecture (Hill 1962), there was

‘…no biological knowledge to support (or to refute) Pott’s observation in the 18th century of the excess of cancer in chimney sweeps. It was lack of biological knowledge in the 19th that led to a prize essayist writing on the value and the fallacy of statistics to conclude, amongst other “absurd” associations, that “it could be no more ridiculous for the strange who passed the night in the steerage of an emigrant ship to ascribe the typhus, which he there contracted, to the vermin with which bodies of the sick might be infected.” And coming to nearer times, in the 20th century there was no biological knowledge to support the evidence against rubella.’

(7) Coherence: On the other hand the cause-and-effect interpretation of our data should not seriously conflict with the generally known facts of the natural history and biology of the disease – in the expression of the Advisory Committee to the Surgeon-General it should have coherence.

Those criteria should be borne in mind before we dismiss an argument purely on the grounds that it is based only on correlations.


  1. Pierce R. Butler says

    … I could conduct informal causal experiments by eating and not eating broccoli and seeing if the heartburn came …

    Or you could try 21st-century “research” with a search engine.

    (Spoiler: Not counting the ad at the top, the first two results say broccoli -> heartburn, the third mentions broccoli only in the headline[?], the fourth has as headline “Broccoli Battles Heartburn”. No wonder this century is such a mess already…).


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