‘Correlation does not imply causation‘ is a phrase used in science and statistics to emphasize that a relationship between two variables does not automatically imply that one causes the other. The opposite belief, ‘correlation proves causation,’ is one of several questionable cause logical fallacies by which two events that occur together are claimed to have a cause-and-effect relationship.

The fallacy is also known as ‘cum hoc ergo propter hoc’ (Latin for ‘with this, therefore because of this’) and ‘false cause.’ It is a common fallacy in which it is assumed that, because two things or events occur together, one must be the cause of the other. By contrast, the fallacy, ‘post hoc ergo propter hoc,’ requires that one event occur after the other, and so may be considered a related fallacy.

In a widely studied example, numerous epidemiological studies showed that women who were taking combined hormone replacement therapy (HRT) also had a lower-than-average incidence of coronary heart disease (CHD), leading doctors to propose that HRT was protective against CHD. But randomized controlled trials showed that HRT caused a small but statistically significant increase in risk of CHD. Re-analysis of the data from the epidemiological studies showed that women undertaking HRT were more likely to be from higher socio-economic groups, with better than average diet and exercise regimens. The use of HRT and decreased incidence of coronary heart disease were coincident effects of a common cause (i.e. the benefits associated with a higher socioeconomic status), rather than cause and effect, as had been supposed.

In logic, the technical use of the word ‘implies’ means ‘to be a sufficient circumstance.’ However, in casual use, the word loosely means ‘suggests’ rather than ‘requires.’ The idea that correlation and causation are connected is certainly true; where there is causation, there is likely to be correlation. Indeed, correlation is used when inferring causation; the important point is that such inferences are not always correct because there are other possibilities. Edward Tufte, in a criticism of the brevity of ‘correlation does not imply causation,’ deprecates the use of ‘is’ to relate correlation and causation (as in ‘Correlation is not causation’), citing its inaccuracy as incomplete.While it is not the case that correlation is causation, simply stating their nonequivalence omits information about their relationship. Tufte suggests that the shortest true statement that can be made about causality and correlation is one of the following: ‘Empirically observed covariation is a necessary but not sufficient condition for causality’ or ‘Correlation is not causation but it sure is a hint.’

There is a relation between this subject-matter and the ‘Ecological fallacy’ (committed when a correlation observed at the population level is assumed to apply at the individual level), described in a 1950 paper by William S. Robinson. Robinson shows that ecological correlations, where the statistical object is a group of persons (i.e. an ethnic group), does not show the same behavior as individual correlations, where the objects of inquiry are individuals: ‘The relation between ecological and individual correlations which is discussed in this paper provides a definite answer as to whether ecological correlations can validly be used as substitutes for individual correlations. They cannot.’ ‘(a)n ecological correlation is almost certainly not equal to its corresponding individual correlation.’

Philosopher David Hume argued that causality is based on experience, and experience similarly based on the assumption that the future models the past, which in turn can only be based on experience – leading to circular logic. In conclusion, he asserted that causality is not based on actual reasoning: only correlation can actually be perceived. In order for a correlation to be established as causal, the cause and the effect must be connected through an impact mechanism in accordance with known laws of nature.

Intuitively, causation seems to require not just a correlation, but a counterfactual dependence. Suppose that a student performed poorly on a test and guesses that the cause was his not studying. To prove this, one thinks of the counterfactual – the same student writing the same test under the same circumstances but having studied the night before. If one could rewind history, and change only one small thing (making the student study for the exam), then causation could be observed (by comparing outcomes). Because one cannot rewind history and replay events after making small controlled changes, causation can only be inferred, never exactly known. This is referred to as the ‘Fundamental Problem of Causal Inference’ – it is impossible to directly observe causal effects.

A major goal of scientific experiments and statistical methods is to approximate as best as possible the counterfactual state of the world. For example, one could run an experiment on identical twins who were known to consistently get the same grades on their tests. One twin is sent to study for six hours while the other is sent to the amusement park. If their test scores suddenly diverged by a large degree, this would be strong evidence that studying (or going to the amusement park) had a causal effect on test scores. In this case, correlation between studying and test scores would almost certainly imply causation. Well-designed experimental studies replace equality of individuals as in the previous example by equality of groups. This is achieved by randomization of the subjects to two or more groups. Although not a perfect system, the likeliness of being equal in all aspects rises with the number of subjects placed randomly in the treatment/placebo groups. From the significance of the difference of the effect of the treatment vs. the placebo, one can conclude the likeliness of the treatment having a causal effect on the disease.

When experimental studies are impossible and only pre-existing data are available, as is usually the case for example in economics, ‘regression analysis’ (a statistical tool to show in what way a number of independent variables influence a dependent variable) can be used. Factors other than the potential causative variable of interest are controlled for by including them as ‘regressors’ (independent variables) in addition to the regressor representing the variable of interest. False inferences of causation due to reverse causation (or wrong estimates of the magnitude of causation due the presence of bidirectional causation) can be avoided by using explanators (regressors) that are necessarily exogenous, such as physical explanators like rainfall amount (as a determinant of, say, futures prices), lagged variables whose values were determined before the dependent variable’s value was determined, instrumental variables for the explanators (chosen based on their known exogeneity), etc. Spurious correlation due to mutual influence from a third, common, causative variable, is harder to avoid: the model must be specified such that there is a theoretical reason to believe that no such underlying causative variable has been omitted from the model; in particular, underlying time trends of both the dependent variable and the independent (potentially causative) variable must be controlled for by including time as another independent variable.

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