Correlation does not imply causation  

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Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.
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Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.

In statistics, many statistical tests calculate correlations between variables and when two variables are found to be correlated, it is tempting to assume that this shows that one variable causes the other. That "correlation proves causation" is considered a questionable cause logical fallacy when two events occurring together are taken to have established a cause-and-effect relationship. This fallacy is also known as cum hoc ergo propter hoc, Latin for "with this, therefore because of this", and "false cause". A similar fallacy, that an event that followed another was necessarily a consequence of the first event, is the post hoc ergo propter hoc (Latin for "after this, therefore because of this.") fallacy.

For example, in a widely studied case, numerous epidemiological studies showed that women 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 (ABC1), 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 a direct cause and effect, as had been supposed.

As with any logical fallacy, identifying that the reasoning behind an argument is flawed does not imply that the resulting conclusion is false. In the instance above, if the trials had found that hormone replacement therapy does in fact have a negative incidence on the likelihood of coronary heart disease the assumption of causality would have been correct, although the logic behind the assumption would still have been flawed. Indeed, a few go further, using correlation as a basis for testing a hypothesis to try to establish a true causal relationship; examples are the Granger causality test, convergent cross mapping, and Liang-Kleeman information flow.

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