The conventional dictum that "correlation does not imply causation" means that correlation cannot be used to infer a causal relationship between the variables. This dictum does not imply that correlations cannot indicate the potential existence of causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with identity relations (tautology) where no causal process exists. Consequently, establishing a correlation between two variables is not a sufficient condition to establish a causal relationship (in either direction). Many statistical tests calculate correlation between variables. A few go further and calculate the likelihood of a true causal relationship. Examples include the Granger causality test and convergent cross mapping.
The assumption that correlation proves causation is considered a "questionable cause logical fallacy," in that two events occurring together are taken to have 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. " Consider the following:
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.
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 caused a decrease in coronary heart disease, but not to the degree suggested by the epidemiological studies, the assumption of causality would have been correct, although the logic behind the assumption would still have been flawed.
General Pattern
For any two correlated events A and B, the following relationships are possible:
- A causes B;
- B causes A;
- A and B are consequences of a common cause, but do not cause each other;
- There is no connection between A and B; the correlation is coincidental.
Less clear-cut correlations are also possible. For example, causality is not necessarily one-way; in a predator-prey relationship, predator numbers affect prey, but prey numbers (e.g., food supply) also affect predators.
The cum hoc ergo propter hoc logical fallacy can be expressed as follows:
- A occurs in correlation with B.
- Therefore, A causes B.
In this type of logical fallacy, one makes a premature conclusion about causality after observing only a correlation between two or more factors. Generally, if one factor (A) is observed to only be correlated with another factor (B), it is sometimes taken for granted that A is causing B, even when no evidence supports it. This is a logical fallacy because there are at least five possibilities:
- A may be the cause of B.
- B may be the cause of A.
- Some unknown third factor C may actually be the cause of both A and B.
- There may be a combination of the above three relationships. For example, B may be the cause of A at the same time as A is the cause of B (contradicting that the only relationship between A and B is that A causes B). This describes a self-reinforcing system.
- The "relationship" is a coincidence or so complex or indirect that it is more effectively called a coincidence (i.e., two events occurring at the same time that have no direct relationship to each other besides the fact that they are occurring at the same time). A larger sample size helps to reduce the chance of a coincidence, unless there is a systematic error in the experiment.
In other words, there can be no conclusion made regarding the existence or the direction of a cause and effect relationship only from the fact that A and B are correlated. Determining whether there is an actual cause and effect relationship requires further investigation, even when the relationship between A and B is statistically significant, a large effect size is observed, or a large part of the variance is explained.
Greenhouse Effect
The greenhouse effect is a well-known cause-and-effect relationship. While well-established, this relationship is still susceptible to logical fallacy due to the complexity of the system.