Correlation Does Not Imply Causation

Source
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 this. This is a logical fallacy because there are at least five possibilities:

1. A may be the cause of B.
2. B may be the cause of A.
3. Some unknown third factor C may actually be the cause of both A and B.
4. 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.
5. 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.

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Fallacy Examples-

B causes A (reverse causation)
The more firemen fighting a fire, the bigger the fire is going to be.
Therefore firemen cause fire.

Third factor C (the common-causal variable) causes both A and B
Spurious relationship
Sleeping with one's shoes on is strongly correlated with waking up with a headache.
Therefore, sleeping with one's shoes on causes headache.

The above example commits the correlation-implies-causation fallacy, as it prematurely concludes that sleeping with one's shoes on causes headache. A more plausible explanation is that both are caused by a third factor, in this case alcohol intoxication, which thereby gives rise to a correlation.

A causes B and B causes A
Increased pressure results in increased temperature.
Therefore pressure causes temperature.

The ideal gas law, PV=nRT describes the direct relationship between pressure and temperature (along with other factors) to show that there is a direct correlation between the two properties. Given a fixed mass, an increase in temperature will cause an increase in pressure; likewise, increased pressure will cause an increase in temperature. This demonstrates (possiblility 4) in that the two are directly proportional to each other and not independent functions.

Coincidence
With a decrease in the number of pirates, there has been an increase in global warming over the same period.
Therefore, global warming is caused by a lack of pirates.

Since the 1950s, both the atmospheric CO2 level and crime levels have increased sharply.
Hence, atmospheric CO2 causes crime.