Section 7.6 Correlation and Causation
If variables A and B are correlated, the apparent correlation might be due to random sampling, or it might be the result of non-representative sampling, or it might indicate a real correlation between quantities in the population.
If the correlation is real, there are three possible explanations: A causes B, or B causes A, or some other set of factors causes both A and B. These explanations are called "causal relationships".
Correlation alone does not distinguish between these explanations, so it does not tell you which ones are true. This rule is often summarized with the phrase "Correlation does not imply causation," which is so pithy it has its own Wikipedia page.
So what can you do to provide evidence of causation?
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Use time. If A comes before B, then A can cause B but not the other way around. The order of events can help us infer the direction of causation, but it does not preclude the possibility that something else causes both A and B.
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Use randomness. If you divide a large sample into two groups at random and compute the means of almost any variable, you expect the difference to be small. If the groups are nearly identical in all variables but A and B, you can rule out the possibility that something else causes both A and B.
These ideas are the motivation for the randomized controlled trial, in which subjects are assigned randomly to two (or more) groups: a treatment group that receives some kind of intervention, like a new medicine, and a control group that receives no intervention, or another treatment whose effects are known. A randomized controlled trial is the most reliable way to demonstrate a causal relationship, and the foundation of evidence-based medicine.
Unfortunately, controlled trials are sometimes impossible or unethical. An alternative is to look for a natural experiment, where similar groups are exposed to different conditions due to circumstances beyond the control of the experimenter.
Identifying and measuring causal relationships is the topic of a branch of statistics called causal inference.
