What is the difference between correlation and causation?
- Correlation is a descriptive relationship, all it says is: here are a bunch of measurements of two (or more variables) and there is a specific numerical relationship - that the correlation coefficient is above a certain threshold. Formally speaking correlation doesn't indicate anything more than that.
- Causation on the other hand is an explanatory relationship: It provides a logical relationship between the variables, which one should be able to generalize to cases beyond the set of measurements from which we drew the initial hypothesis. In your Pirates/Temperature graph case, one would have to provide a detailed explanation/mechanism of what is it exactly about pirates that kept global temperatures down? Was it their clothes? Was it their funny accents? Was it their obnoxious parrots? Then would should be able to generalize this mechanism to other cases. Let's suppose that on deeper analysis, I decide that it was their accents that were keeping global temperatures low. I could then generalize this by having normal non pirate people speak with a pirate accent, and measure the effect it has on global warming.
Causation is hard to define concisely, but here is a recent definition from Nancy Cartwright (See also this post):
C causes E if and only if C increases the probability of E in every situation which is otherwise causally homogeneous with respect to E. (Causal Laws and Effective Strategies, 423)
The key word here is "in every situation". Correlation doesn't involve every situation, only the data we have right now. Causation on the other hand involves every situation, including many not included in our current data. Notice the probabilistic definition, to allow for statements like "Smoking causes cancer" to be true even though we have met one or two heavy smokers who don't have cancer.
For example, how do we know when we're dealing with correlation only and not also causation here?
This is still an open question. Resolving it would have serious implications for philosophy of science (with regards to the demarcation problem and to the problem of induction).
To some extent, Karl Popper's falsificationism was an attempt to solve this problem. Before that, Logical positivists verificationism was problematic exactly because correlation doesn't imply causation, and one could verify all sorts of pseudo-scientific theories, provided they dug up enough suitably correlated data-points. However, falsification doesn't solve the problem completely.
Now given that two data sets A & B are correlated, lets look at the possibilities of why they are correlated:
- Their correlated is purely coincidental.
- A causes B.
- B causes A.
- A and B are both caused by a third variable C.
Falsification allows us to rule out cases like (1): In your pirate example, if one had a way of suddenly increasing the number of pirates, or managed to get everybody to starting speaking like a pirate for a few years, they would notice that after 10 or 20 years, the global average temperatures still didn't go down, so we can safely throw away the hypothesis that pirates have some causal relationship on global temperatures.
But that leaves us with (2), (3) and (4). For these, there are some heuristics for discerning between the possibilities, but no sure fire method.
- Time is a factor: In any real world physical case, cause always precedes effects, so if your data is timestamped with enough precision, you could use that as an indicator. This would allows to decide between (2) and (3), but still leaves us undecided between which ever one of those we select and (4).
- Generalization: If your explanatory theory explains only the two variables you have and nothing else, than you might want to dig deeper. A good theory should be able to explain other related questions, not just those posed by the data at hand. For example, although a causal relationship between pirates and climate might explain the data you have in your graph, it won't ever explain anything else. Explaining climate change using the concept of greenhouse gases also explains other phenomena, for example why greenhouses work, and why Venus is so much warmer than Earth.
- Does your explanation fit with other already existing theories? Is there a more general pattern that these theories follow? If pirates are linked to climate change, can we also link the decline in gladiators to water pollution?