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Salemicus comments on Taking "correlation does not imply causation" back from the internet - Less Wrong Discussion
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There's also e): A causes B within our sample, but A does not cause B generally, or in the sense that we care about.
For example, suppose a teacher gives out a gold star whenever a pupil does a good piece of work, and this causes the pupil to work harder. Suppose also that this effect is greatest on mediocre pupils and least on the best pupils - but the best pupils get most of the gold stars, naturally.
Now suppose an educational researcher observes the class, and notes the correlation between receiving a gold star, and increased effort. This is genuine causation. He then concludes that the teacher should give out more gold stars, regardless of whether the pupil does a good piece of work or not, and focus the stars on mediocre pupils. This change made, the gold stars no longer cause increased effort. The causation disappears! Changing the way the teacher hands out the gold stars changes the relationship between gold stars and effort. So although there was genuine causation in the original sample, there is no general causation, or causation in the sense we care about; we can't treat the gold stars as an exogenous variable.
See also the Lucas Critique.
That's because you have cause and effect reversed: The extra effort causes the gold stars, not the other way around.
No, the gold stars cause extra effort after they are given out. This is part of the hypothetical.
The pupils work harder after they are given a gold star because they see their good work is appreciated. But if the gold stars are given out willy-nilly, then the pupils no longer feel proud to get one, and so they lose their ability to make students work harder.
As Robert Lucas would put it, the relationship is not robust to changes in the policy regime.