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gwern comments on Living Forever is Hard, or, The Gompertz Curve - Less Wrong
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Yes, I see you did use the word "proven". That does lesson the force of my comment but not by very much. I'm curious actually what you mean by proven outside a mathematical context. In this context, the correlation is extremely strong, and there's no obvious alternate hypothesis. I'm not completely sure what you consider in this context to be an acceptable amount of evidence, but the medical consensus seems clear, and the links Gwern gave show that the correlation exists even when one tries to control for other variables.
I think he wants randomized controlled trials. Which of course no ethics board would ever approve because avoiding aspirin in kids is not that hard or expensive, and the cost of confirming it would be too high.
("You want to test whether slamming your face into the wall causes pain, and doesn't just correlate with it? Why on earth?")
We have a plausible mechanism (confirmed by massive amounts of science) by which slamming your face into the wall causes pain; for the link between Aspirin and Reye's syndrome there is no such plausible mechanism.