cupholder comments on Case study: abuse of frequentist statistics - Less Wrong
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Sorry. You are quite right, and I was sloppy. I had in mind the implicit idea that holding the choices of statistical test and data collection procedure constant, different p-values suggest how strongly one should reject the null hypothesis, and I should have made that explicit. It is absolutely true that if I just ask someone, "Test A gave me p = 0.008 and Test B gave me p = 0.4, which test's null hypothesis is worse off?", the correct answer is "how should I know?"
Yep. I think this is an example of the frequentist encapsulating what a Bayesian would call priors in their sampling assumptions.