Fiducialize statistical significance: transforming p-values into conservative posterior probabilities and Bayes factors

Abstract

One remedy to the misuse of p-values transforms them to bounds on Bayes factors. With a prior probability of the null hypothesis, such a bound gives a lower bound on the posterior probability. Unfortunately, knowing a posterior probability is above some number cannot ensure that the null hypothesis is improbable enough to warrant its rejection. For example, if the lower bound is 0.0001, that implies that the posterior probability is at least 0.0001 but does not imply it is lower than 0.05 or even 0.9. A fiducial argument suggests an alternative estimate of the posterior probability that the null hypothesis is true. In the case that the prior probability of the null hypothesis is 50%, the estimated posterior probability is about p ⁡ ( ln ⁡ p ) 2 for low p. In other cases, each occurrence of p in the formula is the p-value calibrated by multiplying it by the prior odds of the null hypothesis. In the absence of a prior, p ⁡ ( ln ⁡ p ) 2 also serves as an asymptotic Bayes factor. Since the fiducial estimate of the posterior probability is greater than the lower bounds, its use in place of a bound leads to more stringent hypothesis testing. Making that replacement in a rationale for 0.005 as the significance level reduces the level to 0.001.