Interval estimation, point estimation, and null hypothesis significance testing calibrated by an estimated posterior probability of the null hypothesis

Much of the blame for failed attempts to replicate reports of scientific findings has been placed on ubiquitous and persistent misinterpretations of the p value. An increasingly popular solution is to transform a two-sided p value to a lower bound on a Bayes factor. Another solution is to interpret a one-sided p value as an approximate posterior probability.

Combining the two solutions results in confidence intervals that are calibrated by an estimate of the posterior probability that the null hypothesis is true. The combination also provides a point estimate that is covered by the calibrated confidence interval at every level of confidence. Finally, the combination of solutions generates a two-sided p value that is calibrated by the estimate of the posterior probability of the null hypothesis. In the special case of a 50% prior probability of the null hypothesis and a simple lower bound on the Bayes factor, the calibrated two-sided p value is about (1 - abs(2.7 p ln p)) p + 2 abs(2.7 p ln p) for small p. The calibrations of confidence intervals, point estimates, and p values are proposed in an empirical Bayes framework without requiring multiple comparisons.