Approximate Objective Bayes Factors From P-Values and Sample Size: The 3p√n Rule

In 1936, Sir Harold Jeffreys proposed an approximate objective Bayes factor that quantifies the degree to which the point-null hypothesis H0 outpredicts the alternative hypothesis H1. This approximate Bayes factor (henceforth JAB01) depends only on sample size and on how many standard errors the maximum likelihood estimate is away from the point under test. We revisit JAB01 and introduce a piecewise transformation that clarifies the connection to the frequentist two-sided p-value. Specifically, if p ≤ .10 then JAB01 ≈ 3p√n; if .10 .50 then JAB01 ≈ p^(1/4)√n. These transformation rules present p-value practitioners with a straightforward opportunity to obtain Bayesian benefits such as the ability to monitor evidence as data accumulate without reaching a foregone conclusion. Using the JAB01 framework we derive simple and accurate approximate Bayes factors for the t-test, the binomial test, the comparison of two proportions, and the correlation test.