Extracting Bayesian Evidence from Frequentist p-Values

The $p$-value and the Bayes factor are measures of evidence that are often considered to be philosophically and mathematically incompatible: The $p$-value quantifies conflict between data and $H_0$ ("surprise"), whereas the Bayes factor quantifies the relative predictive accuracy of $H_0$ versus $H_1$ ("evidence"). We revisit Jeffreys's Approximate Bayes factor (JAB) -- a simple, largely overlooked approximation dating back to the 1930s -- which connects these two paradigms for objective hypothesis testing of the existence of an effect. Under a unit-information prior the approximation requires only the $p$-value and the effective sample size $n_\text{eff}$. We clarify the core assumptions and boundary conditions for the application of JAB and show across 704 published $t$-tests and 39 comparisons of proportions that JAB approximates objective Bayes factors remarkably well. The connection between $p$-values and JAB has a practical implication: The evidence implied by a $p$-value depends strongly on $n_\text{eff}$. Conventional verbal labels for $p$-values (e.g., "strong surprise" for .001 < $p$ < .01) correspond to similarly graded Bayes factors only around $n_\text{eff} \approx 8$; for larger samples the same $p$-value implies weaker evidence. In moderately sized to large samples, $p > .10$ can amount to moderate or even strong evidence for $H_0$. JAB offers a cheap, sample-size-sensitive supplement to $p$-values, computable from routinely reported statistics, that remains valid even under optional stopping.