On the Relation between Likelihood Ratios and p -values for Testing Success Probabilities of Bernoulli Trials

It is well known that there is no direct one-to-one relation between p-values and likelihood ratios or Bayes factors, since their relation crucially involves the sample size n. We investigate their (asymptotic) relation in a coin-tossing context where the hypotheses of interest address the success probability of the coin, and where detailed computations are possible. This leads to useful insights in the nature of p-values and likelihood ratios. Our results imply, for instance, that under mild conditions, a p-value of 0.05 cannot correspond to a likelihood ratio larger than 7.5, for any hypothesis versus a null hypothesis that the success probability has a specific value. We also show it is unlikely one can obtain a large likelihood ratio by tossing a fair coin until the number of heads deviates from the mean by several standard deviations.