Are P-values and Bayes factors valid measures of evidential strength?

P-values and Bayes factors are commonly used as measures of the evidential strength of the data collected in hypothesis tests. It is not clear, however, that they are valid measures of that evidential strength; that is, whether they have the properties that we intuitively expect a measure of evidential strength to have. I argue here that measures of evidential strength should be stochastically ordered by both the effect size and the sample size. I consider the case that the data are normally distributed and show that, for that case, P-values are valid measures of evidential strength while Bayes factors are not. Specifically, I show that in a sharp Null hypothesis test the Bayes factor is stochastically ordered by the sample size only if the effect size or the sample size is sufficiently large. This lack of stochastic ordering lies at the root of the Jeffreys-Lindley paradox.