Calibrating the evidence in experiments with applications to meta-analysis

How much more evidence is there in a ‘highly significant’ p-value of 0.01 relative to one ‘just significant’ at 0.05? Why does the replication of an experiment lead, on average, to a higher p-value than the one just obtained? To answer such questions one must go beyond the traditional p-value which is conditional on the data and thus interpretable only in the context of the experiment just performed. One can achieve this by considering the random p-value which has a uniform distribution under the null hypothesis but a highly skewed distribution under alternative hypotheses.