What is the right way to use p-values?

P-values have been blamed for much of the replication crisis. Proposals range from lowering thresholds (Benjamin et al.'s 'p < 0.005'), to replacing p-values with Bayes factors, to abandoning significance testing altogether (Amrhein et al.).

How to communicate statistical evidence cleanly without rituals that distort inference is a methodological challenge with cultural as well as mathematical dimensions.