p-values for model evaluation

Deciding whether a model provides a good description of data is often based on a goodness-of-fit criterion summarized by a p-value. Although there is considerable confusion concerning the meaning of p-values, leading to their misuse, they are nevertheless of practical importance in common data analysis tasks. We motivate their application using a Bayesian argumentation. We then describe commonly and less commonly known discrepancy variables and how they are used to define p-values. The distribution of these are then extracted for examples modeled on typical data analysis tasks, and comments on their usefulness for determining goodness-of-fit are given.