Summary: There are two distinct definitions of “\(P\)-value” for evaluating a proposed hypothesis or model for the process generating an observed dataset. The original definition starts with a measure of the divergence of the dataset from what was expected under the model, such as a sum of squares or a deviance statistic. A \(P\)-value is then the ordinal location of the measure in a reference distribution computed from the model and the data, and is treated as a unit-scaled index of compatibility between the data and the model. In the other definition, a \(P\)-value is a random variable on the unit interval whose realizations can be compared to a cutoff \(\alpha\) to generate a decision rule with known error rates under the model and specific alternatives. It is commonly assumed that realizations of such decision \(P\)-values always correspond to divergence \(P\)-values. But this need not be so: Decision \(P\)-values can violate intuitive single-sample coherence criteria where divergence \(P\)-values do not. It is thus argued that divergence and decision \(P\)-values should be carefully distinguished in teaching, and that divergence \(P\)-values are the relevant choice when the analysis goal is to summarize evidence rather than implement a decision rule.
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