P values for composite null models

The problem of investigating compatibility of an assumed model with the dat a is investigated in the situation when the assumed model has unknown param eters. The most frequently used measures of compatibility are p values, bas ed on statistics T for which large values are deemed to indicate incompatib ility of the data and the model. When the null model has unknown parameters . ?, values are not uniquely defined. The proposals for computing a p value in such a situation include the plug-in and similar p values on the freque ntist side, and the predictive and posterior predictive p values on the Bay esian side. We propose two alternatives, the conditional predictive p value and the partial posterior predictive p value, and indicate their advantage s from both Bayesian and frequentist perspectives.