Hierarchical models defined by means of directed, acyclic graphs are a power- ful and widely used tool for Bayesian analysis of problems of varying degrees of complexity. A simulation based method for model criticism in such models has been suggested by O'Hagan in the form of a con ict measure based on contrasting separate local information sources about each node in the graph. This measure is however not well calibrated. In order to rectify this, alter- native mutually similar tail probability based measures have been proposed independently, and have been proved to be uniformly distributed under the assumed model in quite general normal models with known covariance matri- ces. In the present paper, exploiting the property of pivotality, we extend this result to a variety of models. An advantage of this is that computationally costly pre-calibration schemes needed for some other suggested methods can be avoided. Another advantage is that non-informative prior distributions can be used when performing model criticism.