Bayesian Analysis Using Power Priors with Application to Pediatric Quality of Care

We illustrate how power prior distributions can be used to incorporate historical data into a Bayesian analysis when there is uncertainty about the similarity between the current and historical studies. We compare common specifi cations of the power prior and explore whether it is preferable to condition on the power parameter, a0, or to treat it as a random variable with a prior distribution of its own. We show that there are two natural ways of formulating the power prior for random a0. The fi rst approach excludes the historical data in all but extreme cases and may therefore be of limited practical use. The second approach, called the normalized power prior (NPP), provides a measure of congruence between the current and historical data, so that the historical data are downweighted more substantially as the studies diverge. While this is an intuitively appealing property, our experience suggests that in real world problems involving large datasets and models with several parameters, the NPP may lead to considerably more downweighting than desired. We therefore advise practitioners to consider whether such attenuation is desirable, or whether it is more appropriate to assign a 0 a fi xed value based on expert opinion about the relevance of the historical data to the current analysis. We also extend the power prior to hierarchical regression models that allow covariate effects to differ across studies. We apply these methods to a pair of studies designed to improve delivery of care in pediatric clinics.