Power prior distributions for regression models

We propose a general class of prior distributions for arbitrary regression models. We discuss parametric and semiparametric models. The prior specification for the regression coefficients focuses on observable quantities in that the elicitation is based on the availability of historical data $D_0$ and a scalar quantity $a_0$ quantifying the uncertainty in $D_0$. Then $D_0$ and $a_0$ are used to specify a prior for the regression coefficients in a semiautomatic fashion. The most natural specification of $D_0$ arises when the raw data from a similar previous study are available. The availability of historical data is quite common in clinical trials, carcinogenicity studies, and environmental studies, where large data bases are available from similar previous studies. Although the methodology we present here is quite general, we will focus only on using historical data from similar previous studies to construct the prior distributions. The prior distributions are based on the idea of raising the likelihood function of the historical data to the power $a_0$, where $0 \le a_0 \le 1$. We call such prior distributions power prior distributions. We examine the power prior for four commonly used classes of regression models. These include generalized linear models, generalized linear mixed models, semiparametric proportional hazards models, and cure rate models for survival data. For these classes of models, we discuss the construction of the power prior, prior elicitation issues, propriety conditions, model selection, and several other properties. For each class of models, we present real data sets to demonstrate the proposed methodology.