A Hierarchical Prior for Generalized Linear Models Based on Predictions for the Mean Response

• Activity Number: 4 - Recent Advancements in Prior Elicitation and Computational Tools for Bayesian Design and Analysis
• Type: Invited
• Date/Time: Sunday, August 7, 2022 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Matthew A. Psioda* and Joseph G Ibrahim and Ethan Alt
• Companies: University of North Carolina at Chapel Hill and University of North Carolina and Harvard University
• Keywords: hierarchical models; generalized linear models; bayesian inference; hyperprior; prior elicitation; conjugate prior

Abstract

There has been increased interest in using prior information in statistical analyses. For example, in rare diseases, it can be difficult to establish efficacy based solely on data from a prospective study due to low sample sizes. In such cases, an informative prior for the treatment effect may be elicited. We develop a novel extension of the conjugate prior of Chen and Ibrahim (2003) that enables practitioners to elicit a prior prediction for the mean response for generalized linear models, treating the prediction as random. We refer to the hierarchical prior as the hierarchical prediction prior (HPP). For independent and identically distributed settings and for the normal linear model, this hyperprior can be seen to be a conjugate prior. We also develop an extension of the HPP for situations where summary statistics from a previous study are available, drawing comparisons with the power prior. The HPP allows for discounting based on the quality of individual level predictions, and simulation results suggest that, compared to the conjugate prior and the power prior, the HPP provides efficiency gains (e.g., lower MSE) in cases where predictions are incompatible with the data.