A hierarchical prior for generalized linear models based on predictions for the mean response

TL;DR

This paper introduces a hierarchical prediction prior for generalized linear models that incorporates prior predictions for the mean response, improving robustness and efficiency in statistical inference, especially with limited data.

Contribution

It extends the conjugate prior framework to include prior predictions as random, offering a new hierarchical prior that adapts based on prediction quality and prior-data compatibility.

Findings

The hierarchical prediction prior improves robustness to prior-data conflict.

The method achieves lower mean squared error in simulations.

Applications demonstrate enhanced inference stability.

Abstract

There has been increased interest in using prior information in statistical analyses. For example, in rare diseases, it can be difficult to establish treatment efficacy based solely on data from a prospective study due to low sample sizes. To overcome this issue, 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. For i.i.d. settings and the normal linear model, we derive cases for which the hyperprior is a conjugate prior. We also develop an extension of the HPP in 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, having the potential to provide efficiency gains (e.g., lower MSE) where predictions are incompatible with the data. An efficient Markov chain Monte Carlo algorithm is developed. Applications illustrate that inferences under the HPP are more robust to prior-data conflict compared to selected non-hierarchical priors.