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

Area of Science:

• Statistics
• Biostatistics
• Statistical Modeling

Background:

• Prior information integration is crucial in statistical analyses, particularly for rare diseases with small sample sizes.
• Eliciting informative priors for treatment effects can overcome limitations of prospective studies.
• Existing methods may struggle with prior-data conflict.

Purpose of the Study:

• To develop a novel extension of the conjugate prior for generalized linear models, enabling random prior prediction of mean response.
• To introduce the hierarchical prediction prior (HPP) for enhanced statistical analysis.
• To extend the HPP for scenarios involving summary statistics from previous studies.

Main Methods:

• Developed a hierarchical prior (HPP) extending the conjugate prior of Chen and Ibrahim (2003).
• Derived conditions for conjugate hyperpriors in normal linear models and independent and identically distributed settings.
• Created an extension of the HPP for utilizing summary statistics from prior studies.
• Developed an efficient Monte Carlo Markov chain algorithm for implementation.

Main Results:

• The HPP allows for discounting based on the quality of individual predictions.
• Simulations show HPP offers efficiency gains (e.g., lower mean squared error) when predictions are incompatible with data, outperforming conjugate and power priors.
• Inferences under HPP demonstrate greater robustness to prior-data conflict compared to non-hierarchical priors.

Conclusions:

• The hierarchical prediction prior (HPP) provides a robust and efficient method for incorporating prior information in statistical modeling.
• HPP is particularly valuable in situations with potential prior-data conflict, enhancing reliability of study findings.
• The developed methods and algorithms facilitate practical application of HPP in diverse research settings.