Quantification of empirical determinacy: the impact of likelihood weighting on posterior location and spread in Bayesian meta-analysis estimated with JAGS and INLA

TL;DR

This paper introduces a method to quantify how much data influences the posterior distribution in Bayesian meta-analysis, using the concept of empirical determinacy, and demonstrates its application with case studies and simulations.

Contribution

It develops a novel approach to measure empirical determinacy in Bayesian hierarchical models, implemented in the R package ed4bhm, applicable beyond NNHM.

Findings

Quantified the impact of data on posterior location and spread.

Demonstrated the method's application in case studies and simulations.

Provided insights into how modeling choices affect empirical determinacy.

Abstract

The popular Bayesian meta-analysis expressed by Bayesian normal-normal hierarchical model (NNHM) synthesizes knowledge from several studies and is highly relevant in practice. Moreover, NNHM is the simplest Bayesian hierarchical model (BHM), which illustrates problems typical in more complex BHMs. Until now, it has been unclear to what extent the data determines the marginal posterior distributions of the parameters in NNHM. To address this issue we computed the second derivative of the Bhattacharyya coefficient with respect to the weighted likelihood, defined the total empirical determinacy (TED), the proportion of the empirical determinacy of location to TED (pEDL), and the proportion of the empirical determinacy of spread to TED (pEDS). We implemented this method in the R package \texttt{ed4bhm} and considered two case studies and one simulation study. We quantified TED, pEDL and pEDS under different modeling conditions such as model parametrization, the primary outcome, and the prior. This clarified to what extent the location and spread of the marginal posterior distributions of the parameters are determined by the data. Although these investigations focused on Bayesian NNHM, the method proposed is applicable more generally to complex BHMs.