Bayesian hierarchical modeling based on multisource exchangeability

Abstract

Bayesian hierarchical models produce shrinkage estimators that can be used as the basis for integrating supplementary data into the analysis of a primary data source. Established approaches should be considered limited, however, because posterior estimation either requires prespecification of a shrinkage weight for each source or relies on the data to inform a single parameter, which determines the extent of influence or shrinkage from all sources, risking considerable bias or minimal borrowing. We introduce multisource exchangeability models (MEMs), a general Bayesian approach for integrating multiple, potentially non-exchangeable, supplemental data sources into the analysis of a primary data source. Our proposed modeling framework yields source-specific smoothing parameters that can be estimated in the presence of the data to facilitate a dynamic multi-resolution smoothed estimator that is asymptotically consistent while reducing the dimensionality of the prior space. When compared with competing Bayesian hierarchical modeling strategies, we demonstrate that MEMs achieve approximately 2.2 times larger median effective supplemental sample size when the supplemental data sources are exchangeable as well as a 56% reduction in bias when there is heterogeneity among the supplemental sources. We illustrate the application of MEMs using a recently completed randomized trial of very low nicotine content cigarettes, which resulted in a 30% improvement in efficiency compared with the standard analysis.

Fig. 1.

Each MEM is a combination of supplemental sources assumed exchangeable with the primary cohort, in order to estimate the parameters of interest, , and is contained within each box for . Within a box the solid arrows and the observables, , represent which supplemental sources are assumed exchangeable with the primary cohort within the given MEM. The dashed arrows represent that the posterior model weights for each MEM, , are used in calculating the weighted average of each MEM’s posterior distribution, , to be used for posterior inference, .

Fig. 2.

Median effective supplemental sample size using MEM with priors, CP, and SHM under each scenario. Dashed vertical lines are used to represent assumed observed values of the supplemental group means for each scenario.

Fig. 3.

Plots demonstrating bias versus shrinkage trade-offs using the methods of CP, SHM, and MEM with source-inclusion priors. Note that CP overlaps for all four scenarios.

Fig. 4.

Plot comparing percent change in mean estimation from the standard model with no borrowing and the percent reduction in the posterior standard deviation for the difference between the treatment and control groups in Table 1 for the standard approach with no borrowing to the CP, SHM, and MEM with source-inclusion priors approaches.