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It is often of interest to combine available estimates of a similar quantityfrom multiple data sources. When the corresponding variances of each estimateare also available, a model should take into account the uncertainty of theestimates themselves as well as the uncertainty in the estimation of variances.In addition, if there exists a strong association between estimates and theirvariances, the correlation between these two quantities should also beconsidered. In this paper, we propose a bivariate hierarchical Bayesian modelthat jointly models the estimates and their estimated variances assuming acorrelation between these two measures. We conduct simulations to explore theperformance of the proposed bivariate Bayesian model and compare it to othercommonly used methods under different correlation scenarios. The proposedbivariate Bayesian model has a wide range of applications. We illustrate itsapplication in three very different areas: PET brain imaging studies,meta-analysis, and small area estimation.

Bivariate Hierarchical Bayesian Model for Combining Summary Measures and their Uncertainties from Multiple Sources