How vague is vague? How informative is informative? Reference analysis for Bayesian meta‐analysis

Abstract

Meta‐analysis provides important insights for evidence‐based medicine by synthesizing evidence from multiple studies which address the same research question. Within the Bayesian framework, meta‐analysis is frequently expressed by a Bayesian normal‐normal hierarchical model (NNHM). Recently, several publications have discussed the choice of the prior distribution for the between‐study heterogeneity in the Bayesian NNHM and used several “vague” priors. However, no approach exists to quantify the informativeness of such priors, and thus, we develop a principled reference analysis framework for the Bayesian NNHM acting at the posterior level. The posterior reference analysis (post‐RA) is based on two posterior benchmarks: one induced by the improper reference prior, which is minimally informative for the data, and the other induced by a highly anticonservative proper prior. This approach applies the Hellinger distance to quantify the informativeness of a heterogeneity prior of interest by comparing the corresponding marginal posteriors with both posterior benchmarks. The post‐RA is implemented in the freely accessible R package ra4bayesmeta and is applied to two medical case studies. Our findings show that anticonservative heterogeneity priors produce platykurtic posteriors compared with the reference posterior, and they produce shorter 95% credible intervals (CrI) and optimistic inference compared with the reference prior. Conservative heterogeneity priors produce leptokurtic posteriors, longer 95% CrI and cautious inference. The novel post‐RA framework could support numerous Bayesian meta‐analyses in many research fields, as it determines how informative a heterogeneity prior is for the actual data as compared with the minimally informative reference prior.

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

“…Following Roos et al [2021], we consider the Bhattacharyya coefficient (BC) [Bhattacharyya, 1943] between two probability density functions, which is given by…”

“…In general, any density can be approximated to the first order by a normal distribution [Hjort andGlad, 1995, Johnson et al, 1994]. Following Roos et al [2021], we approximate the marginal posterior distribution π(ψ | y) for ψ ∈ {µ, τ, θ 1 , . .…”

“…Recently, Roos et al [2021] considered Bayesian NNHM and proposed a two-dimensional method for sensitivity assessment based on the second derivative of the Bhattacharyya coefficient (BC). In one dimension, they used a factor f to scale the within-study standard deviation provided by the data.…”

“…Alternatively, perturbation of the total number of observations could be expressed by a weighted likelihood with w = 1/f 2 [Bissiri et al, 2016, Holmes andWalker, 2017]. An open question remains whether the applicability of the scaling proposed by Roos et al [2021] could be generalized to weighting other likelihoods, or extended to general-purpose Bayesian estimation techniques such as the Bayesian numerical approximation INLA [Rue et al, 2009], and Bayesian MCMC sampling with JAGS [Plummer, 2016].…”

“…We expect that likelihood weighting will affect both the location and the spread of the marginal posterior distributions. Because the method proposed by Roos et al [2021] quantifies the total impact of the scaling on the posteriors without focusing on location and spread, a refined method is needed. Ideally, such a refined method should quantify what proportion of the total impact of the data on the posterior affects the location and what proportion affects the spread.…”