Which Coefficients Need Correction? Design-Effect-Ratio Diagnostics and Selective Calibration for Bayesian Meta-Analysis With Dependent Effect Sizes

Bayesian meta-analysis with dependent effect sizes has recently acquired a robust reporting option: the posterior from a convenient working model can be calibrated so that its covariance matches a declared cluster-robust variance target. The correction is currently applied to the entire coefficient block at once, as if every coefficient required it to the same degree. This premise fails in a predictable way. Coefficients differ systematically in how much of the within-study dependence they inherit, and estimands identified by comparisons within studies are protected twice, by cancellation of shared sampling errors and by hierarchical shrinkage, so that blanket rescaling narrows exactly the intervals that were already wide enough. We propose the design-effect ratio, the ratio of the declared target variance to the working-posterior variance of a pre-specified estimand, as a two-sided diagnostic: estimands whose posterior understates the target are calibrated, and the remainder are left alone, never narrowed. Exact decompositions, transferred from Bayesian survey inference through a recently established likelihood correspondence, explain both regimes and show that coefficient-level ratios cannot stand in for contrast-level ratios. In two simulation studies with known truth and a re-analysis of a 117-study synthesis of brief alcohol interventions, the selective workflow holds near-nominal coverage where the frequentist benchmark undercovers (.975 against .904 for the weakest slope), avoids the coverage loss that blanket calibration incurs on protected estimands, and uses up to one fifth less interval width. We conclude that the decision to calibrate should be made estimand by estimand. An R package implements the workflow.