A Diagnostic-Guided Hierarchical Bayesian Framework for Meta-Analytic Correlation Matrices: From Cellwise Estimation to Latent Co-Moderation

<p>Meta-analytic syntheses of correlation matrices typically estimate each cell independently and then repair the assembled matrix. That ordering creates compounding problems: cellwise heterogeneity estimates are least stable exactly where evidence is thinnest, independently generated cells routinely violate positive-definiteness, and the tendency of correlations to rise and fall together across studies is ignored entirely. We develop a hierarchical Bayesian framework that reverses the derivation: each study’s population correlation matrix is parameterized natively on the positive-definite manifold, cellwise means and heterogeneity are derived from the joint posterior, and the between-study covariance of cells receives a low-rank factor-analytic structure: co-heterogeneity — regularized by increasing shrinkage, with effective rank assessed from posterior eigenvalues and sensitivity refits. A tiered ladder of models with accompanying diagnostics recommends the simplest structure a given dataset supports. Across 1,225 simulation replications in 13 conditions, coverage of 95% credible intervals was generally near nominal across conditions (.920–.985), central-matrix recovery correlates .99 with truth, and on data with no co-heterogeneity the estimated between-study covariance is nearly structureless (mean absolute off-diagonal correlation .08, versus .25 under genuine structure). A preliminary sensitivity study quantifies the cost of ignoring within-study sampling covariance. Applied to a cross-cultural meta-analysis of the Rosenberg Self-Esteem Scale (113 samples), between-study variation among the 36 directly reported matrices concentrates overwhelmingly on a single dimension (.83 of Fisher-z between-study variance; .76 under an Olkin–Siotani-corrected sensitivity fit, the two specifications bracketing the estimand); including the 77 matrices reconstructed from published factor loadings dilutes the share to .63 and exposes matrix provenance as the dataset’s largest structural moderator. Directly reported and reconstructed matrices should not be treated as exchangeable when the estimand is between-study matrix variation.</p>