Mode-Matched Inverse Gamma Priors for Variance Components in Bayesian Multilevel Models

We introduce a strategy for specifying informative inverse-gamma (IG) priors for variance components in Bayesian multilevel models (MLMs), derived via transformations from chi-square to gamma to inverse-gamma distributions. A Monte Carlo simulation compared frequentist (maximum likelihood) estimation and Bayesian estimation using uninformative, weakly informative, and strongly informative variance priors across varied conditions (clusters J = 10, 30, 100; cluster sizes M = 5, 30; intraclass correlations 0.01, 0.20, 0.40; levels of explained variance at L1 and L2 R^2 = 0, 0.2, 0.4). The simulation results indicated that strongly informative IG priors (with hyperparameters set so the prior mode equals a plausible true variance) yielded more accurate and stable variance estimates with reduced bias and narrower credible intervals than flat/uninformative or weak IG(0.01, 0.01) priors. In an empirical example using the TIMSS 2019 Grade 8 science achievement data, both the full sample (273 schools) and small subsample (30 schools) were analyzed. The small-sample analysis with an informative variance prior anchored near the full-sample variance while considerably reducing the uncertainty of estimates. Findings suggest that carefully calibrated informative variance priors improve the precision and accuracy of parameter estimates, particularly when the number of higher-level units is limited.