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

The present study introduces a strategy for specifying informative inverse-gamma (IG) priors for variance components in Bayesian multilevel models by matching the prior mode to a substantively plausible or empirically grounded target while keeping prior strength explicit. A Monte Carlo simulation compared maximum likelihood with Bayesian baseline, IG(0.01, 0.01), and mode-matched inverse-gamma priors across 162 two-level random-intercept conditions. In true-mode benchmark conditions, VarIGab improved estimation of the between-level residual variance, especially with small or moderate numbers of clusters. It reduced RMSE and mean absolute relative bias relative to the Bayesian baseline and IG(0.01, 0.01) priors while producing narrower intervals than the baseline. In a disjoint reference-sample illustration, the sample-calibrated prior reduced RMSE and interval width relative to the baseline but showed less favorable relative-bias performance when the number of clusters was very small. Variance-prior specification had little effect on fixed-effect point estimates among Bayesian approaches. A TIMSS 2019 U.S. Grade 8 illustration used a non-overlapping 243-school reference sample to calibrate prior modes for a separate 30-school target analysis. The findings illustrate both the potential and limitations of reference-informed inverse-gamma priors and underscore the importance of transparent calibration and sensitivity analysis.