APPLICATION OF GIBBS SAMPLING TO NESTED VARIANCE-COMPONENTS MODELS WITH HETEROGENEOUS WITHIN-GROUP VARIANCE

Bayesian analysis of hierarchically structured data with random interc ept and heterogeneous within-group (Level-1) variance is presented. In ferences about all parameters, including the Level-1 variance and inte rcept for each group, are bused on their marginal posterior distributi ons approximated via the Gibbs sampler. Analysis of artificial data wi th varying degrees of heterogeneity and varying Level-2 sample sizes i llustrates the likely benefits of using a Bayesian approach to model h eterogeneity of variance (Bayesian). Results are compared to those bas ed on now-standard restricted maximum likelihood with homogeneous Leve l-1 variance (RML/Hom). Bayes/Het provides sensible interval estimates for Level-1 variances and their heterogeneity, and, relatedly, for ea ch group's intercept. RML/Hom inferences about Level-2 regression coef ficients appear surprisingly robust to heterogeneity, and conditions u nder which such robustness can be expected are discussed. Application is illustrated in a reanalysis of High School and Beyond data. It appe ars informative and practically feasible to obtain approximate margina l posterior distributions for all Level-1 and Level-2 parameters when analyzing large- or small-scale survey darn. A key advantage of the Ba yes approach is that inferences about any parameter appropriately refl ect uncertainty about all remaining parameters.