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Application of Gibbs Sampling to Nested Variance Components Models With Heterogeneous Within-Group Variance

University of Alabama at Birmingham

Stephen W. Raudenbush

University of Michigan

Bayesian analysis of hierarchically structured data with random intercept and heterogeneous within-group (Level-1) variance is presented. Inferences about all parameters, including the Level-1 variance and intercept for each group, are based on their marginal posterior distributions approximated via the Gibbs sampler Analysis of artificial data with varying degrees of heterogeneity and varying Level-2 sample sizes illustrates the likely benefits of using a Bayesian approach to model heterogeneity of variance (Bayes/Het). Results are compared to those based on now-standard restricted maximum likelihood with homogeneous Level-1 variance (RML/Hom). Bayes/Het provides sensible interval estimates for Level-1 variances and their heterogeneity, and, relatedly, for each group’s intercept. RML/Hom inferences about Level-2 regression coefficients appear surprisingly robust to heterogeneity, and conditions under which such robustness can be expected are discussed. Application is illustrated in a reanalysis of High School and Beyond data. It appears informative and practically feasible to obtain approximate marginal posterior distributions for all Level-1 and Level-2 parameters when analyzing large- or small-scale survey data. A key advantage of the Bayes approach is that inferences about any parameter appropriately reflect uncertainty about all remaining parameters.

Key Words: Keywords: Bayesian inference • Gibbs sampler • heterogeneity of variance • hierarchical models

Journal of Educational and Behavioral Statistics, Vol. 23, No. 2, 93-116 (1998)
DOI: 10.3102/10769986023002093


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