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

Rafa M. Kasim and Stephen W. Raudenbush
Journal of Educational and Behavioral Statistics
Vol. 23, No. 2 (Summer, 1998), pp. 93-116
Stable URL: http://www.jstor.org/stable/1165316
Page Count: 24
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Application of Gibbs Sampling to Nested Variance Components Models with Heterogeneous Within-Group Variance
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Abstract

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.

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