Point estimates for variance-structure parameters in Bayesian analysis of hierarchical models

Markov chain Monte Carlo (MCMC) made Bayesian analysis feasible for hierarchical models, but the literature about their variance parameters is sparse. This is particularly so for point estimators of variance-structure parameters, which are useful for simplifying tables and sample-size calculations, and as "plug-in" estimators in complex calculations. This paper uses simulation experiments to compare three such point estimators, the posterior mode, median, and mean, for three parameterizations of the variance structure, as precisions, standard deviations, and variances. We first consider simple linear regression, where fairly explicit expressions are possible, and then three more complex models: crossed random effects, smoothed analysis of variance (SANOVA), and the conditional autoregressive (CAR) model with two classes of neighbor relations. We illustrate the latter results using periodontal data. The posterior mean often performs poorly in terms of bias and mean-squared error, and should be avoided. The posterior median never performs worse than the mean and often performs far better. The surprise is that, on the whole, the posterior mode performs best regardless of the variance structure's parameterization, although the potential for multi-modality may make it unattractive for general use.