Comparison of Resampling and Bayesian Approaches in Variance Component Estimation of a Hierarchical Univariate Mixed Effect Model

The purpose of the study is to investigate the relative performance of two estimation procedures, a semi-frequentist estimation technique (via a Bootstrapped the restricted maximum likelihood: Bootstrap-REML) and Bayesian method (via a Gibbs sampler), for estimation of variance components of a two level hierarchical linear mixed model. For this purpose one variable named X was generated using R simulation with the structure of two level nested designs which showed Gaussian distribution. The variable X contains 10000 data, with an average of 0 and variances of 100 and. For this data, five different scenarios were created according to the rate of variance components and analyzes were carried out. All of the estimations and definitions of autocorrelation, changes of the total variance and estimation biases were performed for the posterior distributions and bootstrapped parameter distributions of all the scenarios. In general, the results obtained with both methods are close to each other, although the bias of the results obtained with the Gibbs sampling method was found less and autocorrelation was not found for Gibbs sampling estimates. In conclusion, according to the results of this study, it is not possible to say that using the Bootstrap-REML estimator under Gaussian distribution and balanced data is a good alternative to Bayesian Gibbs sampler.Perhaps different results may be obtained from another study using unbalanced data, non-normally distributed data and high sample sizes.