A non-iterative sampling Bayesian method for linear mixed models with normal independent distributions

Abstract

In this paper, we utilize normal/independent (NI) distributions as a tool for robust modeling of linear mixed models (LMM) under a Bayesian paradigm. The purpose is to develop a non-iterative sampling method to obtain i.i.d. samples approximately from the observed posterior distribution by combining the inverse Bayes formulae, sampling/importance resampling and posterior mode estimates from the expectation maximization algorithm to LMMs with NI distributions, as suggested by Tan et al. [33]. The proposed algorithm provides a novel alternative to perfect sampling and eliminates the convergence problems of Markov chain Monte Carlo methods. In order to examine the robust aspects of the NI class, against outlying and influential observations, we present a Bayesian case deletion influence diagnostics based on the Kullback–Leibler divergence. Further, some discussions on model selection criteria are given. The new methodologies are exemplified through a real data set, illustrating the usefulness of the proposed methodology.

Keywords:

• Gibbs algorithms
• inverse Bayes formulae
• MCMC
• linear mixed models
• normal/independent distributions
• sampling/importance resampling

Acknowledgements

The authors are grateful to two anonymous referees for the valuable comments and suggestions that lead to the paper’s significant improvement. V.H. Lachos wishes to acknowledge the support of the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) through Grant Number 2008/201384-6 and the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP-Brazil) through Grant Number 2008/11455-0. C. R. B. Cabral and C. A. Abanto-Valle acknowledge the partial financial support from CNPq and CAPES.