Bayesian analysis of censored linear regression models with scale mixtures of normal distributions

Abstract

As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyzes might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Student-t, Pearson type VII, slash and the contaminated normal distributions, as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. The likelihood function is utilized to compute not only some Bayesian model selection measures but also to develop Bayesian case-deletion influence diagnostics based on the q-divergence measure. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with applications using real and simulated data.

Keywords:

• Bayesian modeling
• censored regression models
• MCMC
• scale mixtures of normal distributions
• Bayesian diagnostics

Acknowledgements

We thank the editor, associate editor and two referees whose constructive criticism led to improved presentation and quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

The research of V. H. Lachos was supported by Grant [305054/2011-2] from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) and Grant [2014/02938-9] from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP-Brazil). A.M. Garay was supported by Grant [161119/2012-3] from CNPq and by Grant [2013/21468-0] from FAPESP-Brazil. Heleno Bolfarine was supported by CNPq and C.R. B. Cabral was supported by CNPq (via BPPesq and the Universal and CT-Amazônia Projects), CAPES (via PROCAD 2007 Project) and FAPEAM (via Universal Amazonas Project).