Bayesian analysis of joint mean and covariance models for longitudinal data

Abstract

Efficient estimation of the regression coefficients in longitudinal data analysis requires a correct specification of the covariance structure. If misspecification occurs, it may lead to inefficient or biased estimators of parameters in the mean. One of the most commonly used methods for handling the covariance matrix is based on simultaneous modeling of the Cholesky decomposition. Therefore, in this paper, we reparameterize covariance structures in longitudinal data analysis through the modified Cholesky decomposition of itself. Based on this modified Cholesky decomposition, the within-subject covariance matrix is decomposed into a unit lower triangular matrix involving moving average coefficients and a diagonal matrix involving innovation variances, which are modeled as linear functions of covariates. Then, we propose a fully Bayesian inference for joint mean and covariance models based on this decomposition. A computational efficient Markov chain Monte Carlo method which combines the Gibbs sampler and Metropolis–Hastings algorithm is implemented to simultaneously obtain the Bayesian estimates of unknown parameters, as well as their standard deviation estimates. Finally, several simulation studies and a real example are presented to illustrate the proposed methodology.

Keywords:

• joint mean and covariance models
• Cholesky decomposition
• Bayesian analysis
• Gibbs sampler
• Metropolis–Hastings algorithm

Acknowledgements

The authors would like to thank the editor and the two referees for helpful comments, which lead to improvement of an earlier version of this paper. This work is supported by grants from the National Natural Science Foundation of China (11301485, 11271039, 11261025, 11101015); Startup Foundation for Talents in Zhejiang Agriculture and Forest University(2013FR079); Funding Project of Science and Technology Research Plan of Beijing Education Committee (JC006790201001); The Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions (CIT&TCD201304058).