Semi-parametric Bayesian estimation of mixed-effects models using the multivariate skew-normal distribution

In this paper, we develop a semi-parametric Bayesian estimation approach through the Dirichlet process (DP) mixture in fitting linear mixed models. The random-effects distribution is specified by introducing a multivariate skew-normal distribution as base for the Dirichlet process. The proposed approach efficiently deals with modeling issues in a wide range of non-normally distributed random effects. We adopt Gibbs sampling techniques to achieve the parameter estimates. A small simulation study is conducted to show that the proposed DP prior is better at the prediction of random effects. Two real data sets are analyzed and tested by several hypothetical models to illustrate the usefulness of the proposed approach.

[1]  Andrew Thomas,et al.  The BUGS project: Evolution, critique and future directions , 2009, Statistics in medicine.

[2]  Ernesto San Martín,et al.  Linear mixed models with skew-elliptical distributions: A Bayesian approach , 2008, Comput. Stat. Data Anal..

[3]  A. Azzalini,et al.  The multivariate skew-normal distribution , 1996 .

[4]  Dipak K. Dey,et al.  Robust linear mixed models with skew-normal independent distributions from a Bayesian perspective , 2009 .

[5]  E. Frees,et al.  Demand for Services: Determinants of Tax Preparation Fees , 1999 .

[6]  G. Verbeke,et al.  The effect of misspecifying the random-effects distribution in linear mixed models for longitudinal data , 1997 .

[7]  M Davidian,et al.  Linear Mixed Models with Flexible Distributions of Random Effects for Longitudinal Data , 2001, Biometrics.

[8]  M. Genton,et al.  On fundamental skew distributions , 2005 .

[9]  Dipankar Bandyopadhyay,et al.  Linear mixed models for skew-normal/independent bivariate responses with an application to periodontal disease. , 2010, Statistics in medicine.

[10]  Lancelot F. James,et al.  Gibbs Sampling Methods for Stick-Breaking Priors , 2001 .

[11]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[12]  Michael A. West,et al.  Hierarchical priors and mixture models, with applications in regression and density estimation , 2006 .

[13]  K. Mardia.  Measures of multivariate skewness and kurtosis with applications , 1970 .

[14]  Sophia Rabe-Hesketh,et al.  Multilevel and Longitudinal Modeling Using Stata , 2005 .

[15]  C. Antoniak.  Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems , 1974 .

[16]  G. Verbeke,et al.  A Linear Mixed-Effects Model with Heterogeneity in the Random-Effects Population , 1996 .

[17]  R. Arellano-Valle,et al.  Skew-normal Linear Mixed Models , 2005, Journal of Data Science.

[18]  R. Dorazio.  On selecting a prior for the precision parameter of Dirichlet process mixture models , 2009 .

[19]  R. Arellano-Valle,et al.  Bayesian Inference for Skew-normal Linear Mixed Models , 2007 .

[20]  H. Bolfarine,et al.  Skew scale mixtures of normal distributions: Properties and estimation , 2011 .

[21]  S. Sahu,et al.  A new class of multivariate skew distributions with applications to bayesian regression models , 2003 .

[22]  George Casella,et al.  Estimation in Dirichlet random effects models , 2010, 1002.4756.

[23]  Tsung-I Lin,et al.  Robust linear mixed models using the skew t distribution with application to schizophrenia data , 2010, Biometrical journal. Biometrische Zeitschrift.

[24]  A. L. Pretorius,et al.  Bayesian estimation in animal breeding using the Dirichlet process prior for correlated random effects , 2003, Genetics Selection Evolution.

[25]  P. Müller,et al.  Bayesian Inference in Semiparametric Mixed Models for Longitudinal Data , 2010, Biometrics.

[26]  J. Sethuraman.  A CONSTRUCTIVE DEFINITION OF DIRICHLET PRIORS , 1991 .

[27]  M. Escobar.  Estimating Normal Means with a Dirichlet Process Prior , 1994 .

[28]  S. MacEachern.  Estimating normal means with a conjugate style dirichlet process prior , 1994 .

[29]  Lancelot F. James,et al.  Approximate Dirichlet Process Computing in Finite Normal Mixtures , 2002 .

[30]  Peter Müller,et al.  CENTER-ADJUSTED INFERENCE FOR A NONPARAMETRIC BAYESIAN RANDOM EFFECT DISTRIBUTION. , 2011, Statistica Sinica.

[31]  E. Lesaffre,et al.  Smooth Random Effects Distribution in a Linear Mixed Model , 2004, Biometrics.

[32]  F. Quintana,et al.  Nonparametric Bayesian modelling using skewed Dirichlet processes , 2009 .

[33]  M. Escobar,et al.  Bayesian Density Estimation and Inference Using Mixtures , 1995 .