Robust linear mixed models with skew-normal independent distributions from a Bayesian perspective

Linear mixed models were developed to handle clustered data and have been a topic of increasing interest in statistics for the past 50 years. Generally, the normality (or symmetry) of the random effects is a common assumption in linear mixed models but it may, sometimes, be unrealistic, obscuring important features of among-subjects variation. In this article, we utilize skew-normal/independent distributions as a tool for robust modeling of linear mixed models under a Bayesian paradigm. The skew-normal/independent distributions is an attractive class of asymmetric heavy-tailed distributions that includes the skew-normal distribution, skew-t, skew-slash and the skew-contaminated normal distributions as special cases, providing an appealing robust alternative to the routine use of symmetric distributions in this type of models. The methods developed are illustrated using a real data set from Framingham cholesterol study.

[1]  R. Arellano-Valle,et al.  LIKELIHOOD BASED INFERENCE FOR SKEW-NORMAL INDEPENDENT LINEAR MIXED MODELS , 2010 .

[2]  Barnes.  Discussion of the Paper , 1961, Public health papers and reports.

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

[4]  Aki Vehtari,et al.  Discussion on the paper by Spiegelhalter, Best, Carlin and van der Linde , 2002 .

[5]  T. Louis,et al.  Bayes and Empirical Bayes Methods for Data Analysis. , 1997 .

[6]  Bradley P. Carlin,et al.  Markov Chain Monte Carlo conver-gence diagnostics: a comparative review , 1996 .

[7]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[8]  Hong Chang,et al.  Model Determination Using Predictive Distributions with Implementation via Sampling-Based Methods , 1992 .

[9]  John K Kruschke,et al.  Bayesian data analysis. , 2010, Wiley interdisciplinary reviews. Cognitive science.

[10]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[11]  W. Michael Conklin,et al.  Monte Carlo Methods in Bayesian Computation , 2001, Technometrics.

[12]  M. Kendall.  Theoretical Statistics , 1956, Nature.

[13]  D. Dey,et al.  A General Class of Multivariate Skew-Elliptical Distributions , 2001 .

[14]  Geert Molenberghs,et al.  Random Effects Models for Longitudinal Data , 2010 .

[15]  M. Genton,et al.  Robust Likelihood Methods Based on the Skew‐t and Related Distributions , 2008 .

[16]  Alan E. Gelfand,et al.  Model Determination using sampling-based methods , 1996 .

[17]  Tsung-I Lin,et al.  Estimation and prediction in linear mixed models with skew‐normal random effects for longitudinal data , 2008, Statistics in medicine.

[18]  D. Spiegelhalter,et al.  Bayesian measures of model omplexity and t , 2001 .

[19]  Purushottam W. Laud,et al.  Predictive Model Selection , 1995 .

[20]  Jack C. Lee,et al.  Robust mixture modeling using the skew t distribution , 2007, Stat. Comput..

[21]  Jerry Nedelman,et al.  Book review: “Bayesian Data Analysis,” Second Edition by A. Gelman, J.B. Carlin, H.S. Stern, and D.B. Rubin Chapman & Hall/CRC, 2004 , 2005, Comput. Stat..

[22]  Arjun K. Gupta,et al.  A multivariate skew normal distribution , 2004 .

[23]  Arjun K. Gupta.  Multivariate skew t-distribution , 2003 .

[24]  Bradley P. Carlin,et al.  BAYES AND EMPIRICAL BAYES METHODS FOR DATA ANALYSIS , 1996, Stat. Comput..

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

[26]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

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

[28]  Marc G. Genton,et al.  Skew-elliptical distributions and their applications : a journey beyond normality , 2004 .

[29]  P. Embrechts.  Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality , 2005 .

[30]  M. Genton,et al.  The multivariate skew-slash distribution , 2006 .

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

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

[33]  Bradley P. Carlin,et al.  BAYES AND EMPIRICAL BAYES METHODS FOR DATA ANALYSIS , 1996, Stat. Comput..

[34]  A. Azzalini,et al.  Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution , 2003, 0911.2342.

[35]  Linear Mixed Effects Models with Flexible Generalized Skew-Elliptical Random Effects , 2004 .

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

[37]  J. Q. Smith,et al.  1. Bayesian Statistics 4 , 1993 .

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

[39]  Ying Nian Wu,et al.  Efficient Algorithms for Robust Estimation in Linear Mixed-Effects Models Using the Multivariate t Distribution , 2001 .

[40]  Daniel Gianola,et al.  Robust Linear Mixed Models with Normal/Independent Distributions and Bayesian MCMC Implementation , 2003 .

[41]  Aki Vehtari.  Discussion to "Bayesian measures of model complexity and fit" by Spiegelhalter, D.J., Best, N.G., Carlin, B.P., and van der Linde, A. , 2002 .

[42]  K. Lange,et al.  Normal/Independent Distributions and Their Applications in Robust Regression , 1993 .

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