Original Article

Bayesian Analysis of Non-normal and Non-independent Mixed Model UsingSkew-Normal/Independent Distributions

Abstract

The  main  assumptions  in  liner  mixed  model  are normality  and  independency  of  random  effect component.   Unfortunately,   these   two  assumptions   might   be  unrealistic   in  some   situations. Therefore, in this paper, we will discuss about the analysis of Bayesian analysis of non-normal and non-independent mixed model using skew-normal/independent distributions, and finally, this methodology is illustrated through an application  to a triglyceride data from Isfahan’s Mobarakeh Steel Company Cohort Study.

Ahrens W, Pigeot I. Handbook of epidemiology. New York, NY: Springer Science & Business Media; 2005.

Diggle P, Heagerty P, Liang K, Zeger S.Analysis of longitudinal data. 2nd ed. Oxford, UK: OUP Oxford; 2002.

Fitzmaurice G, Laird N, Ware JH. Applied longitudinal analysis. Hoboken, NJ: John Wiley & Sons; 2004.

Browne WJ, Draper D. A comparison of Bayesian and likelihood-based methods for fitting multilevel models. Bayesian Anal 2006; 1(3): 473-514.

Luke DA. Multilevel modeling. 3rd ed.Washington DC: SAGE Publications; 2004.

Butler SM, Louis TA. Random effects models with non-parametric priors. Stat Med 1992; 11(14-15): 1981-2000.

Verbeke G, Lesaffre E. The effect of misspecifying the random-effects distribution in linear mixed models for longitudinal data. Computational Statistics & Data Analysis 1997; 23(4): 541-56.

Ghidey W, Lesaffre E, Verbeke G. Acomparison of methods for estimating the random effects distribution of a linear mixed model. Stat Methods Med Res 2010; 19(6):575-600.

Lin TI, Lee JC. On modelling data from degradation sample paths over time. Australian & New Zealand Journal of Statistics 2003; 45(3): 257-70.

Lee JC, Lin TI, Lee KJ, Hsu YL. Bayesian analysis of Box–Cox transformed linear mixed models with ARMA (p,qp,q) dependence. Journal of Statistical Planning and Inference 2005; 133(2): 435-51.

Jara A, Quintana F, San Martín E. Linear mixed models with skew-elliptical distributions: A Bayesian approach. Computational Statistics & Data Analysis 2008; 52(11): 5033-45.

Davidian M, Gallant AR. The nonlinear mixed effects model with a smooth random effects density. Biometrika 1993; 80(3): 475-88.

Magder LS, Zeger SL. A smooth nonparametric estimate of a mixing distribution using mixtures of Gaussians. Journal of the American Statistical Association 1996; 91(435): 1141-51.

Verbeke G, Lesaffre E. A linear mixed- effects model with heterogeneity in the random-effects population. Journal of the American Statistical Association 1996;91(433): 217-21.

Kleinman KP, Ibrahim JG. A semi- parametric Bayesian approach to generalized linear mixed models. Stat Med 1998; 17(22):2579-96.

Aitkin M. A general maximum likelihood analysis of variance components in generalized linear models. Biometrics 1999;55(1): 117-28.

Jiang J. Conditional inference about generalized linear mixed models. Ann Statist 1999; 27(6): 1974-2007.

Tao H, Palta M, Yandell BS, Newton MA.An estimation method for the semiparametric mixed effects model. Biometrics 1999; 55(1):102-10.

Zhang D, Davidian M. Linear mixed models with flexible distributions of random effects for longitudinal data. Biometrics 2001; 57(3):795-802.

Ghidey W, Lesaffre E, Eilers P. Smooth random effects distribution in a linear mixed model. Biometrics 2004; 60(4): 945-53.

Pinheiro JC, Liu C, Wu YN. Efficient algorithms for robust estimation in linear mixed-effects models using the multivariate t distribution. Journal of Computational and Graphical Statistics 2001; 10(2): 249-76.

Zhou T, He X. Three-step estimation in linear mixed models with skew-t distributions. Journal of Statistical Planning and Inference 2008; 138(6): 1542-55.

Rosa GJM, Padovani CR, Gianola D. Robust linear mixed models with normal/independent distributions and Bayesian MCMC implementation. Biometrical Journal 2003; 45(5): 573-90.

Lin TI, Lee JC. A robust approach to t linear mixed models applied to multiple sclerosis data. Stat Med 2006; 25(8): 1397-412.

Lin TI, Lee JC. Bayesian analysis of hierarchical linear mixed modeling using the multivariate t distribution. Journal of Statistical Planning and Inference 2007;137(2): 484-95.

Lange K, Sinsheimer JS.Normal/independent distributions and their applications in robust regression. Journal of Computational and Graphical Statistics 1993;2(2): 175-98.

Ma Y, Genton MG. A flexible class of skew- symmetric distributions. Scand J Statist 2004;31: 459-68.

Arellano-Valle RB, Bolfarine H, Lachos VH.Bayesian inference for skew-normal linear mixed models. Journal of Applied Statistics 2007; 34(6): 663-82.

Lachos VH, Dey DK, Cancho VG. Robust linear mixed models with skew-normal independent distributions from a Bayesian perspective. Journal of Statistical Planning and Inference 2009; 139(12): 4098-110.

Bandyopadhyay D, Lachos VH, Abanto-Valle CA, Ghosh P. Linear mixed models for skew-normal/independent bivariate responses with an application to periodontal disease. Stat Med 2010; 29(25): 2643-55.

Azzalini A. A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 1985; 12(2): 171-8.

Nadarajah S, Kotz S. Skewed distributions generated by the normal kernel. Statistics & Probability Letters 2003; 65(3): 269-77.

Bandyopadhyay D, Lachos VH, Abanto-Valle CA, Ghosh P. Labra, Linear mixed models for skew-normal/independent bivariate responses with an application to periodontal disease. Stat Med 2010; 29(25): 2643-55.

de Deleeuw J, Goldstein H, Meijer E.Handbook of multilevel analysis. New York, NY: Springer; 2007.

Hox J. Multilevel Analysis: Techniques and applications. 2nd ed. London, UK: Routledge;2010.

Hobert JP, Casella G. The effect of improper priors on Gibbs sampling in hierarchical linear mixed models. Journal of the American Statistical Association 1996;91(436): 1461-73.

Zhao Y, Staudenmayer J, Coull BA, Wand MP. General design Bayesian generalized linear mixed models. Statistical Science 2006; 21(1): 35-51.

Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A. Bayesian measures of model complexity and fit. J R Statist Soc B 2002;64(4): 583-639.

Ntzoufras I. Bayesian modeling using win BUGS. Hoboken, NJ: Wiley; 2009.

Pati AK, Chandrawanshi A, Reinberg A.Shift work: Consequences and management. Current Science 2001; 81(1): 32-52.

Sharifian A, Farahani S, Pasalar P, Gharavi M, Aminian O. Shift work as an oxidative stressor. J Circadian Rhythms 2005; 3: 15.

Poss J, Custodis F, Werner C, Weingartner O, Bohm M, Laufs U. Cardiovascular disease and dyslipidemia: beyond LDL. Curr Pharm Des 2011; 17(9): 861-70.

Cziraky MJ, Watson KE, Talbert RL.Targeting low HDL-cholesterol to decrease residual cardiovascular risk in the managed care setting. J Manag Care Pharm 2008; 14(8Suppl): S3-28.

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IssueVol 1 No 3/4 (2015) QRcode
SectionOriginal Article(s)
Keywords
multilevel modeling bayesian analysis normal independe nt distributions triglycerides

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How to Cite
1.
Gholami-Fesharaki M, Kazemnejad A. Bayesian Analysis of Non-normal and Non-independent Mixed Model UsingSkew-Normal/Independent Distributions. JBE. 2015;1(3/4):112-120.