Skip to main content

Advertisement

SpringerLink
Log in

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

  • Original Paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Antoniak CE (1974) Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann Stat 2:1152–1174

    Article  MathSciNet  MATH  Google Scholar 

  • Arellano-Valle RB, Genton MG (2005) Fundamental skew distributions. J Multivar Anal 96:93–116

    Article  MathSciNet  MATH  Google Scholar 

  • Arellano-Valle RB, Bolfarine H, Lachos VH (2005) Skew-normal linear mixed models. J Data Sci 3: 415–438

    Google Scholar 

  • Arellano-Valle RB, Bolfarine H, Lachos VH (2007) Bayesian inference for skew-normal linear mixed models. J Appl Stat 34(6):663–682

    Article  MathSciNet  Google Scholar 

  • Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83:715–726

    Article  MathSciNet  MATH  Google Scholar 

  • Bandyopadhyay D, Lachos VH, Abanto-Valle CA, Ghosh P (2010) Linear mixed models for skew-normal/independent bivariate responses with an application to periodontal disease. Stat Med 29(25):2643–2655

    Article  MathSciNet  Google Scholar 

  • Dorazio RM (2009) On selecting a prior for the precision parameter of Dirichlet process mixture models. J Stat Plan Inf 139:3384–3390

    Article  MathSciNet  MATH  Google Scholar 

  • Escobar MD (1994) Estimating normal means with a Dirichlet process prior. J Am Stat Assoc 89(425): 268–277

    Google Scholar 

  • Escobar MD, West M (1995) Bayesian density estimation and inference using mixtures. J Am Stat Assoc 90:577–588

    Article  MathSciNet  MATH  Google Scholar 

  • Ferguson TS (1973) A Bayesian analysis of some nonparametric problems. Ann Stat 1:209–230

    Article  MathSciNet  MATH  Google Scholar 

  • Ferreira CDS, Bolfarine H, Lachos VH (2011) Skew scale mixtures of normal distributions: properties and estimation. Stat Methodol 8:154–171

    Article  MathSciNet  MATH  Google Scholar 

  • Frees EW (2004) Longitudinal and panel data: analysis and applications for the social sciences. Cambridge University Press, Cambridge pp: 65–67 and 81–86

  • Frischmann PJ, Frees EW (1999) Demand for services: determinants of tax preparation fees. J Am Tax Assoc 21(Supplement):1–23

    Article  Google Scholar 

  • Ghidey W, Lesaffre E, Eilers P (2004) Smooth random effects distribution in a linear mixed model. Biometrics 60:945–953

    Google Scholar 

  • Ho HJ, Lin TI (2010) Robust linear mixed models using the skew t distribution with application to schizophrenia data. Biometr J 52(4):449–469

    Article  MathSciNet  MATH  Google Scholar 

  • Iglesias PL, Orellana Y, Quintana FA (2009) Nonparametric Bayesian modeling using skewed Dirichlet processes. J Stat Plan Inf 139:1203–1214

    Article  MathSciNet  MATH  Google Scholar 

  • Ishwaran H, James LF (2001) Gibbs sampling methods for stick-breaking priors. J Am Stat Assoc 96(453):161–173

    Article  MathSciNet  MATH  Google Scholar 

  • Ishwaran H, James LF (2002) Approximate Dirichlet process computing in finite normal mixtures: smoothing and prior information. J Comput Graph Stat 11:508–532

    Article  MathSciNet  Google Scholar 

  • Jara A, Quintana F, SanMartin E (2008) Linear mixed models with skew-elliptical distributions: a Bayesian approach. Comput Stat Data Anal 52:5033–5045

    Article  MathSciNet  MATH  Google Scholar 

  • Kyung M, Gill J, Casella G (2010) Estimation in Dirichlet random effects models. Ann Stat 38:919–1009

    Article  MathSciNet  Google Scholar 

  • Lachos VH, Dey DK, Cancho VG (2009) Robust linear mixed models with skew-normal independent distributions from a Bayesian perspective. J Stat Plan Inf 139:4098–4110

    Article  MathSciNet  MATH  Google Scholar 

  • Li Y, Lin X, Müller P (2010) Bayesian inference in Semiparametric mixed models for longitudinal data. Biometrics 66(1):70–78

    Article  MathSciNet  MATH  Google Scholar 

  • Li Y, Müller P, Lin X (2011) Center-adjusted inference for a nonparametric Bayesian random effect distribution. Stat Sin 21(3):1201–1223

    Article  MATH  Google Scholar 

  • Lunn D, Spiegelhalter D, Thomas A, Best N (2009) The BUGS project: evolution, critique and future directions (with discussion). Stat Med 28:3049–3082

    Article  MathSciNet  Google Scholar 

  • MacEachern SN (1994) Estimating normal means with a conjugate style Dirichlet process prior. Commun Stat B 23:727–741

    Article  MathSciNet  MATH  Google Scholar 

  • Mardia KV (1970) Measures of multivariate skewness and kurtosis with applications. Biometrika 57: 519–530

    Google Scholar 

  • Van Der Merwe AJ, Pretorius AL (2003) Bayesian estimation in animal breeding using Dirichlet process prior for correlated random effects. Genet Sel Evol 35:137–158

    Article  Google Scholar 

  • Quintana FA (2010) Linear regression with a dependent skewed Dirichlet process. Chil J Stat 1(2):35–49

    MathSciNet  MATH  Google Scholar 

  • Rabe-Hesketh S, Skrondal A (2005) Multilevel and Longitudinal Modeling Using Stata. Stata Press,

  • Sahu SK, Dey DK, Branco MD (2003) A new class of multivariate skew distributions with applications to Bayesian regression models. Can J Stat 31:129–150

    Article  MathSciNet  MATH  Google Scholar 

  • Sethuraman J (1994) A constructive definition of Dirichlet priors. Stat Sin 4:639–650

    MathSciNet  MATH  Google Scholar 

  • Verbeke G, Lesaffre E (1996) A linear mixed-effects model with heterogeneity in the random-effects population. J Am Stat Assoc 91:217–221

    Article  MATH  Google Scholar 

  • Verbeke G, Lesaffre E (1997) The effect of misspecifying the random effects distribution in linear mixed models for longitudinal data. Comput Stat Data Anal 23:541–556

    Article  MathSciNet  MATH  Google Scholar 

  • West M, Müller P, Escobar MD (1994) Hierarchical priors and mixture models, with applications in regression and density estimation. In: Freeman PR, Smith AFM (eds) Aspects of uncertainty: a tribute to D.V. Lindley. Wiley, New York, pp 363–386

    Google Scholar 

  • Zhang D, Davidian M (2001) Liner mixed models with flexible distributions of random effects for longitudinal data. Biometrics 57:795–802

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Authors are grateful to the office of Graduate Studies of the University of Isfahan for their support. We also gratefully acknowledge referees for their invaluable comments.

Author information

Authors and Affiliations

  1. Department of Statistics, University of Isfahan, Isfahan, Iran

    Reyhaneh Rikhtehgaran & Iraj Kazemi

Authors
  1. Reyhaneh Rikhtehgaran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Iraj Kazemi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Iraj Kazemi.

Appendix

Appendix

Lemma 1

(Arellano-Valle and Genton 2005) Let \(\varvec{y}\sim N_{m}\left( \varvec{\mu },\varvec{\varSigma }\right) \). Then for any fixed \(k\)-dimensional vector \(\varvec{a}\) and \(k\times m\) matrix \(\varvec{D}\),

$$\begin{aligned} E\left( \varPhi _{k}\left( \varvec{a}+\varvec{Dy}|\varvec{\mu }_0 ,\varvec{ \varSigma }_0 \right) \right) =\varPhi _{k}\left( \varvec{a}~|~\varvec{\mu }_0 - \varvec{D\mu },\varvec{\varSigma }_0 +\varvec{D\varSigma D}^{\prime }\right). \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rikhtehgaran, R., Kazemi, I. Semi-parametric Bayesian estimation of mixed-effects models using the multivariate skew-normal distribution. Comput Stat 28, 2007–2027 (2013). https://doi.org/10.1007/s00180-012-0392-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-012-0392-3

Keywords

Search

Navigation