Statistics and Computing

, Volume 29, Issue 3, pp 571–583 | Cite as

Semiparametric Bayesian analysis for longitudinal mixed effects models with non-normal AR(1) errors

  • Junshan Shen
  • Hanjun Yu
  • Jin Yang
  • Chunling LiuEmail author


This paper studies Bayesian inference on longitudinal mixed effects models with non-normal AR(1) errors. We model the nonparametric zero-mean noise in the autoregression residual with a Dirichlet process (DP) mixture model. Applying the empirical likelihood tool, an adjusted sampler based on the Pólya urn representation of DP is proposed to incorporate information of the moment constraints of the mixing distribution. A Gibbs sampling algorithm based on the adjusted sampler is proposed to approximate the posterior distributions under DP priors. The proposed method can easily be extended to address other moment constraints owing to the wide application background of the empirical likelihood. Simulation studies are used to evaluate the performance of the proposed method. Our method is illustrated via the analysis of a longitudinal dataset from a psychiatric study.


Autocorrelation Dirichlet process mixture models Empirical likelihood Pólya urn representation Random effects 



The authors appreciate the Associate Editor and three referees for their constructive comments and guidance to the current version. The authors thank Prof. Yuichi Kitamura for providing their slide of the manuscript and Prof. Min-Hui Chen for his suggestions during the revision. The first author was partially supported by the National Natural Science Foundation of China (11171230, 11471024); the research of the first two authors was supported by the Scientific Research Level Improvement Quota Project of Capital University of Economics and Business; the research of the third author was supported by the postgraduate studentship of Hong Kong Polytechnic University; and the research of the fourth author was partially supported by the National Natural Science Foundation of China 11401502, the Hong Kong Polytechnic University fund G-UB01, and the General Research Fund of Hong Kong 15327216.


  1. Alquier, P., Friel, N., Everitt, R., Boland, A.: Noisy monte carlo: convergence of markov chains with approximate transition kernels. Stat. Comput. 26(1–2), 29–47 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Antoniak, C.E.: Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Stat. 2, 1152–1174 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Arnau, J., Bono, R., Blanca, M.J., Bendayan, R.: Using the linear mixed model to analyze nonnormal data distributions in longitudinal designs. Behav. Res. Methods 44, 1224–1238 (2012)CrossRefGoogle Scholar
  4. Bartolucci, F., Bacci, S.: Longitudinal analysis of self-reported health status by mixture latent auto-regressive models. J. R. Stat. Soc. Ser. C 63, 267–288 (2014)MathSciNetCrossRefGoogle Scholar
  5. Blackwell, D., MacQueen, J.B.: Ferguson distributions via pólya urn schemes. Ann. Stat. 1, 353–355 (1973)CrossRefzbMATHGoogle Scholar
  6. Brunner, L.J., Lo, A.Y.: Bayes methods for a symmetric unimodal density and its mode. Ann. Stat. 17, 1550–1566 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Chi, E.M., Reinsel, G.C.: Models for longitudinal data with random effects and AR(1) errors. J. Am. Stat. Assoc. 84, 452–459 (1989)MathSciNetCrossRefGoogle Scholar
  8. Choi, H.: Expert information and nonparametric Bayesian inference of rare events. Bayesian Anal. 11(2), 421–445 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Damsleth, E., El-Shaarawi, A.: Arma models with double-exponentially distributed noise. J. R. Stat. Soc. Ser. B (Methodol.) 51, 61–69 (1989)MathSciNetGoogle Scholar
  10. Escobar, M.D.: Estimating normal means with a Dirichlet process prior. J. Am. Stat. Assoc. 89, 268–277 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Escobar, M.D., West, M.: Bayesian density estimation and inference using mixtures. J. Am. Stat. Assoc. 90, 577–588 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Fan, T.H., Wang, Y.F., Zhang, Y.C.: Baysian model selection in linear mixed effects models with autoregressive (p) errors using mixture priors. J. Appl. Stat. 41, 1814–1829 (2014)MathSciNetCrossRefGoogle Scholar
  13. Ferguson, T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat. 1, 209–230 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Geweke J (1992) Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics. Oxford Univ. Press, pp 169–193Google Scholar
  15. Goldstein, H., Healy, M.J., Rasbash, J.: Multilevel time series models with applications to repeated measures data. Stat. Med. 13, 1643–1655 (1994)CrossRefGoogle Scholar
  16. Griffin, J.E.: An adaptive truncation method for inference in Bayesian nonparametric models. Stat. Comput. 26, 423–441 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Hedeker, D., Gibbons, R.D.: Longitudinal Data Analysis. Wiley, London (2006)zbMATHGoogle Scholar
  18. Hoff, P.D.: Constrained nonparametric estimation via mixtures. Ph.D. thesis, Department of Statistics, University of Wisconsin (2000)Google Scholar
  19. Hoff, P.D.: Nonparametric estimation of convex models via mixtures. Ann. Stat. 31, 174–200 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Kitamura, Y., Otsu, T.: Bayesian analysis of moment condition models using nonparametric priors. Technical Report, Yale University (2011)Google Scholar
  21. Kleinman, K.P., Ibrahim, J.G.: A semiparametric Bayesian approach to the random effects model. Biometrics 54, 921–938 (1998)CrossRefzbMATHGoogle Scholar
  22. Laird, N.M., Ware, J.H.: Random-effects models for longitudinal data. Biometrics 38, 963–974 (1982)CrossRefzbMATHGoogle Scholar
  23. Lazar, N.A.: Bayesian empirical likelihood. Biometrika 90(2), 319–326 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Lee, J.C., Niu, W.F.: On an unbalanced growth curve model with random effects and AR(1) errors from a Bayesian and the ML points of view. J. Stat. Plan. Inference 76, 41–55 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Li, Y., Müller, P., Lin, X.: Center-adjusted inference for a nonparametric Bayesian random effect distribution. Stat. Sin. 21, 1201–1223 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Luo, Y., Lian, H., Tian, M.: Bayesian quantile regression for longitudinal data models. J. Stat. Comput. Simul. 82, 1635–1649 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. MacEachern, S.N., Müller, P.: Estimating mixture of Dirichlet process models. J. Comput. Graph. Stat. 7, 223–238 (1998)Google Scholar
  28. Neal, R.M.: Markov chain sampling methods for Dirichlet process mixture models. J. Comput. Graph. Stat. 9, 249–265 (2000)MathSciNetGoogle Scholar
  29. Owen, A.B.: Empirical Likelihood. Chapman & Hall/CRC, London (2001)CrossRefzbMATHGoogle Scholar
  30. Reich, B.J., Bondell, H.D., Wang, H.J.: Flexible Bayesian quantile regression for independent and clustered data. Biostatistics 11, 337–352 (2010)CrossRefGoogle Scholar
  31. Reisby, N., Gram, L.F., Bech, P., Nagy, A., Petersen, G.O., Ortmann, J., Ibsen, I., Dencker, S.J., Jacobsen, O., Krautwald, O.: Imipramine: clinical effects and pharmacokinetic variability. Psychopharmacology 54, 263–272 (1977)CrossRefGoogle Scholar
  32. Roberts, G., Rosenthal, J., Schwartz, P.: Convergence properties of perturbed markov chains. J. Appl. Probab. 35(1), 1–11 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Sethuraman, J.: A constructive definition of Dirichlet priors. Stat. Sin. 4, 639–650 (1994)MathSciNetzbMATHGoogle Scholar
  34. Shin, M.: Bayesian generalized method of moments. Technical Report, University of Illinois (2014)Google Scholar
  35. Tiku, M.L., Wong, W.K., Vaughan, D.C., Bian, G.: Time series models in non-normal situations: symmetric innovations. J. Time Ser. Anal. 21, 571–596 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  36. Wang, W.L., Fan, T.H.: Estimation in multivariate \(t\) linear mixed models for multiple longitudinal data. Stat. Sin. 21, 1857–1880 (2011)MathSciNetzbMATHGoogle Scholar
  37. Yang, M., Dunson, D.B., Baird, D.: Semiparametric Bayes hierarchical models with mean and variance constraints. Comput. Stat. Data Anal. 54, 2172–2186 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of StatisticsCapital University of Economics and BusinessFengtai, BeijingPeople’s Republic of China
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHung Hom, KowloonHong Kong

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