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

Abstract

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.

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References

  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)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google Scholar 

  5. Blackwell, D., MacQueen, J.B.: Ferguson distributions via pólya urn schemes. Ann. Stat. 1, 353–355 (1973)

    Article  MATH  Google Scholar 

  6. Brunner, L.J., Lo, A.Y.: Bayes methods for a symmetric unimodal density and its mode. Ann. Stat. 17, 1550–1566 (1989)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  Google Scholar 

  8. Choi, H.: Expert information and nonparametric Bayesian inference of rare events. Bayesian Anal. 11(2), 421–445 (2016)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Google Scholar 

  10. Escobar, M.D.: Estimating normal means with a Dirichlet process prior. J. Am. Stat. Assoc. 89, 268–277 (1994)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  Google Scholar 

  13. Ferguson, T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat. 1, 209–230 (1973)

    MathSciNet  Article  MATH  Google 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 169–193

  15. Goldstein, H., Healy, M.J., Rasbash, J.: Multilevel time series models with applications to repeated measures data. Stat. Med. 13, 1643–1655 (1994)

    Article  Google Scholar 

  16. Griffin, J.E.: An adaptive truncation method for inference in Bayesian nonparametric models. Stat. Comput. 26, 423–441 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  17. Hedeker, D., Gibbons, R.D.: Longitudinal Data Analysis. Wiley, London (2006)

    Google Scholar 

  18. Hoff, P.D.: Constrained nonparametric estimation via mixtures. Ph.D. thesis, Department of Statistics, University of Wisconsin (2000)

  19. Hoff, P.D.: Nonparametric estimation of convex models via mixtures. Ann. Stat. 31, 174–200 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  20. Kitamura, Y., Otsu, T.: Bayesian analysis of moment condition models using nonparametric priors. Technical Report, Yale University (2011)

  21. Kleinman, K.P., Ibrahim, J.G.: A semiparametric Bayesian approach to the random effects model. Biometrics 54, 921–938 (1998)

    Article  MATH  Google Scholar 

  22. Laird, N.M., Ware, J.H.: Random-effects models for longitudinal data. Biometrics 38, 963–974 (1982)

    Article  MATH  Google Scholar 

  23. Lazar, N.A.: Bayesian empirical likelihood. Biometrika 90(2), 319–326 (2003)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  26. Luo, Y., Lian, H., Tian, M.: Bayesian quantile regression for longitudinal data models. J. Stat. Comput. Simul. 82, 1635–1649 (2012)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Google Scholar 

  29. Owen, A.B.: Empirical Likelihood. Chapman & Hall/CRC, London (2001)

    Google 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)

    Article  Google 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)

    Article  Google Scholar 

  32. Roberts, G., Rosenthal, J., Schwartz, P.: Convergence properties of perturbed markov chains. J. Appl. Probab. 35(1), 1–11 (1998)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  34. Shin, M.: Bayesian generalized method of moments. Technical Report, University of Illinois (2014)

  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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

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.

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Correspondence to Chunling Liu.

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Shen, J., Yu, H., Yang, J. et al. Semiparametric Bayesian analysis for longitudinal mixed effects models with non-normal AR(1) errors. Stat Comput 29, 571–583 (2019). https://doi.org/10.1007/s11222-018-9824-4

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Keywords

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