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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
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)
Antoniak, C.E.: Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Stat. 2, 1152–1174 (1974)
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)
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)
Blackwell, D., MacQueen, J.B.: Ferguson distributions via pólya urn schemes. Ann. Stat. 1, 353–355 (1973)
Brunner, L.J., Lo, A.Y.: Bayes methods for a symmetric unimodal density and its mode. Ann. Stat. 17, 1550–1566 (1989)
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)
Choi, H.: Expert information and nonparametric Bayesian inference of rare events. Bayesian Anal. 11(2), 421–445 (2016)
Damsleth, E., El-Shaarawi, A.: Arma models with double-exponentially distributed noise. J. R. Stat. Soc. Ser. B (Methodol.) 51, 61–69 (1989)
Escobar, M.D.: Estimating normal means with a Dirichlet process prior. J. Am. Stat. Assoc. 89, 268–277 (1994)
Escobar, M.D., West, M.: Bayesian density estimation and inference using mixtures. J. Am. Stat. Assoc. 90, 577–588 (1995)
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)
Ferguson, T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat. 1, 209–230 (1973)
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
Goldstein, H., Healy, M.J., Rasbash, J.: Multilevel time series models with applications to repeated measures data. Stat. Med. 13, 1643–1655 (1994)
Griffin, J.E.: An adaptive truncation method for inference in Bayesian nonparametric models. Stat. Comput. 26, 423–441 (2016)
Hedeker, D., Gibbons, R.D.: Longitudinal Data Analysis. Wiley, London (2006)
Hoff, P.D.: Constrained nonparametric estimation via mixtures. Ph.D. thesis, Department of Statistics, University of Wisconsin (2000)
Hoff, P.D.: Nonparametric estimation of convex models via mixtures. Ann. Stat. 31, 174–200 (2003)
Kitamura, Y., Otsu, T.: Bayesian analysis of moment condition models using nonparametric priors. Technical Report, Yale University (2011)
Kleinman, K.P., Ibrahim, J.G.: A semiparametric Bayesian approach to the random effects model. Biometrics 54, 921–938 (1998)
Laird, N.M., Ware, J.H.: Random-effects models for longitudinal data. Biometrics 38, 963–974 (1982)
Lazar, N.A.: Bayesian empirical likelihood. Biometrika 90(2), 319–326 (2003)
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)
Li, Y., Müller, P., Lin, X.: Center-adjusted inference for a nonparametric Bayesian random effect distribution. Stat. Sin. 21, 1201–1223 (2011)
Luo, Y., Lian, H., Tian, M.: Bayesian quantile regression for longitudinal data models. J. Stat. Comput. Simul. 82, 1635–1649 (2012)
MacEachern, S.N., Müller, P.: Estimating mixture of Dirichlet process models. J. Comput. Graph. Stat. 7, 223–238 (1998)
Neal, R.M.: Markov chain sampling methods for Dirichlet process mixture models. J. Comput. Graph. Stat. 9, 249–265 (2000)
Owen, A.B.: Empirical Likelihood. Chapman & Hall/CRC, London (2001)
Reich, B.J., Bondell, H.D., Wang, H.J.: Flexible Bayesian quantile regression for independent and clustered data. Biostatistics 11, 337–352 (2010)
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)
Roberts, G., Rosenthal, J., Schwartz, P.: Convergence properties of perturbed markov chains. J. Appl. Probab. 35(1), 1–11 (1998)
Sethuraman, J.: A constructive definition of Dirichlet priors. Stat. Sin. 4, 639–650 (1994)
Shin, M.: Bayesian generalized method of moments. Technical Report, University of Illinois (2014)
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)
Wang, W.L., Fan, T.H.: Estimation in multivariate \(t\) linear mixed models for multiple longitudinal data. Stat. Sin. 21, 1857–1880 (2011)
Yang, M., Dunson, D.B., Baird, D.: Semiparametric Bayes hierarchical models with mean and variance constraints. Comput. Stat. Data Anal. 54, 2172–2186 (2010)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11222-018-9824-4