Computational Statistics

, Volume 34, Issue 1, pp 71–87 | Cite as

Pseudo-Bayesian D-optimal designs for longitudinal Poisson mixed models with correlated errors

  • Hong-Yan Jiang
  • Rong-Xian YueEmail author
Original Paper


This paper is concerned with the problem of pseudo-Bayesian D-optimal designs for the first-order Poisson mixed model for longitudinal data with time-dependent correlated errors. A standard approximate covariance matrix of the parameter estimation is obtained based on the quasi-likelihood method. Furthermore, to overcome the dependence of pseudo-Bayesian D-optimal designs on the choice of the prior mean, a hierarchical pseudo-Bayesian D-optimal designs based on the hierarchical prior distribution of unknown parameters is proposed. The results show that the optimal number of time points depends on both the interclass autoregressive coefficients and different cost constraints. The relative efficiency of equidistant designs compared with the hierarchical pseudo-Bayesian D-optimal designs is also discussed.


Quasi-likelihood method Estimating equation Repeated measures Hierarchical design Longitudinal data 


  1. Abebe HT, Tan FES, Breukelen VGJP, Berger MPF (2014) Bayesian D-optimal designs for the two parameter logistic mixed effects model. Comput Stat Data Anal 71(1):1066–1076MathSciNetCrossRefzbMATHGoogle Scholar
  2. Atkinson AC, Donev AN, Tobias RD (2007) Optimal experimental designs, with SAS. Oxford University Press, New YorkzbMATHGoogle Scholar
  3. Casella G, Berger RL (2001) Statistical inference, 2nd edn. Duxbury Press, LondonzbMATHGoogle Scholar
  4. Chi EM, Reinsel GC (1989) Models for longitudinal data with random effects and AR(1) errors. J Am Stat Assoc 84(406):452–459MathSciNetCrossRefGoogle Scholar
  5. Fang K-T, Wang Y (1994) Number-theoretic methods in statistics. Chapman and Hall, LondonCrossRefzbMATHGoogle Scholar
  6. Fedorov VV, Leonov SL (2014) Optimal design for nonlinear response models. Chapman and Hall/CRC Press, Boca RatonzbMATHGoogle Scholar
  7. Ford I, Torsney B, Wu CFJ (1992) The use of a canonical form in the construction of locally optimal designs for non-linear problems. J R Stat Soc Ser B 54(2):569–583zbMATHGoogle Scholar
  8. Khuri AI, Mukherjee B, Sinha BK, Ghosh M (2006) Design issues for generalized linear models: a review. Stat Sci 21(3):376–399MathSciNetCrossRefzbMATHGoogle Scholar
  9. Li W, Nachtsheim CJ (2000) Model-robust factorial designs. Technometrics 42:345–352CrossRefGoogle Scholar
  10. Liang KY, Zeger SL (1986) Longitudinal data analysis using generalized linear models. Biometrika 78:13–22MathSciNetCrossRefzbMATHGoogle Scholar
  11. Lin TI, Wang WL (2013) Multivariate skew-normal at linear mixed models for multi-outcome longitudinal data. Stat Model 13(3):199–221MathSciNetCrossRefGoogle Scholar
  12. McCullagh P, Nelder JA (1989) Generalized linear models, 2nd edn. Chapman and Hall, New YorkCrossRefzbMATHGoogle Scholar
  13. Mcgree JM, Eccleston JA (2012) Robust designs for poisson regression models. Technometrics 54(1):64–72MathSciNetCrossRefGoogle Scholar
  14. Mcgree JM, Duffull SB, Eccleston JA, Ward LC (2007) Optimal designs for studying bioimpedance. Physiol Meas 28(12):1465CrossRefGoogle Scholar
  15. Molenberghs G, Verbeke G (2005) Models for discrete longitudinal data. Springer, New YorkzbMATHGoogle Scholar
  16. Niaparast M (2010) Optimal designs for mixed effects poisson regression models. Ph.D. thesis, Magdeburg University, GermanyGoogle Scholar
  17. Niaparast M, Schwabe R (2013) Optimal design for quasi-likelihood estimation in poisson regression with random coefficients. J Stat Plan Inference 143:296–306MathSciNetCrossRefzbMATHGoogle Scholar
  18. Ouwens MJNM, Tan FES, Berger MPF (2006) A maximin criterion for the logistic random intercept model with covariates. J Stat Plan Inference 136(3):962–981MathSciNetCrossRefzbMATHGoogle Scholar
  19. Roy A (2006) Estimating correlation coefficient between two variables with repeated observations using mixed effects model. Biom J 48(2):286–301MathSciNetCrossRefGoogle Scholar
  20. Russell KG, Woods DC, Lewis SM, Eccleston JA (2009) D-optimal designs for poisson regression models. Stat Sin 19(2):2831–2845MathSciNetzbMATHGoogle Scholar
  21. Ryan EG, Drovandi CC, Pettitt AN (2015) Simulation-based fully bayesian experimental design for mixed effects models. Comput Stat Data Anal 92:26–39MathSciNetCrossRefzbMATHGoogle Scholar
  22. Tan FES, Berger MPF (1999) Optimal allocation of time points for the random effects model. Commun Stat Simul Comput 28(2):517–540MathSciNetCrossRefzbMATHGoogle Scholar
  23. Tekle FB (2008) D-optimal designs for prospective cohort studies. Ph.D. thesis, Department of Methodology and Statistics, University of Maastricht, The NetherlandsGoogle Scholar
  24. Tekle FB, Tan FES, Berger MPF (2008) Maximin D-optimal designs for binary longitudinal responses. Comput Stat Data Anal 52(12):5253–5262MathSciNetCrossRefzbMATHGoogle Scholar
  25. Wang W-L, Fan T-H (2010) ECM-based maximum likelihood inference for multivariate linear mixed models with autoregressive errors. Comput Stat Data Anal 54(5):1328–1341MathSciNetCrossRefzbMATHGoogle Scholar
  26. Wang Y, Myers RH, Smoth EP, Ye K (2006) D-optimal designs for poisson regression models. J Stat Plan Inference 136:2831–2845MathSciNetCrossRefzbMATHGoogle Scholar
  27. Wedderburn RWM (1974) Quasi-likelihood functions, generalized linear models, and the gauss-newton method. Biometrika 61(3):439–447MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and ScienceShanghai Normal UniversityShanghaiChina
  2. 2.Department of Mathematics and PhysicsHuaiyin Institute of TechnologyHuaianChina
  3. 3.Scientific Computing Key Laboratory of Shanghai UniversitiesShanghaiChina

Personalised recommendations