Modified Robust Design Criteria for Poisson Mixed Models

  • Hongyan Jiang
  • Rongxian YueEmail author


The maximin D-optimal design (MMD-optimal design) and hypercube design (HCD-optimal design) are two robust designs which overcome the problem of design dependence on the unknown parameters. This article considers the robust designs for Poisson mixed models. Given the prior knowledge of the fixed effects parameters, a modification of the two robust design criteria is proposed by applying the number-theoretic methods. The simulated annealing algorithm is used to find the optimal exact designs. The results show that the modified optimal designs perform better in the relative D-efficiency and programming time.



The work is supported by the NSFC grants 11971318, 11871143.


  1. 1.
    Atkinson, A.C., Cox, D.C.: Planning experiments for discriminating between models. J. Roy. Statist. Soc. Ser. B 36, 321–348 (1974)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Atkinson, A.C., Donev, A.N., Tobias, R.D.: Optimal Experimental Designs, With SAS. Oxford university Press, New York (2007)Google Scholar
  3. 3.
    Bohachevsky, I.O., Johnson, M.E., Stein, M.L.: Generalized simulated annealing for function optimization. Technometrics 28, 209–217 (1986)CrossRefGoogle Scholar
  4. 4.
    Dette, H., Haines, L.M., Imhof, L.A.: Maximin and Bayesian optimal designs for regression models. Statist. Sinica 17, 463–480 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Fang, K.T., Wang, Y.: Number-Theoretic Methods in Statistics. Chapman and Hall, London (1994)CrossRefGoogle Scholar
  6. 6.
    Haines, L.M.: The application of the annealing algorithm to the construction of exact optimal designs for linear regression models. Technometrics 29, 439–447 (1987)zbMATHGoogle Scholar
  7. 7.
    Haines, L.M.: A geometric approach to optimal design for one-parameter non-linear models. J. Roy. Statist. Soc. Ser. B 57, 575–598 (1995)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Khuri, A.I., Mukherjee, B., Sinha, B.K., Ghosh, M.: Design issue for generalized linear models: a review. Statist. Sci. 21, 376–399 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Foo, L.K., Duffull, S.: Methods of robust design of nonlinear models with an application to pharmacokinetics. J. Biopharm. Statist. 20, 886–902 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    McCullagh, P.M., Nelder, J.A.S.: Generalized Linear Models, 2nd edn. Chapman and Hall, London (1989)CrossRefGoogle Scholar
  11. 11.
    McGree, J.M., Duffull, S.B., Eccleston, J.A.: Optimal design for studying bioimpedance. Physiol. Meas. 28, 1465–1483 (2007)CrossRefGoogle Scholar
  12. 12.
    Niaparast, M.: On optimal design for a Poisson regression model with random intercept. Statist. Probab. Lett. 79, 741–747 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Niaparast, M.: On optimal design for mixed effects Poisson regression models. Ph.D. thesis, Otto-von-Guericke University, Magdeburg (2010)Google Scholar
  14. 14.
    Niaparast, M., Schwabe, R.: Optimal design for quasi-likelihood estimation in Poisson regression with random coefficients. J. Statist. Plann. Inference 143, 296–306 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wang, Y.P.: Optimal experimental designs for the Poisson regression model in toxicity studies. Ph.D. thesis, Virginia Polytechnic Institute and State University (2002)Google Scholar
  16. 16.
    Wedderburn, R.W.M.: Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika 61, 439–447 (1974)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Wong, W.K.: A unified approach to the construction of minimax designs. Biometrika 79, 611–619 (1992)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhou, M., Wang, W.J.: Representative points of Student’s \(t_{n}\) distribution and their application in statistical simulation (in Chinese). Acta. Math. Appl. Sinica. Chin. Ser. 39, 620–640 (2016)zbMATHGoogle Scholar
  19. 19.
    Zhou, Y.D., Fang, K.T.: FM-criterion for representative points (in Chinese). Sci. Chin. Math. 49, 1009–1020 (2019)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Huaiyin Institute of TechnologyJiangsuChina
  2. 2.College of Mathematics and ScienceShanghai Normal UniversityShanghaiChina

Personalised recommendations