Performance in population models for count data, part II: A new SAEM algorithm

  • Radojka SavicEmail author
  • Marc Lavielle


Analysis of count data from clinical trials using mixed effect analysis has recently become widely used. However, algorithms available for the parameter estimation, including LAPLACE and Gaussian quadrature (GQ), are associated with certain limitations, including bias in parameter estimates and the long analysis runtime. The stochastic approximation expectation maximization (SAEM) algorithm has proven to be a very efficient and powerful tool in the analysis of continuous data. The aim of this study was to implement and investigate the performance of a new SAEM algorithm for application to count data. A new SAEM algorithm was implemented in MATLAB for estimation of both, parameters and the Fisher information matrix. Stochastic Monte Carlo simulations followed by re-estimation were performed according to scenarios used in previous studies (part I) to investigate properties of alternative algorithms (Plan et al., 2008, Abstr 1372 [http://wwwpage-meetingorg/?abstract=1372]). A single scenario was used to explore six probability distribution models. For parameter estimation, the relative bias was less than 0.92% and 4.13% for fixed and random effects, for all models studied including ones accounting for over- or under-dispersion. Empirical and estimated relative standard errors were similar, with distance between them being <1.7% for all explored scenarios. The longest CPU time was 95 s for parameter estimation and 56 s for SE estimation. The SAEM algorithm was extended for analysis of count data. It provides accurate estimates of both, parameters and standard errors. The estimation is significantly faster compared to LAPLACE and GQ. The algorithm is implemented in Monolix 3.1, (beta-version available in July 2009).


Nonlinear mixed effects model Count data Maximum likelihood estimation Fischer information matrix SAEM algorithm MONOLIX 



Radojka Savic was financially supported by a Postdoc grant from the Swedish Academy of Pharmaceutical Sciences (Apotekarsocieteten). We thank Dr. Shasha Jumbe for valuable comments on the manuscript.


  1. 1.
    Frame B, Miller R, Lalonde RL (2003) Evaluation of mixture modeling with count data using NONMEM. J Pharmacokinet Pharmacodyn 30:167–183PubMedCrossRefGoogle Scholar
  2. 2.
    Troconiz IF, Plan EL, Miller R, Karlsson MO (2008) Modelling overdispersion and Markovian features in count data. American Conference on Pharmacometrics, Tucson, AZGoogle Scholar
  3. 3.
    Beal SL, Sheiner LB, Boeckmann AJ (1989–2006) NONMEM users guides, icon development solutions, Ellicott City, Maryland, USAGoogle Scholar
  4. 4.
    Ette EI, Williams PJ (2007) Pharmacometrics: the science of quantitative pharmacology. Wiley, HobokenGoogle Scholar
  5. 5.
    Wang Y (2007) Derivation of various NONMEM estimation methods. J Pharmacokinet Pharmacodyn 34:575–593PubMedCrossRefGoogle Scholar
  6. 6.
    SAS Institute Inc SAS Online Doc. SAS/STAT User’s Guide, The NLMIXED Procedure.
  7. 7.
    Plan EL, Maloney A, Troconiz IF, Karlsson MO (2008) Maximum likelihood approximations: performance in population models for count data, p. 17, Abstr 1372 [http://wwwpage-meetingorg/?abstract=1372]. Accessed January 2009
  8. 8.
    Clausen WHO, Ronn BB, Skoovgard IM (2009) Maximum likelihood estimation in nonlinear mixed effect models: Adaptive Gaussian Quadrature by sparse grid sampling, p 18, Abstr 1584 [http://wwwpage-meetingorg/?abstract=1584]. Accessed June 2009
  9. 9.
    Jonsson S, Kjellsson MC, Karlsson MO (2004) Estimating bias in population parameters for some models for repeated measures ordinal data using NONMEM and NLMIXED. J Pharmacokinet Pharmacodyn 31:299–320PubMedCrossRefGoogle Scholar
  10. 10.
    Verbeke G (2006) Mixed models for the analysis of categorical repeated measures, p 15, Abstr 930 [http://wwwpage-meetingorg/?abstract=930]. Accessed June 2009
  11. 11.
    Bauer RJ, Guzy S, Ng C (2007) A survey of population analysis methods and software for complex pharmacokinetic and pharmacodynamic models with examples. AAPS J 9:E60–E83PubMedCrossRefGoogle Scholar
  12. 12.
    Kuhn E, Lavielle M (2005) Maximum likelihood estimation in nonlinear mixed effects models. Comput Stat Data Anal 49:1020–1038CrossRefGoogle Scholar
  13. 13.
    MONOLIX 2.4 User Guide, Vol. Accessed January 2009
  14. 14.
    Girard P, Mentre F (2005) A comparison of estimation methods in nonlinear mixed effects models using a blind analysis, p 14, Abstr 834 [http://wwwpage-meetingorg/?abstract=834], Pamplona, Spain
  15. 15.
    Brendel K, Comets E, Laffont C, Laveille C, Mentre F (2006) Metrics for external model evaluation with an application to the population pharmacokinetics of gliclazide. Pharm Res 23:2036–2049PubMedCrossRefGoogle Scholar
  16. 16.
    Karlsson MO, Holford N (2008) A tutorial on visual predictive checks, p 17, Abstr 1434 [http://wwwpage-meetingorg/?abstract=1434], Marseille. Accessed January 2009
  17. 17.
    Savic RM, Karlsson MO (2009) Importance of shrinkage in empirical bayes estimates for diagnostics: problems and solutions. AAPS J (in press)Google Scholar
  18. 18.
    Wahlby U, Jonsson EN, Karlsson MO (2001) Assessment of actual significance levels for covariate effects in NONMEM. J Pharmacokinet Pharmacodyn 28:231–252PubMedCrossRefGoogle Scholar
  19. 19.
    Lavielle M, Mentre F (2007) Estimation of population pharmacokinetic parameters of saquinavir in HIV patients with the MONOLIX software. J Pharmacokinet Pharmacodyn 34:229–249PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.UMR 738 INSERM - Université Paris Diderot - Paris 7ParisFrance
  2. 2.Laboratoire de MathématiquesINRIA Saclay and University Paris-Sud 11OrsayFrance

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