Maximum Likelihood Estimation Methods: Performances in Count Response Models Population Parameters
Elodie Plan (1), Alan Maloney (2), Iñaki F. Trocóniz (3), Mats O. Karlsson (1)
(1) Division of Pharmacokinetics and Drug Therapy, Department of Pharmaceutical Biosciences, Faculty of Pharmacy, Uppsala University, Uppsala, Sweden; (2) Exprimo NV, Lummen, Belgium; (3) Department of Pharmacy and Pharmaceutical Technology, School of Pharmacy, University of Navarra, Pamplona, Spain.
Objectives: Count responses represent a full class of common pharmacodynamic outcomes, there is a need for greater knowledge in Non-Linear Mixed Effects approach for modelling discrete observations.
Laplacian approximation iteratively determines the 2nd derivative of the maximum of the Likelihood [1] based on 1 quadrature point, a value that can be increased using the Gaussian Quadrature (GQ).
The aim of this study was to explore different probability distribution models and the accuracy of the estimation of their population parameters using different methods (LAPLACE and GQ) and programs (NONMEM VI and SAS(NLMIXED)).
Methods: This methodological study was performed through stochastic Monte Carlo simulations followed by re-estimations. All simulations (100 data sets for each probability distribution) were performed in NONMEM; parameters values used were derived from a real case study on 551 epileptic patients [2].
Count models investigated during this study were: Poisson (PS), Poisson with Markovian features (PMAK), Poisson with a mixture distribution for individual observations (PMIX), Zero Inflated Poisson (ZIP), Generalized Poisson (GP) and Inverse Binomial (INB) [3]. Estimations of the simulated data sets were completed with LAPLACE in NONMEM and LAPLACE/GQ in SAS.
Performances were evaluated by computing: (i) the absolute value of the relative bias (AVB) and (ii) the root mean square error (RMSE) of the estimated population parameters compared to the true ones.
Results: With LAPLACE in NONMEM, the AVB in fixed effects was < 3.4 % in all models; it was even < 0.8 % in PS, PMAK and PMIX. The RMSE was 3.9 – 10.4 % with lowest values for the mean count parameter (λ).
The estimation of the random effect of λ resulted in an AVB ranged 0.3 – 8.2 % in all models. The magnitude of the random effect of the dispersion parameter present in ZIP, GP and INB showed the largest bias (-25.9, -15.7 and -21.9 % respectively).
Analysis with GQ resulted in an adjustment of parameter bias (> -5.3 %).
Conclusions: The PS count model parameters were accurately estimated in NONMEM; addition of Markovian features or Mixture distribution did not harm the relative estimation error profile, but inclusion of a parameter allowing for inequality between λ and variance of the counts was associated with a marked bias in the estimation of its variability.
References:
[1] Wang Y. Derivation of various NONMEM estimation methods. Journal of Pharmacokinetics and pharmacodynamics. 34:575-93 (2007)
[2] Trocóniz IF, Plan EL, Miller R, Karlsson MO. Modelling Overdispersion and Markovian Features in Count Data. American Conference on Pharmacometrics, Tucson, Arizona. (2008)
[3] Del Castilloa J, Pérez-Casany M. Overdispersed and underdispersed Poisson generalizations. Journal of Statistical Planning and Inference. 134,2:486-500 (2005)
