Estimation of Individual Parameters of a Mixed–Effects Dose-Toxicity Model for Ordinal Data
Inès Paule (1), Pascal Girard (1), Michel Tod (1,2)
(1) Université de Lyon, Lyon, France ; (2) EA3738 CTO, Faculté de Médecine Lyon-Sud, Université Lyon 1, Oullins, France; (3) Hôpital Croix-Rousse, Hospices Civils de Lyon, Lyon, France.
Objectives: This work addresses the issue of estimating the individual effects (empirical Bayes estimates (EBEs)) of mixed-effects models for ordered categorical data (e.g. undesirable effects).
Methods: Different algorithms for estimating the EBEs were compared by a simulation study: 2 local optimizers: the quasi-Newton method (as implemented in NONMEM software) and the simplex; a global optimization method Recursive Random Search (RRS) and the estimation of full posterior distribution by MCMC sampling (in WinBUGS). The first three methods provide single maximum a posteriori (MAP) point estimates (modes), which were compared to means of the posterior distributions given by MCMC sampling. The comparison is made in terms of accuracy and precision of point estimates, as well as approximate run time. The model used in this simulation study is a kinetic-pharmacodynamic (KPD) mixed-effects proportional odds Markov model for ordinal data . A sensitivity analysis investigated the impact of the richness of data, of the extent of variability of the random effects, and of the strength of the relationship between the outcome (toxicity grades) and the explanatory variable (dose).
Results: All tested optimisation algorithms gave similar results (those of the RRS and the simplex were almost identical, with the simplex being 60 times faster). The EBEs had some bias and quite low precision. Means of posterior distributions were more accurate estimates than modes. Sensitivity analysis showed a significant impact of data richness and especially of the identifiability of the model, that is a weak relationship between the outcome (toxicity grade) and the explanatory variable (drug dose) impedes the correct estimation of EBEs.
Conclusions: Correct EBEs of categorical data models may be obtained only in particularly informative conditions and only for individuals having some toxicity, and the quality of estimates is insensitive to the estimation method used. In those conditions, the simplex gives as good results as a global optimizer, and is much faster.
 Henin, E. et al. Clin. Pharmacol. Ther. 85, 4, 418-425 (2009)