A proposal for implementation of the Markov property into a continuous time transition state model in NONMEM
N. Snelder (2), C. Diack (2), N. Benson (1), B. Ploeger (2), P. van der Graaf (1)
(1) Pfizer Global Research & Development, Sandwich, UK; (2) LAP&P Consultants, Leiden, The Netherlands
Objectives: Repeated measures of ordered categorical data are typically described in NONMEM using proportional odds models. These models often result in an adequate description of the data. However, a limitation of proportional odds models is the underlying assumption that, given the random effects, the observations are independent. In specific situations this may result in an overestimation of inter-individual variability, resulting in individual profiles that are physiologically implausible [1]. When dependency between observations is an issue, Markov models are better suited than proportional odds models since they inherently assume that future events depend on present events. Initially, this property was implemented in a hybrid model, a proportional odds model where the probabilities are dependent on the preceding stage through a first order Markov element [2]. As this model is discrete in time the sampling scheme can play an important role. Therefore, we have further developed this model by implementing the Markov property in a continuous time transition state model in NONMEM using the Kolmogorov backward equations [3].
Methods: Categorical measurements from the monosodium iodoacetate induced arthritis model were described by a proportional odds model and by a transition state model including the Markov property. The transition state model was implemented using ordinary differential equations such that each differential equation represented the probability for a certain category. The Markov property was implemented using the Kolmogorov backward equations. Subsequently, simulations were performed with both models and a comparison was made between the simulated and original data.
Results: Both models resulted in a comparable description of the population data. However, the transition state model with the Markov property resulted in a better prediction of the individual data as demonstrated by simulations.
Conclusion: The Markov property was successfully implemented in a transition state model in NONMEM using the Kolmogorov backward equations. Simulations with this model resulted in physiologically plausible individual profiles. Moreover, as the transition state model including the Markov property is continuous in time the influence of the sampling scheme on the parameter estimates is less as compared to the hybrid model. As a result, this model has better properties to simulate observations from different sampling schemes.
References
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[3] Sheldon M. Ross: Introduction to probability models. Seventh edition, 2000.
