Parameter estimation in non-linear mixed effects models using interval analysis
Andrew C. Hooker (1) and Warwick Tucker (2)
(1) Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden. (2) Department of Mathematics, University of Bergen, Bergen, Norway.
Objectives: Parameter estimation for non-linear mixed effects models is an important aspect of model based drug development. Typically, these models are fit to data using a maximum likelihood approach requiring basic statistical assumptions, and often the models are linearized resulting in model misspecification. Further, the solution to these maximum likelihood based methods often require good initial estimates for parameters as they are prone to getting trapped in local optima. In this work we present a method of parameter estimation based on interval analysis (IA) to circumvent these problems . Our method encloses the set of parameters for a given model that are consistent with the data set, and does not rely on maximum likelihood. As a result, no statistical assumptions are made, and model linearization is not needed. Further, the resulting parameter sets consistent with the data can be interpreted as confidence intervals for parameter estimates, so no asymptotic assumptions based on the Fisher information matrix, or bootstrap methodologies, are needed to compute standard errors. Finally, interval methods are global search methods that guarantee that no solutions within the search space are lost.
Methods: Both maximum likelihood (using the FOCE with interaction method in NONMEM ) and IA methods (implemented in C++) were used to fit a variety of simulated pharmacometric data. The parameter interval estimates computed with IA were then compared to the parameter estimates and standard errors computed with maximum likelihood.
Results: Parameters with no between subject variability were shown to be in good agreement between the two methods. The IA method tended to have a larger range for possible parameter values as the method provides intervals for the parameter estimates that correspond to all the possible solutions that would allow the model to fit the data (compared to confidence intervals in the maximum likelihood approach).
Conclusions: Interval analysis is a suitable foundation for parameter estimation in non-linear mixed effects models. The proposed method reduces the assumptions and approximations needed to estimate model parameters, and the method guarantees finding all parameter values that are consistent with the model. Because of this, IA methods are a natural choice in investigating, among others, model identifiability, the presence of inter-individual variability and model misspecification.
 Tucker, W., Z. Kutalik, and V. Moulton, Estimating parameters for generalized mass action models using constraint propagation. Math Biosci, 2007. 208(2): p. 607-20.