Statistical Inference For Non-linear Mixed Effects Models Involving Ordinary Differential Equations

Abstract

In the context of nonlinear mixed effect modeling, 'within subject mechanisms' are often represented by a system of nonlinear ordinary differential equations (ODE), whose parameters characterize the different characteristics of the underlying population. These models are useful because they offer a flexible framework where parameters for both individuals and population can be estimated by combining information across all subjects. Estimating parameters for these models becomes challenging in the absence of any analytical solution for the system of ODEs involved in the modeling.

In this thesis we proposed two estimation approaches (i) Bayesian Euler's Approximation Method (BEAM) and (ii) Splines Euler's Approximation Method (SEAM). While we proposed SEAM only for the fixed effect models, BEAM is described for fixed as well as mixed effects models. Both of these approaches involve the likelihood approximation based on the naive Euler's numerical approximation method, thereby providing an analytic closed form approximation for the mean function. SEAM combines the Euler's approximation with Spline interpolation to obtain the parameter estimates for each subject separately. On the other hand, BEAM combines the likelihood approximation with the existing Bayesian hierarchical modeling framework to obtain the parameter estimates.

For illustration purposes, we presented the real data analyses and simulation studies for both fixed and mixed effects models and compared the results with estimates from the NLS method (fixed effects model) and from the NLME method (mixed effects model). For both type of models, proposed methodologies provide competitive results in terms of estimation accuracy and efficiency. The Bayesian Euler's approximation method was also used to estimate parameters involved in an HIV model, for which an analytical closed form mean function is not available.