Description
Longitudinal (panel) studies focus on the models and data that arise from repeated measurements taken from a group of subjects. In this dissertation, we consider mixed effects modeling approaches with nonparametric estimation methods to model the dynamics in the longitudinal data, and explore the study in three focused research topics. Firstly, motivated by establishing a reliable quantitative investigation of the inequalities in developing countries, we use China’s health care expenditure data as an example, and adapt a semi-parametric varying-coefficients spatial panel data model. A profile likelihood based estimation procedure with a fully iterated two-step local estimation method is proposed to estimate the varying coefficients, spatial specific effects, and spatial autoregressive parameter. Simulation studies are conducted to examine the performance of the proposed estimators. The simulation results indicate our methods work very well. Comparison of the estimated coefficients indicates that the semi-parametric spatial varying coefficient panel data model surpasses the parametric one, confirming that the estimated coefficients are time-varying and can better describe the impact of those important factors. We secondly investigate human immunodeficiency virus (HIV) dynamic modeling. HIV dynamic models have been introduced to characterize short-term acquired immune deficiency syndrome (AIDS) treatment. However, in long-term HIV dynamics, viral load will typically increase over the treatment period due to the development of drug resistance. Time-varying drug resistance can be incorporated into the ordinary differential equations (ODEs) model, but often has no close form solution. In clinical practice, only viral load and CD4+ T-cell counts can be censored with measurement errors due to technical constraints. In this study, we consider a mixed-effects nonlinear parametric model describing short-term longitudinal AIDS treatment, along with a set of mixed-effects ODE models fitting long-term longitudinal AIDS clinical trial data. We propose a multi-stage estimation procedure to estimate the mean constant dynamic parameters and time-varying curve functions, and also quantify individual heterogeneity among subjects. Bootstrap confidence intervals for the constant dynamic parameters are calculated to evaluate the estimation procedure. The finite sample properties of the proposed estimator are studied via simulations and the methodology is also applied to a longitudinal AIDS clinical data set. The results suggest that the proposed estimation procedure is effective and appropriate to estimate both individual constant dynamic parameters and time-varying curve functions in long-term HIV dynamic models. Finally, we propose three deterministic linear dynamic models in general forms describing longitudinal dynamic systems, which are designed to be applicable across multiple disciplines. The proposed models are characterized by a set of ODEs in which all state variables in the system are used to interpret each of their derivatives. In this study, we consider multi-stage smoothing-based and mixed effects modeling approaches to estimate both unknown functional varying coefficients and constant parameters. Further, we develop a cross validation based method to identify the parametric and nonparametric components in the proposed models, and show two computational algorithms for illustrations. A simulation study of the FitzHugh-Nagumo system, a biophysical two dimension model of neuronal spike generation, is conducted. The results indicate the importance of data quality for achieving reliable estimates of dynamic parameters. Nonetheless, good estimates can still be obtained from large noise data given a large number of time points with sufficient subjects.
