Model Diagnostics for the Nonlinear Mixed Effects Model with Balanced Longitudinal Data

CHISWELL, KAREN ELIZABETH. Model Diagnostics for the Nonlinear Mixed Effects Model with Balanced Longitudinal Data. (Under the direction of Dr. John Monahan.) Longitudinal data, where the response for each experimental unit is measured repeatedly on several occasions, are common in biomedical (e.g., pharmacokinetic) and agricultural (e.g., growth and yield) studies. When each unit is measured at the same time points, the data are termed balanced. The methods developed in this dissertation assume balanced longitudinal data with a normally distributed response variable. The nonlinear mixed effects model (NLMM) provides a useful statistical framework for characterizing longitudinal data when a nonlinear regression function describes the trend for each experimental unit or subject. In a simple NLMM with two stages, a nonlinear model describes the expected trajectory for each subject at the first stage. Coefficients in the first stage model are specific to each subject. At the second stage these subject-specific coefficients are modeled as random variables (random effects) from a probability distribution. Withinsubject variability (e.g., due to measurement errors or fluctuations in within-subject response) is modeled at the first stage. Variation between subjects is characterized by the second stage model for the random effects, some of which may be trivial (the coefficient value is the same for all subjects). Our methods explore the structure of variation in the longitudinal data using traditional tools of multivariate analysis (principal component analysis, factor analysis, and multivariate regression). Assuming that preliminary estimates of the NLMM model parameters are available, a first-order linearization of the subject specific model allows us to use the multivariate results to check assumptions about the variance structure defined by the NLMM. In particular, we propose methods for identifying the non-trivial random effects, for assessing the presence of within-subject heteroscedasticity (non-constant variance), and for testing evidence of withinsubject autocorrelation. In simulation studies our methods demonstrated high power to correctly identify random effects, to detect heteroscedasticity, and to find strong AR(1) structure in within-subject residuals. We applied our methods to two examples. Data in the first example were collected from a gas uptake experiment in rats. Our methods identified significant evidence of within-subject heteroscedasticity, and suggested that between-subject variation could be explained by random effects on the two rate parameters in an open two-compartment model, or by the Vmax metabolic parameter and rat body weight in the PBPK model. In the second example, pharmacokinetic data were collected from human subjects dosed orally with theophylline. Our analysis identified random effects on the absorption rate and clearance parameters of the open one-compartment model, and suggested heteroscedasticity in within-subject errors. Model Diagnostics for the Nonlinear Mixed Effects Model with Balanced Longitudinal Data by Karen Elizabeth Chiswell A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy