Statistical theory and robust methodology for nonlinear models with application to toxicology

Nonlinear regression models are commonly used in dose-response studies, especially when researchers are interested in determining various toxicity characteristics of a chemical or a drug. There are several issues one needs to pay attention to when fitting nonlinear models for toxicology data, such as structure for the error variance in the model and the presence of potential influential and outlying observations. In this dissertation I developed robust statistical methods for analyzing nonlinear regression models, which are based on robust M-estimation and preliminary test estimation (PTE) procedures. In the first part of this research the M-estimation methods in heteroscedastic nonlinear models are considered for two cases. In one case, the error variance is proportional to some known function of mean response, while in the other case the error variance is modeled as a polynomial function of dose. The asymptotic properties of the proposed M-procedures and the asymptotic efficiency of the proposed M-estimators are provided. In the second part I consider PTE-based methodology using M-methods for estimating the regression parameters. Based on the outcome of the preliminary test, the proposed methodology determines the appropriate error variance structure for the data and accordingly chooses the suitable estimation procedure. Since the resulting methodology uses M-estimators, it is expected to be robust to outliers and influential observations, although such issues have not been explored in this dissertation. Consequently, one does not have to pre-specify the error structure for the variances not does the user have to perform model diagnostics to choose a method of estimation. Some asymptotic results will be given to obtain the asymptotic covariance matrix of the PTE. Finally numerical studies are presented to illustrate the methodology. The results of the numerical studies suggest that the PTE using M-methods performs well and is robust to the error variance structure