Nonlinear models with thresholds for longitudinal data

Nonlinear models with thresholds for longitudinal data have application in the fields of toxicology and clinical medicine. The practicality of using nonlinear models for longitudinal data has been enhanced by the recent commercial availability of software that allows for maximum likelihood based solutions for model parameters. However, the use of maximum likelihood methods requires assumption of a specific distribution for the response variable being modeled. Furthermore, these procedures require derivatives of the nonlinear functions with respect to the parameters. Threshold models are not smooth functions at the threshold; the derivative of the model function does not exist there.; When using least squares methods to obtain solutions, it is not necessary to make distributional assumptions. Use of a direct search algorithm to find parameters that minimize a residual sum of squares criterion obviates the need to take derivatives. For these reasons, least squares procedures using the Nelder-Mead algorithm were used to solve for mean-model parameter estimates in this dissertation. Regardless of the distribution of the response variable, in non-threshold models, the parameter estimates are asymptotically normal. However, it has been demonstrated that convergence of threshold parameter estimates to normal random variables is slow compared to the other parameter estimates. This dissertation shows how the Nelder-Mead algorithm using least squares as a criterion can be used to estimate model parameters for both the situation where population-averaged inference is the focus and where subject-specific inference is of interest. The procedure used for subject-specific inference was also used without a threshold in the model to analyze a data set that has been analyzed by others using different methods, including maximum likelihood. This analysis confirms that the least squares procedure performs comparably to other methods with respect to parameter estimation. Methods for making inference about the threshold parameters in light of their non-normal distributions are provided.