ALGORITHM DEVELOPMENT & SOFTWARE DESIGN FOR POPULATION KINETIC ANALYSIS

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The specific are: 1. Develop convergent numerical algorithms for the parametric and nonparametric estimators in population kinetic analysis to be implemented in Project 2. 2. Analyze the statistical properties of the estimators in Specific Aim 1: consistency, asymptotic normality, asymptotic confidence regions and hypothesis testing. 3. Investigate efficiency and robustness of the estimators in Specific Aim I via simulation studies. For the parametric case, we will develop four algorithms: a "true" maximum likelihood (ML) algorithm, a Global Two Stage (GTS) algorithm, a NONMEM type algorithm, and a Lindstrom-Bates type algorithm. The ML algorithm is based on Monte Carlo integration for evaluating the objective function. The GTS, NONMEM, and Lindstrom-Bates type algorithms are all based on the extended least squares (ELS) method. For the nonparametric case, we will develop a Mallet type algorithm for mixed effects models. For the parametric case, consistency is a difficult issue. Consistency means that the estimated values converge to the true values as the number of subjects gets arbitrarily large. It is important to note that the original estimation procedures ot'NONMEM and Lindstrom-Bates are not consistent for 2eneral nonlinear models. The only algorithm that is consistent relative to the true parameter values is the true maximum likelihood algorithm. For the class of ELS algorithms we develop, there is a generalized notion of consistency, which means that the estimated values converge to the values that best approximate the model. We will investigate the required theory for the generalized consistency and asymptotic normality of these algorithms. The formulas for the asymptotic confidence intervals and hypothesis testing will follow from the same theory. For the nonparametric case, the consistency of the method, relative to the true values of the model, has already been established. What remains then is the determination of the asymptotic confidence intervals for estimated parameters such as means, medians, trimmed means, etc. At present these results have not been derived for the nonparametric case. We will use the theory of maximum likelihood estimation in infinite dimensional spaces for this purpose. Efficiency of an (unbiased) estimator is measured by the generalized variance of estimated values, with the Cramer-Rao lower bound being optimal. Relative efficiency of two estimators compares the corresponding generalized variances. Robustness measures how an algorithm performs when there are violations in the model and/or probability distribution assumptions. By utilizing Monte Carlo simulation studies, these properties can be investigated without requiring 1 asymptotic conditions.