The long-term objectives of this project are to develop methodologies for analyzing unbalanced longitudinal data with between subject variance components and within subject serial correlation. The emphasis is on data where each subject is observed at different unequally spaced times. The development of continuous time series models beyond AR(1) structure for the within subject errors in longitudinal data which includes random subject effects is one of the specific aims of this proposal.
A second aim i s the development of nonparametric procedures for longitudinal data. Instead of using polynomials which have limitations for modelling growth curves, integrated random walk models which produce smoothing splines are a natural nonparametric approach to this problem. The word nonparametric is used in the sense that no specific parametric form is assumed for the growth curve. In many applications, the functional form of a growth curve or dose response curve is nonlinear in the unknown parameters. When there are both within and between components of variance, this produces unique computational problems. Extending the methodology to handle these nonlinear models is a third specific aim of this research. These techniques are important in longitudinal studies involving groups of subjects with different treatments for different exposures to risk factors when it is of interest to determine the effects of treatment or exposure. Too often, unrealistic assumptions are made about the error structure which would invalidate an analysis. A class of error models for within subject errors to be investigated is continuous time ARMA models. These are related to, but quite different from the usual discrete time ARMA models. The basic underlying model is a stochastic differential equation. Putting this model in state space form allows the generation of a discrete time state space model for arbitrary time spacing. The Kalman filter can then be used to calculate exact likelihoods for given values of the unknown parameters, and nonlinear optimization can be used to calculate maximum likelihood estimates of the unknown parameters. The nonparametric approach to be investigated involves using state space integrated random walks to generate smoothing polynomial splines. The attempt will be to develop methodology for representing both the fixed mean curves and the individual growth curves as smoothing polynomial splines. Finally, the parametric approach for nonlinear models will be investigated. Iterative methods using linearization with the linearized coefficients assumed to be random across subjects will be compared with exact maximum likelihood approaches which are very computationally intensive.
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