Abstract:
In many experimental studies, repeated observations are made on each of a number of experimental units with the objective to fit a response curve to the data. Longitudinal data consist of repeated observations on many experimental units. It is reasonable to assume that although the response patterns of the different experimental units may differ, they can all be described by the same functional form. Differences in the response patterns between experimental units are modelled by allowing the parameters of the model to be stochastic. Linear as well as non-linear response functions are considered and it is assumed that the residuals of the models are generated by stationary autoregressive moving average (ARMA) processes. The exact likelihood function of the observations of a random coefficient ARMA process is given as well as an approximation thereof based on numerical integration. It is shown that a Kalman recursive algorithm can be used in situations where the data is incomplete. The concept of marginal maximum likelihood estimation is discussed together with the use of the EM-algorithm to obtain maximum likelihood estimates. Bayes estimators of the coefficients of an ARMA process are given. It is shown how the Gibbs sampler can be used to calculate Bayes estimates. Various models used to describe repeated measurement data are considered. It is assumed that the error terms of these models are generated by an ARMA process with fixed or random coefficients. In repeated measurement experiments more than one related characteristic is often measured at each time point. Vector ARMA models can be used to analyze the change in the response vector over time. It is shown that results applying to the scalar case can be generalized to deal with vectors of measurements. Two distributions in the elliptical class are considered as alternatives to the normal distribution as probability models for the white noise of an ARMA process. The results of two simulation studies are given.
