The long-term objectives of this project are to extend the methodologies developed by the Principal Investigator for analyzing unbalanced longitudinal data with between subject variance components and within subject serial correlation to include multivariate observations. The emphasis is on data where each subject is observed at different unequally spaced times and some elements of the multivariate response vector may be missing. The use of state space representations is a very flexible method for formulating a broad class of multivariate models. The Kalman filter can be used to calculate likelihoods, and nonlinear optimization programs can be used to obtain maximum likelihood methods. The specific aims are to develop methods for modelling serial correlation in the multivariate setting. Continuous time model must be used to handle unequally spaced observations. The basic within subject error model will be multivariate continuous time autoregressions. By integrating the continuous time process over a finite time step, the corresponding discrete time process can be generated at the corresponding sample points. As autoregressive error structures approach the point of being nonstationary, random walk type error structures appear. These are realistic error models for processes that tend to wander around without returning to a stable mean level. In order to generalize the usual assumptions of Gaussian errors, the use of quasi-likelihood in multivariate state space models will also be investigated. Since state space models have random variables in both the state equation and in the observation equation, the use of two multivariate variance-covariance functions is possible. These methods will be developed and applied to a variety of problems in medical research where multivariate methods are necessary to extract the available information from the data. An example is the analysis of the data from a doubly labeled water experiment use to obtain an estimate of energy expenditure for subjects. The cost of the experiment is about $1,000 per subject, and efficient methods of statistical analysis are not available that can account for variance heterogeneity and serial correlation in the bivariate observations. The problem becomes more interesting and challenging if multiple subjects are in a study and random effects and covariates are introduced into the model. The random effects appear nonlinearly.