This grant will investigate the use of state space methods, including state space based EM algorithms, for nonlinear and non-Gaussian longitudinal data models and multivariate longitudinal data models both Gaussian and non-Gaussian. The concentration will be on models with within subject serial correlation for which state space methods are ideally suited. For unequally spaced observations, these serial correlation models are based on continuous time stochastic differential equations, For nonlinear and non- Gaussian models, the modified EM algorithm will be an approximate method. The method will be compared with other approximate methods including generalized estimating equations (GEE), Laplace's approximation, and penalized quasi-likelihood (PQL). These comparisons will be made by simulations, and for real data by comparisons with the Gibbs sampler. The decomposition of time varying covariates into cross sectional and within subject effects with general within and between subject error structures will also be investigated. Orthogonal decompositions with respect to these error structures will be developed for both univariate and multivariate responses. These error structures include autoregressive within subject errors. If these time varying covariates are properly decomposed, including only the between subject component should reduce only the between subject component of variance, and including only the within subject component should reduce only the within subject component of variance. The importance of the decomposition of time varying covariates is not generally appreciated by investigators who simply include the time varying covariate as a fixed effect. This can cause a confounding of the two effects, and greatly affect the interpretation of the analysis.
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