Mixed effects state-space models for longitudinal data with heavy tails

Abstract

The mixed effects state space models approach inherits the modelling power of state-space models for dynamic systems as well as the idea of random effects for longitudinal studies. Although this class of models can be applied in a variety of contexts, there are three assumptions related to the statistical inference proposed for it that limit its potential applications. In this thesis, we focus on expanding the possibilities of these models to consider the use of another distributions, beyond the normal case, in order to model the observational disturbance of the state space structure. Specifically, we propose to use distributions within the scale mixture of normal family in which we find appropriate distributions to model the presence of outliers. In this way, we introduce the mixed effects state space models with scale mixture of normal errors as a broader class of models whither the usual definition of the mixed effects state space model is a special case. Statistical inference for this new class of models is developed under a Bayesian perspective that leads us to propose Markov Chain Monte Carlo algorithms for parameters and states estimation. Through simulation exercises, we investigate computational and estimation properties of the proposed models. The proposal of the mixed effects state space models is motivated by the modelling of the HIV dynamics. Similarly, this thesis focuses on illustrating the usefulness of the proposed models being applied in this context. In addition, we propose an alternative approach to implement the mixed effects state space models with scale mixture normal errors for the case of the non-linear structure of HIV dynamics, since this class of models is restricted to modelling linear dynamics.