Bayesian analysis of multivariate longitudinal data using latent structures with applications to medical data

In biomedical science, sociology, psychology, etc., it is often of interest to understand how multiple variables are associated. In many cases, the observed outcomes are not of direct interest, instead they are considered as manifestations of one or more underlying latent characteristics. When measured repeatedly over time, interest is then often placed on the evolution of the latent characteristics (variables) and/or the effects of covariates on those evolution, rather than on the observed characteristics. 

Compared to statistical models for multivariate observed longitudinal data, models for latent variables have been examined to a lesser extent. This leaves some gaps in the statistical literature for us to propose novel methodologies that are applicable to model multivariate latent variables. An extra motivation for the development of such models is the kind of data that is the topic of research in this doctorate. 

The BelRAI data base consists of data collected on frail individuals of age 65 or older living at home but at risk of institutionalization. These individuals were referred to health care agencies by physicians, social services, or nurses. A survey, called the BelRAI instrument, which is the Belgian version of the interRAI instruments, was filled in at regular intervals. This is instrument provides a comprehensive assessment that evaluates the physical, clinical, psychological, and social condition of an elderly person. The BelRAI data set and the associated research questions, were the triggers for developing our statistical models.

The complexity of the statistical models considered, the Bayesian methodology using Markov chain Monte Carlo techniques seems a natural choice. Indeed the flexibility of the Bayesian approach, but also of the Bayesian software, provides a good tool to model complex data structures in a relatively short amount of time, which would be much harder to achieve when making use of maximum likelihood. In addition, the Bayesian approach is able to simultaneously estimate the latent variables and the model parameters, allowing to incorporate uncertainty in parameter estimation into latent variable estimation in a natural way. 

We have proposed new Bayesian methods to address the clinical research questions. Methodologically, we have covered a wide range of settings for multivariate longitudinal latent variables in balanced and unbalanced designs. We conducted extensive simulation studies to demonstrate the advantages of our proposed models over the existing approaches. Our methods have been also successfully applied to BelRAI data and a publicly available data set involving amyotrophic lateral sclerosis (ALS) patients. But, our methods can also be applied to any dataset with a similar structure in other areas such as sociology, psychology, etc.