Resumen de Bayesian multivariate spatial models for the joint analysis of several diseases

Francisco Palmí Perales

• All is connected in the world. Our daily life is such a mechanism full of connections. For this reason, if we want to develop models that mimic the real procedures, we shall include these connections in our models. Specifically, the results will be more realistic in a joint analysis of different variables than in several individual analyses. This idea is the baseline of multivariate analysis which is a wide research area with relevant results. In particular, although there is some literature of multivariate spatial statistics, there are several interesting topic that are not faced yet. Furthermore, the technology advances and the big amount of data make us to explore new ideas.

  Monte Carlo Makov Chain (Gilks et al., 1996, MCMC) methods have been the main option to analyse Bayesian multivariate spatial models for the last decades. However, these methods are very computationally demanding in a spatial framework, so some researchers have developed other alternatives. For instance, INLA (Rue et al., 2009) is able to fit any model which can be expressed as a latent Gaussian model, which are many of the most used models in practice, in an easy and friendly framework.

  The main objective of this work is to increase the number of multivariate spatial models that INLA can fit. This idea will enable the users to fit multivariate spatial models benefiting from the advantages that INLA offers.

  First of all, a multivariate model for lattice data with shared and specific spatial effects is proposed. This model is able to detect similar spatial or temporal patterns of different variables apart from estimating the spatial and temporal pattern of each variable. This methodology has been applied to a dataset of three different diseases and it has been computed using MCMC and INLA obtaining similar results with both techniques.

  Generalizing the model above, we have proposed another way of increasing the number of multivariate spatial models which can be fitted with INLA. There are models that can not be fitted with INLA, but after fixing some parameters the restultant model can be fitted with INLA. Therefore, focusing on spatial models, we have discussed several ways of fixing this parameters, for example, combining Metropolis-Hastings algorithm with INLA, that is, INLA within MCMC. Furthermore, an extensive description of how to perform multivariate inference has been done remarking that this methodology can not only extend the models that INLA can fit, but also is useful to make multivariate inferences on some of the model parameters.

  Trying to increase the number of INLA-fittable models, we have developed an R package called INLAMSM which uses INLA to fit some traditional multivariate spatial models for lattice data. INLAMSM is on CRAN which is the public repository of R packages. INLAMSM allows the user to include in its model a multivarite spatial effect to account for the multivariate spatial variability in a simple and straightforward way. The implemented models of INLAMSM have been built through the rgeneric latent effect of INLA.

  Following the same idea, we have proposed a novel methodology with shared and specific terms that is able to fit multivariate point processes. These spatial effects are considered as a latent Gaussian field and are estimated using the SPDE approximation (Lindgren et al., 2011). Therefore, this methodology allows the user to estimate the intensity of every point pattern apart from identifying what point patterns have similar spatial trends. Hotspots are also detected by this methodology and three different ways of estimating the effect of exposure to risk factors have been considered.

  A mortality atlas have been developed by applying the developed methodologies for lattice data. This atlas considers different diseases in the Castilla-La Mancha (Spain) region at municipality level. We have used the classical model exposed in Besag et al. (1991) to fit the spatial models and the results are shown and discussed for each disease. Therefore, an interesting and relevant application of the developed methodologies has been shown supporting the relevancy of the developed methodologies.

  Finally, we can state that INLA offers an straightforward framework to fit mutlivariate spatial models. Although there are a good amount of models that INLA can not fit, we have shown that there are different techniques to extend this list, so multivariate spatial models can be tackled with INLA. We have also shown that the developed methods are able to fit multivariate spatial models and also detecting common spatial patters between variables which is a convenient feature, for example, in the analysis of several diseases in an epidemiological context.