Spatial mixed model for areal data on the simplex

On March the fourth, elections took place in Italy for the two Chambers of the Parliament. Many newspapers emphasized the victory of the 5 Star Movement (5SM) and its unprecedented dominance in most of the Southern Regions of Italy. Aim of this contribution is to analyze the electoral results through a rigorous statistical model to evaluate the presence and the possible impact of spatial structures. The response variable is the percentage of votes got by the 5SM in each electoral district. To cope with a bounded continuous outcome lying in the open interval (0,1), a mixture regression model is proposed based on a special mixture of two betas (referred to as flexible beta) sharing the same precision parameter but displaying two distinct component means subject to an inequality constraint. Advantages of this model are its many theoretical properties which are reflected in its computational tractability. Furthermore, the special mixture structure is designed to represent a wide range of phenomena (bimodality, heavy tails and outlying observations). The model is further extended by accounting for spatial correlation through random effects. Intensive simulation studies are performed to evaluate the fit of the proposed regression model. Inferential issues are dealt with by a (Bayesian) Hamiltonian Monte Carlo algorithm.