A flexible Bayesian tool for CoDa mixed models: logistic-normal distribution with Dirichlet covariance

Compositional Data Analysis (CoDa) has gained popularity in recent years. This type of data consists of values from disjoint categories that sum up to a constant. Both Dirichlet regression and logistic-normal regression have become popular as CoDa analysis methods. However, fitting this kind of multivariate models presents challenges, especially when structured random effects are included in the model, such as temporal or spatial effects. To overcome these challenges, we propose the logistic-normal Dirichlet Model (LNDM). We seamlessly incorporate this approach into the \textbf{R-INLA} package, facilitating model fitting, model and model predicting within the framework of Latent Gaussian Models (LGMs). Moreover, we explore metrics like Deviance Information Criteria (DIC), Watanabe Akaike information criterion (WAIC), and cross-validation measure conditional predictive ordinate (CPO) for model selection in \textbf{R-INLA} for CoDa. Illustrating LNDM through a simple simulated example and with an ecological case study on \textit{Arabidopsis thaliana} in the Iberian Peninsula, we underscore its potential as an effective tool for managing CoDa and large CoDa databases.