A semiparametric model for compositional data analysis in presence of covariates on the simplex

Compositional data occur as natural realizations of multivariate observations comprising element proportions of some whole quantity. Such observations predominate in disciplines like geology, biology, ecology, economics and chemistry. Due to unit sum constraint on compositional data, specialized statistical methods are required for analyzing these data. Dirichlet distributions were originally used to study compositional data even though this family of distribution is not appropriate (see Aitchison, 1986) because of their extreme independence properties. Aitchison (1982) endeavored to provide a viable alternative to existing methods by employing Logistic Normal distribution to analyze such constrained data. However this family does not include the Dirichlet class and is therefore unable to address the issue of extreme independence. In this paper generalized Liouville family is investigated to model compositional data which includes covariates. This class permits distributions that admit negative or mixed correlation and also contains non-Dirichlet distributions with non-positive correlation and overcomes deficits in the Dirichlet class. Semiparametric Bayesian methods are proposed to estimate the probability density. Predictive distributions are used to assess performance of the model. The methods are illustrated on a real data set.