A Logistic Normal Mixture Model for Compositions with Essential Zeros

Abstract

Compositions are vectors of nonnegative numbers that sum to a constant, usually one or 100%. They arise in a wide array of fields: geological sampling, budgets,fat/protein/carbohydrate in foods, percentage of the vote acquired by each political party, and more. The usual candidate distributions for modeling compositions -- the Dirichlet and the logistic normal distribution -- have density zero if any component is zero. While statistical methods have been developed for "rounded" zeros, zeros stemming from values below a detection level, and zeros arising from count data, there remain problems with essential zeros, i.e. cases in continuous compositions where a component is truly absent. We develop a model for compositions with essential zeros based on an approach by Aitchison and Kay (2003). It uses a mixture of additive logistic normal distributions of different dimension, related by common parameters. With the requirement of an additional constraint, we develop a likelihood and methods estimating parameters for location and dispersion. We also develop a permutation test for a two-group comparison, and demonstrate the model and test using data from a diabetes study. These results provide the first use of the additive logistic normal distribution formodeling and testing compositional data in the presence of essential zeros.