A Logistic Normal Mixture Model for Compositions with Essential Zeros
AuthorBear, John Stanley
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PublisherThe University of Arizona.
RightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
AbstractCompositions 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.
Degree ProgramGraduate College