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Random Effects Graphical Models and the Analysis of Compositional Data



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Random Effects Graphical Models and the Analysis of Compositional Data Devin S Johnson and Jennifer A Hoeting STARMAP Department of Statistics Colorado State University Developed under the EPA STAR Research Assistance Agreement CR 829095 Motivating Problem Various stream sites in the Mid Atlantic region of the United States were visited in Summer 1994 For each site each observed fish species was crosscategorized according to several traits Environmental variables are also measured at each site e g precipitation chloride concentration Relative proportions are more informative species composition How can we examine complex relationships between and within the covariates and response traits Graphical Models Chain Graphs Graphical models e g log linear models lattice spatial models have been explored for examination of conditional dependencies within multivariate random variables Y1 X1 X2 Y2 Model f y1 y2 x1 x2 f x1 f x2 Probability Model for Individuals Response variables Set of discrete categorical variables Notation y is a specific cell Explanatory variables Set of explanatory variables covariates Notation x refers to a specific explanatory observation Random effects Allows flexibility when sampling many sites Unobserved covariates Notation f f refers to a random effect Probability Model and Extended Chain Graph G Joint distribution f y x f y x f x f Graph illustrating possible dependence relationships for the full model G X1 Y1 1 1 2 X2 Y2 2 Graphical Models for Discrete Compositions Sampling many individuals at a site results in cell counts C y i individuals in cell y at site i 1 S Conditional count likelihood C y i y xi multinomial Ni f y xi i y Joint covariate count likelihood multinomial Ni f y xi i y MVN 1 Parameter estimation Gibbs sampler with hierarchical centering Easier to impliment Improved convergence Fish Species Richness in the Mid Atlantic Highlands 91 stream sites in the Mid Atlantic region of the United States were visited in an EPA EMAP study



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