Biological monitoring: A Bayesian Model for Multivariate Compositional Data

We develop a model to relate a multivariate compositional response to a number of covariates. We propose a new graphical model, called the Random Effects Discrete Regression (REDR) model, which allows for examination of the complex conditional relationships between a set of covariates and multiple discrete response variables. Our approach offers a number of advantages over previous approaches and allows for a wide range of inferences. Relationships between compositional observations can be evaluated through a set of interaction parameters and inference about the influence of covariates is possible through a set of regression coefficients. The model also allows for examination of relationships between the covariates via another set of interactions. Parameter inference via Bayesian methods and MCMC is discussed. The proposed model and MCMC methods are used to examine the relationship between compositional observations of two characteristics of fish species and a number of covariates. These relationships are of interest to the U.S. Environmental Protection Agency for stream monitoring.