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Clipped Latent-Variable Spatial Models for Ordered Categorical Data
Megan Dailey Higgs, Ph.D. Candidate

Department of Statistics, Colorado State University

Tuesday, September 18, 2007
2:00 p.m.
223 Weber

ABSTRACT

A common and important problem arising from the collection of data over space is prediction at new locations.  Models and methods for such prediction are widespread for continuous data, but sparse for categorical data, and especially for ordered categorical data.  Spatial ordered categorical data may result from from studies undertaken in areas such as ecology and the social sciences.  The scarcity of models dealing with both the categorical and spatial nature of such data leads us to the development of such models. Bayesian models have been proposed for the analysis of independent ordered categorical data, relying on the techniques of data augmentation and Gibbs sampling (e.g., Albert and Chib, 1993; Cowles, 1996; Nandram and Chen, 1996; and Chen and Dey, 1997).  Others have developed spatial models for binary and count data by embedding the Gaussian process within the framework of generalized linear mixed models (GLMMs), and then applying Bayesian methods
(e.g., Diggle et al., 1998, and Christensen et al., 2006).  De Oliveira (2000)
proposes the use of a clipped Gaussian random field for prediction of a binary random field.  We combine and extend these methods to develop models for spatial ordered categorical data, investigating the problem from two model formulations.  The first extends the generalized linear spatial model described by Diggle et al. (1998), incorporating the latent spatial variable as a random effect, while the second builds on the idea of a clipped Gaussian random field as proposed by De Oliveira (2000).   Both approaches rely on the use of an underlying latent Gaussian random field and the idea of clipping an underlying continuous distribution.  The models are fit in the Bayesian paradigm using Gibbs sampling and Markov chain Monte Carlo (MCMC) methods.  We assess and compare the models through analytical, graphical, and simulation-based methods.  An in-depth simulation study demonstrates and compares their success in terms of prediction at new locations and estimation of the parameters using a wide variety of simulated data sets.  The models are applied to ordered categorical data describing stream health through an ``index of biotic integrity'' in Montgomery County, Maryland.

 

 


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