Clipped latent-variable spatial models for ordered categorical data

Abstract

Ordered categorical data arise in a variety of scientific disciplines. The ordered categories may be of primary interest, or they may be recorded in an attempt to simplify data collection. When the reason for the categorization is simplification, the categories are typically created based on an underlying continuous variable and a set of threshold values used to define the categories. Information in the continuous variable is clearly lost, but an analysis of ordered categorical data can indirectly involve a latent (unobserved) underlying continuous variable. In fact, this concept creates a convenient platform on which to build models for ordered categorical data, both from an analytical and computational perspective. Latent variable models, coupled with Bayesian inference, has been an active area of research for independent ordered categorical data (e.g. Albert and Chib, 1993; Cowles, 1996; Nandram and Chen, 1996; Chen and Dey, 1997). Such models rely on the techniques of d a ta augmentation and Markov chain Monte Carlo (MCMC) algorithms. The models for independent data have been extended to include correlation in the response(s), focusing on clustered and repeated measures data (e.g. Chen and Dey, 1996; Chib and Greenberg, 1998; and Ishwaran and Gatsonis, 2000). We, however, are interested in incorporating spatial dependence into the model structure, an extension that has not yet been specifically addressed in the literature. Thus, our goal is to develop models for data collected at point-referenced locations over space. Spatial ordered categorical data result from a variety of research areas, such as ecology, epidemiology, and the social sciences. Spatial models for binary and count data have received more recent attention (e.g. Diggle et al., 1998; Gelfand et al., 2000; and Christensen et ah, 2006). Most of the models for binary and count data remain within the convenient context of one-parameter exponential family distributions, embedding the Gaussian process within the framework of generalized linear mixed models (GLMMs). Such models have been coined generalized linear spatial models. An alternative approach to modeling binary data relies on the concept of a clipped Gaussian random field (De Oliveira, 2000). In this dissertation, we extend and combine the modeling strategies for independent ordered categorical data and those for binary and count spatial data to develop two models for spatially dependent point-referenced ordered categorical data. The development of these models involves conceptual model building, as well as the development of computational algorithms. Our proposed models differ in their conceptual origin, in the correlation they induce in the categorical response, and in their generalizability. The first model, termed the Double Latent model, extends the generalized linear spatial model framework described by Diggle et al. (1998), incorporating the latent spatial variable as a random effect. The second model extends the idea of a clipped Gaussian random field as proposed by De Oliveira (2000) and is termed the Clipped Gaussian model. Both approaches rely on the use of an underlying latent Gaussian random field, but differ in how the categorical random field is created from an underlying continuous distribution and at what level the mean structure is defined. They do, however, also have fundamental similarities and can under the probit link assumption be compared as nugget and no-nugget models. Both models are developed under the Bayesian paradigm using Gibbs sampling and MCMC methods, and result in different algorithms used for inference. We assess and compare the models through analytical, graphical, and simulation-based methods. We assess the primary goal of prediction by formulating prediction as a decision problem within the Bayesian paradigm, and testing prediction at new locations with the use of hold-out data sets. An in-depth simulation study is used to investigate and compare the behaviors of the two models, demonstrating their success in terms of prediction and estimation on a large variety of realizations of artificial data created under the assumptions of both models and different sets of parameter values. The usefulness of the models to real problems is demonstrated through an application to ordered categorical data describing stream health through an "index of biotic integrity" in Montgomery County, Maryland.