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Bayesian Probit Regression Models for Spatially-Dependent Categorical Data

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2010, Doctor of Philosophy, Ohio State University, Statistics.

Data augmentation/latent variable methods have been widely recognized for facilitating model fitting in the Bayesian probit regression model. First proposed by Albert and Chib (1993) for independent binary and multi-category data, the latent variable representation of the Bayesian probit regression model allows model fitting to be performed using a simple Gibbs sampler and, for more than two categories, also allows the so-called assumption of irrelevant alternatives required by the logistic regression model to be relaxed (Hausman and Wise, 1978). To accommodate residual spatial dependence, the latent variable specification of the Bayesian probit regression model can be extended to incorporate standard parametric covariance models typically used in analyses of spatially-dependent continuous data, defining what we term the Bayesian spatial probit regression model. In this dissertation, we develop and extend the Bayesian spatial probit regression model by (i) introducing efficient model-fitting algorithms, (ii) deriving classification methods based on the model, and (iii) extending the model to the multi-category spatial setting.

Statistical models for spatial data are notoriously cumbersome to fit, necessitating the availability of fast and efficient model-fitting algorithms. To improve the efficiency of the Gibbs sampler used to fit the Bayesian regression model for independent categorical response variables, Imai and van Dyk (2005) propose introducing a working parameter into the model and compare various data augmentation strategies resulting from different treatments of the working parameter. We build on this work by investigating the efficiency of modified and extended versions of conditional and marginal data augmentation Markov chain Monte Carlo (MCMC) algorithms for the spatial probit regression model, focusing on the special case of binary spatially-dependent response variables.

Within the classification literature, methods that exploit spatial dependence are limited. We show how a spatial classification rule can be derived from the Bayesian spatial probit regression model. In addition, we compare our proposed spatial classifier to various other classifiers in terms of training and test error rates using a land-cover/land-use data set.

When extending the spatial probit regression model to the multi-category setting, care must be taken to ensure that model parameters are estimable and interpretable. Considering three types of categorical and spatial covariate information, we discuss various specifications of the latent variable mean structure and the associated parameter interpretations. Additionally, we explore the specification of the latent variable cross space-category dependence structure and discuss how data augmentation MCMC strategies for fitting the Bayesian spatial probit regression model can be extended to the multi-category setting.

Catherine A. Calder, Ph.D. (Advisor)
L. Mark Berliner, Ph.D. (Committee Member)
Peter F. Craigmile, Ph.D. (Committee Member)
Elizabeth A. Stasny, Ph.D. (Committee Member)
161 p.

Recommended Citations

Citations

  • Berrett, C. (2010). Bayesian Probit Regression Models for Spatially-Dependent Categorical Data [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1285076512

    APA Style (7th edition)

  • Berrett, Candace. Bayesian Probit Regression Models for Spatially-Dependent Categorical Data. 2010. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1285076512.

    MLA Style (8th edition)

  • Berrett, Candace. "Bayesian Probit Regression Models for Spatially-Dependent Categorical Data." Doctoral dissertation, Ohio State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=osu1285076512

    Chicago Manual of Style (17th edition)