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Theses and Dissertations

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Browsing Theses and Dissertations by Subject "Bayesian"

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    Bayesian shape-restricted regression splines
    (Colorado State University. Libraries, 2011) Hackstadt, Amber J., author; Hoeting, Jennifer, advisor; Meyer, Mary, advisor; Opsomer, Jean, committee member; Huyvaert, Kate, committee member
    Semi-parametric and non-parametric function estimation are useful tools to model the relationship between design variables and response variables as well as to make predictions without requiring the assumption of a parametric form for the regression function. Additionally, Bayesian methods have become increasingly popular in statistical analysis since they provide a flexible framework for the construction of complex models and produce a joint posterior distribution for the coefficients that allows for inference through various sampling methods. We use non-parametric function estimation and a Bayesian framework to estimate regression functions with shape restrictions. Shape-restricted functions include functions that are monotonically increasing, monotonically decreasing, convex, concave, and combinations of these restrictions such as increasing and convex. Shape restrictions allow researchers to incorporate knowledge about the relationship between variables into the estimation process. We propose Bayesian semi-parametric models for regression analysis under shape restrictions that use a linear combination of shape-restricted regression splines such as I-splines or C-splines. We find function estimates using Markov chain Monte Carlo (MCMC) algorithms. The Bayesian framework along with MCMC allows us to perform model selection and produce uncertainty estimates much more easily than in the frequentist paradigm. Indeed, some of the work proposed in this dissertation has not been developed in parallel in the frequentist paradigm. We begin by proposing a semi-parametric generalized linear model for regression analysis under shape-restrictions. We provide Bayesian shape-restricted regression spline (Bayes SRRS) models and MCMC estimation algorithms for the normal errors, Bernoulli, and Poisson models. We propose several types of inference that can be performed for the normal errors model as well as examine the asymptotic behavior of the estimates for the normal errors model under the monotone shape-restriction. We also examine the small sample behavior of the proposed Bayes SRRS model estimates via simulation studies. We then extend the semi-parametric Bayesian shape-restricted regression splines to generalized linear mixed models. We provide a MCMC algorithm to estimate functions for the random intercept model with normal errors under the monotone shape restriction. We then further extend the semi-parametric Bayesian shape-restricted regression splines to allow the number and location of the knot points for the regression splines to be random and propose a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm for regression function estimation under the monotone shape restriction. Lastly, we propose a Bayesian shape-restricted regression spline change-point model where the regression function is shape-restricted except at the change-points. We provide RJMCMC algorithms to estimate functions with change-points where the number and location of interior knot points for the regression splines are random. We provide a RJMCMC algorithm to estimate the location of an unknown change-point as well as a RJMCMC algorithm to decide between a model with no change-points and model with a change-point.
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    Spatial probit models for multivariate ordinal data: computational efficiency and parameter identifiability
    (Colorado State University. Libraries, 2013) Schliep, Erin M., author; Hoeting, Jennifer, advisor; Cooley, Daniel, committee member; Lee, Myung Hee, committee member; Webb, Colleen, committee member
    The Colorado Natural Heritage Program (CNHP) at Colorado State University evaluates Colorado's rare and at-risk species and habitats and promotes conservation of biological resources. One of the goals of the program is to determine the condition of wetlands across the state of Colorado. The data collected are measurements, or metrics, representing landscape condition, biotic condition, hydrologic condition, and physiochemical condition in river basins statewide. The metrics differ in variable type, including binary, ordinal, count, and continuous response data. It is common practice to uniformly discretize the metrics into ordinal values and combine them using a weighted-average to obtain a univariate measure of wetland condition. The weights assigned to each metric are based on best professional judgement. The motivation of this work was to improve on the user-defined weights by developing a statistical model to estimate the weights using observed data. The challenges of creating a model that fulfills this requirement are many. First, the observed data are multivariate and consist of different variable types which we wish to preserve. Second, the multivariate response data are not independent across river basin because wetlands at close proximity are correlated. Third, we want the model to provide a univariate measure of wetland condition that can be compared across the state. Lastly, it is of interest to the ecologists to predict the univariate measure of wetland condition at unobserved locations requiring covariate information to be incorporated into the model. We propose a multivariate multilevel latent variable model to address these challenges. Latent continuous response variables are used to model the different types of response variables. An additional latent variable, or common factor, is used as a univariate measure of wetland condition. The mean of the common factor contains observable covariate data in order to predict at unobserved locations. The variance of the common factor is defined by a spatial covariance function to account for the dependence between wetlands. The majority of the metrics reported by the CNHP are ordinal. Therefore, our primary focus is modeling multivariate ordinal response data where binary data is a special case. Probit linear models and probit linear mixed models are examples of models for ordinal response data. Probit models are attractive in that they can be defined in terms of latent variables. Computational efficiency is a major issue when fitting multivariate latent variable models in a Bayesian framework using Markov chain Monte Carlo (MCMC). There is also a high computation cost for running MCMC when fitting geostatistical spatial models. Data augmentation and parameter expansion are both modeling techniques that can lead to optimal iterative sampling algorithms for MCMC. Data augmentation allows for simpler and more feasible simulation from a posterior distribution. Parameter expansion is a method for accelerating convergence of iterative sample algorithms and can enhance data augmentation algorithms. We propose data augmentation and parameter-expanded data augmentation algorithms for fitting MCMC to spatial probit models for binary and ordinal response data. Parameter identifiability is another challenge when fitting multivariate latent variable models due to the multivariate response data, number of parameters, unobserved latent variables, and spatial random effects. We investigate parameter identifiability for the common factor model for multivariate ordinal response data. We extend the common factor model to include covariates and spatial correlation so we can predict wetland condition at unobserved locations. The partial sill and range parameter of a spatial covariance function are difficult to estimate because they are near-nonidentifiable. We propose a new parameterization for the covariance function of the spatial probit model that leads to better mixing and faster convergence of the MCMC. Whereas our spatial probit model for ordinal response data follows the common factor model approach, there are other forms of the spatial probit model. We give a comprehensive comparison of two types of spatial probit models, which we refer to as the first-stage and second-stage spatial probit model. We discuss the implications of fitting each model and compare them in terms of their impact on parameter estimation and prediction at unobserved locations. We propose a new approximation for predicting ordinal response data that is both accurate and efficient. We apply the multivariate multilevel latent variable model to data collected in the North Platte and Rio Grande River Basins to evaluate wetland condition. We obtain statistically derived weights for each of the response metrics with confidence limits. Lastly, we predict the univariate measure of wetland condition at unobserved locations.