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Bayesian shape-restricted regression splines

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dc.contributor.authorHackstadt, Amber J., author
dc.contributor.authorHoeting, Jennifer, advisor
dc.contributor.authorMeyer, Mary, advisor
dc.contributor.authorOpsomer, Jean, committee member
dc.contributor.authorHuyvaert, Kate, committee member
dc.date.accessioned2007-01-03T05:35:54Z
dc.date.available2007-01-03T05:35:54Z
dc.date.issued2011
dc.description.abstractSemi-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.
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.identifierHackstadt_colostate_0053A_10715.pdf
dc.identifier.urihttp://hdl.handle.net/10217/48236
dc.languageEnglish
dc.language.isoeng
dc.publisherColorado State University. Libraries
dc.relation.ispartof2000-2019
dc.rightsCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.
dc.subjectregression splines
dc.subjectBayesian
dc.subjectchange-points
dc.subjectMarkov Chain Monte Carlo
dc.subjectsemi-parametric
dc.subjectshape restrictions
dc.titleBayesian shape-restricted regression splines
dc.typeText
dcterms.rights.dplaThis Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
thesis.degree.disciplineStatistics
thesis.degree.grantorColorado State University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)

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