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In sufficient dimension reduction (Li, 1991; Cook, 1998b), original predictors are replaced by their low-dimensional linear combinations while preserving all of the conditional information of the response given the predictors. Sliced inverse regression [SIR; Li, 1991] and principal Hessian directions [PHD; Li, 1992] are two popular sufficient dimension reduction methods, and both SIR and PHD estimators involve all of the original predictor variables. To deal with the cases when the linear combinations involve only a subset of the original predictors, we propose a Bayesian model averaging (Raftery et al., 1997) approach to achieve sparse sufficient dimension reduction. We extend both SIR and PHD under the Bayesian framework. The superior performance of the proposed methods is demonstrated through extensive numerical studies as well as a real data analysis. / Statistics

Cluster analysis is commonly described as the classification of unlabeled observations into groups such that they are more similar to one another than to observations in other groups. Model-based clustering assumes that the data arise from a statistical (mixture) model and typically a group of many models are fit to the data, from which the `best' model is selected by a model selection criterion (often the BIC in mixture model applications). This chosen model is then the only model that is used for making inferences on the data. Although this is common practice, proceeding in this way ignores a large component of model selection uncertainty, especially for situations where the difference between the model selection criterion for two competing models is relatively insignificant. For this reason, recent interest has been placed on selecting a subset of models that are close to the selected best model and using a weighted averaging approach to incorporate information from multiple models in this set. Model averaging is not a novel approach, yet its presence in a clustering framework is minimal. Here, we use Occam's window to select a subset of models eligible for two types of averaging techniques: averaging a posteriori probabilities, and direct averaging of model parameters. The efficacy of these model-based averaging approaches is demonstrated for a family of generalized hyperbolic mixture models using real and simulated data. / Thesis / Master of Science (MSc)

Motor vehicle crashes are one of the leading causes of death in the United States. Traffic safety research targets at understanding the cause of crash, preventing the crash, and mitigating crash severity. This dissertation focuses on the driver-related traffic safety issues, in particular, on developing and implementing contemporary statistical modeling techniques on driving risk research on Naturalistic Driving Study data. The dissertation includes 5 chapters. In Chapter 1, I introduced the backgrounds of traffic safety research and naturalistic driving study. In Chapter 2, the state-of-practice statistical methods were implemented on individual driver risk assessment using NDS data. The study showed that critical-incident events and driver demographic characteristics can serve as good predictors for identifying risky drivers. In Chapter 3, I developed and evaluated a novel Bayesian random exposure method for Poisson regression models to account for situations where the exposure information needs to be estimated. Simulation studies and real data analysis on Cellphone Pilot Analysis study data showed that, random exposure models have significantly better model fitting performances and higher parameter coverage probabilities as compared to traditional fixed exposure models. The advantage is more apparent when the values of Poisson regression coefficients are large. In Chapter 4, I performed comprehensive simulation-based performance analyses to investigate the type-I error, power and coverage probabilities on summary effect size in classical meta-analysis models. The results shed some light for reference on the prospective and retrospective performance analysis in meta-analysis research. In Chapter 5, I implemented classical- and Bayesian-approach multi-group hierarchical models on 100-Car data. Simulation-based retrospective performance analyses were used to investigate the powers and parameter coverage probabilities among different hierarchical models. The results showed that under fixed-effects model context, complex secondary tasks are associated with higher driving risk. / Ph. D.

In this thesis, a dynamic spatial model for the spread of Chronic Wasting Disease in Colorado mule deer is derived from a system of differential equations that captures the qualitative spatial and temporal behaviour of the disease. These differential equations are incorporated into an empirical Bayesian hierarchical model through the unusual step of deterministic autoregressive updates. Spatial effects in the model are described directly in the differential equations rather than through the use of correlations in the data. The use of deterministic updates is a simplification that reduces the number of parameters that must be estimated, yet still provides a flexible model that gives reasonable predictions for the disease. The posterior distribution generated by the data model hierarchy possesses characteristics that are atypical for many Markov chain Monte Carlo simulation techniques. To address these difficulties, a new MCMC technique is developed that has qualities similar to recently introduced tempered Langevin type algorithms. The methodology is used to fit the CWD model, and posterior parameter estimates are then used to obtain predictions about Chronic Wasting Disease.