▼ The Cox proportional hazards model is commonly used to analyze the exposure-response relationship in occupational cohort studies. This analysis involves identifying cases (those who experience the outcome of interest) and forming risk-sets for each case. The risk-set for a case is the set of cohort members whose failure times are at least as large as the case’s failure time and are under observation immediately before the case’s failure time. Thomas proposed the idea of randomly sampling controls from each risk-set to use for analysis, which results in a nested case-control study. It has been shown that the analysis using the full risk-sets and the analysis using the sampled risk-sets produce asymptotically unbiased results. Also, the asymptotic relative efficiency between analyzing the full risk-sets and using Thomas’ estimator to analyze the sampled risk-sets (sampling m controls per case) is m/(m+1) when there is no exposure-response relationship. A simulation study investigated the non-asymptotic properties of the nested case-control study design and found that the relative efficiency decreased as the number of cases in the cohort decreased, the true exposure-response parameter increased, and the skewness of the exposure distribution of the risk-sets increased. There also appeared to be some bias in a nested case-control study and this bias tended to be away from the null, however, this was not a major issue. In fact, when 10 or more controls were matched with each case, the bias was never more than 10%. A second simulation study compared the estimates obtained from a nested case-control analysis for a given cohort to the estimate obtained from analyzing the full cohort with Cox proportional hazards regression. The nested case-control estimate generally overestimated the full cohort estimate and the size of this discrepancy varied from cohort to cohort. Also, the sample variance of the estimates from a nested case-control study for a given cohort decreased dramatically as the case: control ratio increased. An alternative estimator for a nested case-control study was proposed by Chen and a set of simulations compared the performance of this estimator to that of the traditional Thomas estimator. Chen’s estimator requires the user to define a function named phi. It was shown that the performance of Chen’s estimator is somewhat sensitive to the definition of phi. In particular, if the support of phi was small, Chen’s estimator performed poorly. However, for larger definitions of the support of phi, Chen’s estimator performed comparable, if not better than Thomas’ estimator in terms of the bias and relative efficiency. Finally, a simulation study investigated the effect of classical measurement error on the Cox proportional hazards model. The simulations suggest that the introduction of measurement error may change the perceived shape of the exposure-response curve. In fact, the curve was more likely to level-off in the high exposure range which is commonly seen in occupational cohort studies and this effect became more severe as the magnitude of the error increased. Advisors/Committee Members: Deddens, James.

▼ Gröbner bases are the single most important tool in applicable algebraic geometry. They are used to compute standard representatives in the residue classes of a polynomial ring modulo an ideal and they can be used as a step towards solving a system of polynomial equations. Applications in science and technology are abundant, particularly in cryptography and coding theory. Computation of Gröbner bases is challenging. It has computational complexity double exponential in the worst-case and exponential in average. Although this makes Gröbner bases intractable in many cases, a great deal of effort has been devoted to improve algorithms to compute faster larger Gröbner bases. The concept of mutant polynomials, introduced by Ding in 2006, quantifies the deviation from the average case, leading to faster algorithms that profit on this degeneration. In this dissertation we introduce three algorithms aiming at improving Gröbner bases computation that are inspired by mutant polynomials. The mutantF4 algorithm modifies Faugère's F4 by exploiting the presence of mutant polynomials. The MXL3 algorithm introduces a termination condition based on the absence of mutant polynomials. And the MGB algorithm combines MXL3 with the idea of preempted reduction to avoid storing large sets of polynomials. Each new idea achieved gains in speed and/or memory usage. The MGB algorithm is particularly successful, which was demonstrated by being the first algorithm to be able to compute a Gröbner basis for a random system of 32 quadratic polynomials in 32 variables over GF(2). We also propose LASyz, a method to avoid redundant computation in Gröbner bases computation, that is compatible with mutant algorithms. LASyz is a simple yet effective method to significantly reduce unproductive reductions to zero. It uses linear algebra to maintain a set of generators for the module of syzygies. LASyz is compatible with mutant Gröbner basis algorithms thanks to its simplicity and the use of linear algebra for reducing both polynomials and syzygies. Finally, we establish some theoretical results for mutant polynomials. We define the concept of mutant space, and study its dimension. We demonstrate that the mutant space is trivial in generic situations. We link the concept of mutants to the notions of regular sequences and syzygies. And we conjecture that the dimension of the mutant space constitutes a generic property of sequences of polynomials. Advisors/Committee Members: Ding, Jintai.

▼ This dissertation focuses on Bayesian mediation analysis for survival outcome. It consists of three parts. The first part of this dissertation is Bayesian semi-parametric mediation analysis for survival outcome with incomplete mediator. This work extends (Huang et al., 2004) approach to causal mediation analyses, addresses model misspecification problem that was inherited in the commonly used method (Lin et al., 1997); in addition, it solves the issues of missing mediator under the assumption of MCAR or MAR. The second part considers the problem of assessing mediation effect when both outcome and mediator are censored. This problem is tackled by developing an informative missing data imputation modeling for the censored mediator, and develop a Markov Chain Monte Carlo algorithm to obtain posterior estimates. The third part considers longitudinal mediator problem. The previous work (Taylor et al., 1998, 2001) is extended on joint modeling of longitudinal mediator and the survival outcome. For the longitudinal modeling of mediator, the proposed model considers fixed effects, random effects, and a specified stochastic process with measurement error using Dirichlet process priors on the coefficient parameters. Advisors/Committee Members: Sivaganesan, Siva.

▼ Based on the Merton's problem and the concept of indifference pricing methodology, the author develops a pair of uncoupled partial differential equations to find the fair price of a single illiquid financial security. The equations are developed by two different methods, and the results are consistent. Furthermore, a vector indifference pricing framework is conjectured for multiple securities valuation. The pricing method is applied on financial contract that could not be traded during valuation period and the liquidity premium is revealed. Especially, this method could be applied on private equity valuation problem. Another pricing equation using the concept of consistent pricing is also developed in this paper. Moreover by applying variable transformation technique on the Basic Equity Model, an integral from solution is found on the public equity valuation problem under interest rate risk. Advisors/Committee Members: Stojanovic, Srdjan.

▼ We find exact formulas for the growth of the ideal λAk, where λ is a quadratic element of the algebra of functions over the Galois field Fq for q = 2 and q = 3. More precisely, we calculate dim(λAk), where Ak is the subspace of elements of degree less than or equal to k. The results clarify some of the assertions made in the articles of Yang, Chen, and Courtois [YC], [YCC] regarding the complexity of the XL algorithm. Advisors/Committee Members: Ding, Jintai.

▼ Survival analysis is widely used in many different areas. The classic models, such as Cox proportional hazards model, are frequently used to model univariate survival data. However, in biomedical studies, it is not uncommon that each study subject experience multiple events or subjects are related within some clusters. These data are called multivariate survival data. The statistical methods of these problems need to describe the dependence of observations within a subject or cluster. Frailty model is one way to approach the problem and is commonly used recently. Among the frailties, the gamma frailty is frequently used because of its analytic features. However, the gamma frailty model cannot handle the highly correlated data in some cases. In this thesis, different parametric survival models with gamma frailty and lognormal frailty have been examined in terms of correlation. Overall, lognormal frailty models perform better than gamma frailty models in many survival models. Another approach to solve multivariate survival data problem is via parametric distributions which can directly address the dependence among the data. In this thesis, a bivariate distribution with marginal Weibull distribution is proposed. Some properties of the distribution have been discussed. Weibull model with lognormal frailty, Weibull model with gamma frailty, and the marginal Weibull model are also fitted via Bayesian method and the results are compared. Advisors/Committee Members: Sivaganesan, Siva.

▼ Futures contract is one of the oldest and simplest financial contracts, and the cost of carry model is no doubt the most popular pricing model for futures contracts. However, since it is derived for forward contracts and forward contracts price equals to futures contract only under some specific circumstances, this model has systematic pricing error and fails to capture some important properties of the futures contract, such as the dynamic interaction between the underlying and the futures contract. Another model for futures contracts pricing is the general equilibrium model with stochastic interest rate and volatility. However, this model is based on some relatively strong assumptions and using logarithmic utility of wealth. In this paper, we implement a more general model to derive a closed-form pricing formula for the futures contracts pricing, with stochastic interest rate, dividend rate and appreciation rate. The model shows some different properties comparing with the classic cost of carry model and general equilibrium model. In addition, we use the result of our model to set up a statistical analysis using Standard & Poor’s 500 index and the Standard & Poor’s 500 E-mini futures contracts’ historical data. The statistical analysis results shows that if one choose to use our utility-based pricing framework, then dividend yield rate plays a important role. Our result suggests a promising modification for the classic model. Advisors/Committee Members: Stojanovic, Srdjan.

▼ This dissertation includes two parts: developing a new model for individual pharmacokinetics (PK) and applying a Bayesian three-stage hierarchical model to population PK. As to individual PK, the standard methodology is compartment modeling characterized by physiological mechanisms. Parameters in individual PK are estimated based on data from a single individual. In the individual PK part, the relationship between drug concentration and time for an individual was modeled, and the kinetic parameters for an individual were characterized and quantified. Specifically, a piecewise absorption model without physiological compartment mechanisms was developed and applied for Mycophenolic acid (MPA) data that does not obey a one compartment first-order absorption pattern. In the second part of this dissertation, a Bayesian three-stage hierarchical model was applied to population PK using simulated multi-occasion PK data with both inter-individual variability (IIV) and inter-occasion variability (IOV). This Bayesian approach was applied to three PK models. First, a PK model with independent IOV was studied, and different variances at different occasions were estimated. Second, a PK model with multivariate covariates and correlated and constrained IOV was studied, and unequal constrains in the variance matrix was modeled. Third, a PK model with arbitrary IOV was studied, and four inverse Whishart priors for IOV with different scale matrices were investigated. Based on the result and analysis, a recommendation of choosing the prior distribution was made according to whether or not a reliable source of the covariance matrix exists. For all population PK models, Gibbs sampling and Metropolis-Hasting algorithm were implemented using SAS IML to generate samples from posterior distributions. Advisors/Committee Members: Sivaganesan, Siva.

▼ Narrative comprehension is a fundamental cognitive skill that involves the coordination of different functional brain regions. To investigate the network structure among the brain regions supporting this cognitive function, a Spectral Bayesian Network with Bayesian model averaging is developed based on the spectral density estimation of the functional Magnetic Resonance Imaging (fMRI) time series recorded from multiple brain regions. In this approach, the neural interactions and temporal dependence among different brain regions are measured by spectral density matrices after a Fourier transform of the fMRI signals to the frequency domain. A Bayesian model averaging method is then applied to build the network structure from a set of candidate networks. Using this model, brain networks of three distinct age groups are constructed to assess the dynamic change of network connectivity with respect to age. Networks of multivariate time series are also simulated from vector autoregressive models to compare the performances of the SBN with existing methods in learning network structure from time series data. In addition to the network modeling of the functional interactions among brain regions, the quantification of the functional connectivity between two brain regions is also very important for understanding how the functions of the human brain develop. Using spectral coherence and partial spectral coherence, the overall and direct functional connectivity strengths among the language-related neural circuits are computed based on fMRI time series data collected in 313 children ranging in age from 5 to 18 years in a story comprehension experiment. The age or gender effects on both the pair wise direct link and connection strength are studied to access children’s development of brain functions for story comprehension. In addition, the connectivity differences between the left and right hemispheres, and the connections in both hemispheres that are directly related to the children's story comprehension performance, are also studied using spectral connectivity. The selection of the fMRI preprocessing pipeline plays a critical role in any modeling attempt using fMRI data. The third part of this dissertation focuses a novel residual bootstrap framework for the evaluation of fMRI preprocessing pipelines based on the repeatability of the activation maps generated from a statistical model. Based on the bootstrapping of residual time series from regressions, this framework define image reproducibility using the similarity of the activation maps generated from resampled fMRI images. The multiple-level linear model combined with the residual bootstrap scheme is very flexible and can accommodate non-homogenous populations such as case-control or pediatric data sets. The superior performance of this method is demonstrated through synthetic fMRI data sets with different degrees of motion artifacts, and the evaluation of the preprocessing pipelines from a longitudinal fMRI study. Advisors/Committee Members: Sivaganesan, Siva.

▼ The motivation of my thesis came from a problem I heard on Radiolab, a podcast distributed through National Public Radio. In the podcast, the two hosts asked the question, “What is the probability of flipping seven consecutive tails when flipping a coin a hundred times?” They approximated the probability to being 1/6. From a mathematical point of view, this seems like too simple of an answer because there are 2^100 cases one must consider. In my thesis, I first go about finding an exact probability to this initial question. Afterwards, I show how one can answer a generalized version of this question, where the number of flips and the number of consecutive events are variable. Additionally, I show how to find the probability of consecutive heads or tails occuring. By answering these questions, I learned calculation techniques using matrix multiplication. These methods are shared in the paper. Lastly, I go into some of the underlying mathematics in this matrix multiplication and how it relates to their related recursive sequences. Advisors/Committee Members: Pelikan, Stephan.

▼ In longitudinal studies, we are often interested in simultaneously clustering observations at both subject- and time-levels. Current clustering approaches assume the exchangeability among clustering units, and they are not applicable for our clustering goal. Through the use of a specific base measure, we propose a more suitable method that improves upon the multivariate DP mixture model. A well-known MCMC algorithm, Gibbs sampler, is implemented for the Bayesian posterior distributions and estimates. We compare two kinds of specific base measures from simple to complex. The models are evaluated through simulation studies of multivariate data with different covariance specifications. Performance is assessed by the stationarity, the autocorrelation functions of the Markov chain, the correct classification rates, the 95% credible intervals for parameter estimates, and the CPU time. We illustrate the method with data from a prospective longitudinal study on sleep apnea, tracking the diastolic blood pressure and severity of sleep apnea of 97 children during 24 hours. Advisors/Committee Members: Sivaganesan, Siva.

▼ This thesis concerns the well-posedness and controllability of certain dispersive partial differential equations. The first part focuses on an initial boundary value problem (IBVP) for the Korteweg-de Vries equation posed on a bounded interval with specific nonhomogeneous boundary conditions; this problem was introduced by T. Colin and J.-M. Ghidaglia. The IBVP is shown to be well-posed on the L2-based Sobolev spaces Hs(0,L) for s= 0. Moreover, the existence of global solutions is proved for small data, and these solutions decay exponentially when the boundary data decay exponentially. This result is improved by showing that the IBVP is locally well-posed in Hs(0,L) for s>-3/4. Both results are proved using regularity properties and suitable techniques developed for KdV equations. The boundary controllability of the Korteweg-de Vries equation on a finite interval is also treated in this thesis. The cases of one and two controls are analyzed, and controllability is initially shown for the linear control system. The controllability of the system for one control is guaranteed only if the length of the interval does not belong to a specific set, but no conditions are needed for the case of two controls. These results are then extended to nonlinear cases by using the contraction mapping theorem. Finally, a well-posedness property for the IBVP of the Boussinesq equation posed on the half-line is shown in Hs(R+) for s=0, thanks to recent work on the Schrodinger equation and the regularity properties of its solutions. Advisors/Committee Members: Zhang, Bingyu.

▼ Analyzing multivariate time series has been a very important topic in economics, finance, engineering, social and natural sciences. Compared to univariate models, the multivariate models better represent the dynamics and correlations of the component series. Many popular univariate models such as autoregressive conditional heteroscedasticity (ARCH), generalized ARCH (GARCH), and capital asset pricing model (CAPM), are all under investigations for the extension to their multivariate counterparts. However, when increasing the data dimension, the number of parameters in the multivariate model easily explodes. This brings in various issues such as unsatisfactory estimation efficiency, heavy computational burden, and poor model interpretability, and becomes the bottleneck of high-dimensional time series analysis. In an attempt to address the problem, this dissertation studies a regularization technique for high-dimensional time series by penalty, which simultaneously performs variable selection and parameter estimation. The idea of regularization, including the shrinkage type of estimators, has a long history in statistics. Recent emergence of a large amount of high-dimensional data from various resources has given the old technique renewed attention. Several statisticians in the past decade have made significant contributions to the study of regularization technique in the new context. However, their works are mainly under the framework of independent observations and the extension to the time series settings had remained an unexplored area. This dissertation takes a step forward in filling the gap, and reconstructs several major theorems with regard to the regularization technique in the dependent settings. The established new procedure for analyzing high-dimensional time series data is general in the sense that it readily applies to a large class of stationary multivariate time series models. To demonstrate it, two chapters of the dissertation are dedicated to providing two examples, the first one being the sparse loading full-factor multivariate GARCH model and the second one being the sparse autoregressive model. The second example is extended in a following chapter, to a study of long-order AR approximation to autoregressive fractionally integrated moving average (ARFIMA) models. Advisors/Committee Members: Deddens, James.