Mathematics Theses

Antiretroviral Regimens in HIV-Infected Adults Receiving Medical Care in the United States: Medical Monitoring Project, 2009

Yunfeng Tie — Fri, 26 Apr 2013 12:12:09 PDT

Effective antiretroviral therapy (ART) is essential for viral suppression (VS) in HIV-infected patients. However, there is a lack of nationally representative data on types of ART regimens used and their impact on VS. This thesis used self-reported interview and abstracted medical record from 2009 Medical Monitoring Project (MMP) to study ART regimen type and related health outcomes. Results showed that 88.6% of HIV-infected adults in care was prescribed ART, and about half took regimens designated as ‘preferred’ according to U.S ART guidelines. Among MMP participants prescribed ART, 62.7% achieved durable VS, 77.8% achieved recent VS, 83.5% were 100% dose-adherent, and 17.1% reported side effects. Multivariate regression analyses revealed that although ART was critical for VS, there were minor differences in health outcomes among the major ART classes in the U.S. ART guidelines or six most-commonly used regimens. This study could be potentially useful for future strategic planning of HIV care.

Small Improvement to the Kolmogorov-Smirnov Test

Xing Dong — Fri, 26 Apr 2013 12:07:53 PDT

The Kolmogorov-Smirnov (K-S) test is widely used as a goodness-of-fit test. This thesis consists of two parts to describe ways to improve the classical K-S test in both 1-dimensional and 2-dimensional data. The first part is about how to improve the accuracy of the classical K-S goodness-of-fit test in 1-dimensional data. We replace the p-values estimated by the asymptotic distribution with near-exact p-values. In the second part, we propose two new methods to increase power of the widely used 2-dimensional two-sample Fasano and Franceschini test. Simulation studies show the new methods are significantly more powerful than the Fasano and Franceschini’s test.

Identification of Differential Gene Pathways with Sparse Principal Component Analysis

Yichao Yin — Fri, 26 Apr 2013 12:07:51 PDT

The development of the technology makes it possible to measure large amount of genes expressions simultaneously. Since biological functions are mostly coordinated by multiple genes, called “gene pathway”, it is interesting to identify differential gene pathways which are associated with clinical phenotype. Principal component analysis has been proposed to identify differential gene pathways in several literatures, while sparse principal component analysis (SPCA) has not drawn any attention. We proposed to use SPCA to identify differential gene pathways. The results show that, comparing to PCA, SPCA could identify more differential expressed gene pathways, especially when the higher-order interactions among genes are considered.

Racial Differences in the Prevalence of Depressive Disorders Among U.S. Adult Population

Fang-Di Yang — Fri, 26 Apr 2013 12:07:49 PDT

Depression is a common but serious illness, which may have significant impact on a person’s daily life. This study examined racial/ethnic disparities on the prevalence of lifetime diagnosis of depression and current depression according to 2010 Behavioral Risk Factor Surveillance System (BRFSS) data. Logistic regression models were constructed to calculate odds ratios for comparing prevalence rates of depression across racial/ethnic groups. Significant racial/ethnic differences were found in both the prevalence of lifetime diagnosis of depression and current depression. Regional disparities were also found to be significant in the prevalence of depressive disorders across racial/ethnic groups.

Confidence Interval Estimation for Coefficient of Variation

Shuang Liu — Fri, 26 Apr 2013 12:07:46 PDT

The coefficient of variation (CV) is a helpful quantity to describe the variation in evaluating results from different populations. There are many papers discussing methods of constructing confidence intervals for a single CV, such as exact method and approximation methods for CV when the underlying distribution is a normal distribution. However, the exact method is computationally cumbersome, and approximation methods can't be applied when the underlying distribution is unknown. In this thesis, we propose the generalized confidence interval for CV when the underlying distribution is normal and three empirical likelihood-based non-parametric intervals for CV when the underlying distribution is unknown. Simulation studies are conducted to compare the relative performances of these intervals based on the coverage probability and average interval length. Finally, the application of the proposed methods is demonstrated by using some real examples.

Geršgorin Discs and Geometric Multiplicity

Rachid Marsli — Tue, 04 Dec 2012 10:01:45 PST

If A is an nxn complex matrix and λ is an eigenvalue of A with geometric multiplicity k, then λ is in at least k of the Geršgorin discs D_i of A.

Let k, r, t be positive integers with k ≤ r ≤ t. Then there is a txt complex matrix A and an eigenvalue λ of A such that λ has geometric multiplicity k and algebraic multiplicity t, and λ is in precisely r Geršgorin Discs of A. Some examples and related results are also provided.

The Topology and Algebraic Functions on Affine Algebraic Sets Over an Arbitrary Field

Anthony J. Preslicka — Tue, 04 Dec 2012 10:01:44 PST

This thesis presents the theory of affine algebraic sets defined over an arbitrary field K. We define basic concepts such as the Zariski topology, coordinate ring of functions, regular functions, and dimension. We are interested in the relationship between the geometry of an affine algebraic set over a field K and its geometry induced by the algebraic closure of K. Various versions of Hilbert-Nullstellensatz are presented, introducing a new variant over finite fields. Examples are provided throughout the paper and a question on the dimension of irreducible affine algebraic sets is formulated.

Destination After Entering Foster Care: Road Toward Stability

Dong Yang — Mon, 26 Nov 2012 08:41:36 PST

The duration of children stay in a temporary foster care system needs to be minimal in order to ensure a stable and successful life. However, a time-consuming procedure of investigations is usually taken to decide whether they can reunite with their birth parents. Moreover, if the child fails to reunite with their family, another discharge decision needs to be assessed, leading to even longer time without a normal life. Based on the data from Adoption and Foster Care Analysis and Reporting System (AFCARS), this thesis derives a prediction model to discriminate the children with a tendency of unsuccessful reunification from the rest. An alternative discharge option can therefore be prepared concurrently for the foster youth with high non-reunification probability. The model is obtained by logistic regression and evaluated with receiver operating characteristic (ROC) curve.

A Novel Method for Automated Cell Image Selection

Shuman Guo — Mon, 26 Nov 2012 08:41:34 PST

Retinal pigment epithelium (RPE) is a key site of pathogenesis of age-related macular degeneration (AMD). A key first step toward developing statistical quantifications of RPE morphology is image analysis of RPE flatmount. This thesis work aims to facilitate image analysis by developing a procedure for automated selecting regions with biological information from flatmount images. Our new approach, based on clustering analysis, can extract informative regions from a typical flatmount image of a mouse eye within one minute, a three order magnitude time saving improvement from the current manual procedure. This method is already contributing to the image analysis of RPE flatmounts.

Classification of Genotype and Age of Eyes Using RPE Cell Size and Shape

JIE YU — Mon, 26 Nov 2012 08:41:33 PST

Retinal pigment epithelium (RPE) is a principal site of pathogenesis in age-related macular de-generation (AMD). AMD is a main source of vision loss even blindness in the elderly and there is no effective treatment right now. Our aim is to describe the relationship between the morphology of RPE cells and the age and genotype of the eyes. We use principal component analysis (PCA) or functional principal component method (FPCA), support vector machine (SVM), and random forest (RF) methods to analyze the morphological data of RPE cells in mouse eyes to classify their age and genotype. Our analyses show that amongst all morphometric measures of RPE cells, cell shape measurements (eccentricity and solidity) are good for classification. But combination of cell shape and size (perimeter) provide best classification.

A Comparative Study Between Genotypes and Ages of Eyes Using Morphometric Measures of Retinal Pigment Epithelial Cells

Micheal Shola Folarinde — Thu, 23 Aug 2012 07:11:18 PDT

Aged-related macular degeneration (AMD) is a common eye condition among people older than 65 years and is a leading cause of vision loss. It gradually destroys the macula, the part of the eye that provides sharp, central vision needed for seeing objects clearly. This study aims to test the hypothesis that the morphology of retina pigment epithelium, a key site of AMD pathology, can reflect the various stresses aging and AMD progression impose. We first identify and separate the young and old age group for mouse eyes. Then we classify, the mouse eyes using two genotypes (C57BL/6L, RD10), and two age group (young, old).We show that without dimensional reduction, the cell area and shape measures do not provide good classification of the mouse eyes.

But with the dimension reduction at the eye level, the cell area and shape measures provide excellent classification for mouse genotype and age.

Effect of Risk and Prognosis Factors on Breast Cancer Survival: Study of a Large Dataset with a Long Term Follow-up

Hongwei Wang — Fri, 13 Jul 2012 07:35:07 PDT

The main goal of this study is to seek the effects of some risk and prognostic factors contributing to survival of female invasive breast cancer in United States. The study presents the survival analysis for the adult female invasive breast cancer based on the datasets chosen from the Surveillance Epidemiology and End Results (SEER) program of National Cancer Institute (NCI). In this study, the Cox proportional hazard regression model and logistic regression model were employed for statistical analysis. The odds ratios (OR), hazard ratios (HR) and confidence interval (C.I.) were obtained for the risk and prognosis factors. The study results showed that some risk and prognosis factors, such as the demographic factors (race and age), social and family factor (marital status), biomedical factors (tumor size, disease stage, tumor markers and tumor cell differentiation level etc.) and type of treatment patients received had significant effects on survival of the female invasive breast cancer patients.

Cellular Neural Networks with Switching Connections

Malcom Devoe et al. — Fri, 13 Jul 2012 07:35:06 PDT

Artificial neural networks are widely used for parallel processing of data analysis and visual information. The most prominent example of artificial neural networks is a cellular neural network (CNN), composed from two-dimensional arrays of simple first-order dynamical systems (“cells”) that are interconnected by wires. The information, to be processed by a CNN, represents the initial state of the network, and the parallel information processing is performed by converging to one of the stable spatial equilibrium states of the multi-stable CNN. This thesis studies a specific type of CNNs designed to perform the winner-take-all function of finding the largest among the n numbers, using the network dynamics. In a wider context, this amounts to automatically detecting a target spot in the given visual picture. The research, reported in this thesis, demonstrates that the addition of fast on-off switching (blinking) connections significantly improves the functionality of winner-take-all CNNs. Numerical calculations are performed to reveal the dependence of the probability, that the CNN correctly classifies the largest number, on the switching frequency.

Self-Reported Medical Conditions and Demographic, Behavioral and Dietary Factors Associated with Serum 25(OH)-Vitamin D Concentration in the US Adult Population

William E. Van Fleit III — Fri, 13 Jul 2012 07:32:52 PDT

This research uses data from the 2003-2006 National Health and Nutrition Examination Survey (NHANES) to determine dietary and other factors associated with serum 25(OH)-Vitamin D concentration for 5,474 adults age 20 years and older. After multivariate adjustment, we found that serum 25(OH)-Vitamin D concentration was positively associated with diets high in fruits, vegetables, and lean meats, while diets high in processed foods and high-fat meats were inversely associated with vitamin D level. Serum 25(OH)-Vitamin D concentration was also signifi-cantly associated with age, gender, race/ethnicity, BMI, physical activity, supplementation, and the season of survey administration. Self-reported cardiovascular and kidney disease were significantly associated with serum 25 (OH)-Vitamin D concentration after adjustment for significant confounders.

Estimation Algorithm for Mixture of Experts Recurrent Event Model

Timesha U. Brooks — Tue, 22 Nov 2011 09:27:15 PST

This paper proposes a mixture of experts recurrent events model. This general model accommodates an unobservable frailty variable, intervention effect, influence of accumulating event occurrences, and covariate effects. A latent class variable is utilized to deal with a heterogeneous population and associated covariates. A homogeneous nonparametric baseline hazard and heterogeneous parametric covariate effects are assumed. Maximum likelihood principle is employed to obtain parameter estimates. Since the frailty variable and latent classes are unobserved, an estimation procedure is derived through the EM algorithm. A simulated data set is generated to illustrate the data structure of recurrent events for a heterogeneous population.

Jackknife Empirical Likelihood for the Accelerated Failure Time Model with Censored Data

Maxime K. Bouadoumou — Mon, 18 Jul 2011 06:07:27 PDT

Kendall and Gehan estimating functions are used to estimate the regression parameter in accelerated failure time (AFT) model with censored observations. The accelerated failure time model is the preferred survival analysis method because it maintains a consistent association between the covariate and the survival time. The jackknife empirical likelihood method is used because it overcomes computation difficulty by circumventing the construction of the nonlinear constraint. Jackknife empirical likelihood turns the statistic of interest into a sample mean based on jackknife pseudo-values. U-statistic approach is used to construct the confidence intervals for the regression parameter. We conduct a simulation study to compare the Wald-type procedure, the empirical likelihood, and the jackknife empirical likelihood in terms of coverage probability and average length of confidence intervals. Jackknife empirical likelihood method has a better performance and overcomes the under-coverage problem of the Wald-type method. A real data is also used to illustrate the proposed methods.

Minimum Degree Conditions for Tilings in Graphs and Hypergraphs

Andrew Lightcap — Fri, 15 Jul 2011 09:20:16 PDT

We consider tiling problems for graphs and hypergraphs. For two graphs and , an -tiling of is a subgraph of consisting of only vertex disjoint copies of . By using the absorbing method, we give a short proof that in a balanced tripartite graph , if every vertex is adjacent to of the vertices in each of the other vertex partitions, then has a -tiling. Previously, Magyar and Martin [11] proved the same result (without ) by using the Regularity Lemma.

In a 3-uniform hypergraph , let denote the minimum number of edges that contain for all pairs of vertices. We show that if , there exists a -tiling that misses at most vertices of . On the other hand, we show that there exist hypergraphs such that and does not have a perfect -tiling. These extend the results of Pikhurko [12] on -tilings.

Analysis of Dependently Truncated Sample Using Inverse Probability Weighted Estimator

Yang Liu — Fri, 15 Jul 2011 08:22:55 PDT

Many statistical methods for truncated data rely on the assumption that the failure and truncation time are independent, which can be unrealistic in applications. The study cohorts obtained from bone marrow transplant (BMT) registry data are commonly recognized as truncated samples, the time-to-failure is truncated by the transplant time. There are clinical evidences that a longer transplant waiting time is a worse prognosis of survivorship. Therefore, it is reasonable to assume the dependence between transplant and failure time. To better analyze BMT registry data, we utilize a Cox analysis in which the transplant time is both a truncation variable and a predictor of the time-to-failure. An inverse-probability-weighted (IPW) estimator is proposed to estimate the distribution of transplant time. Usefulness of the IPW approach is demonstrated through a simulation study and a real application.

Interval Estimation for the Correlation Coefficient

Aekyung Jung — Fri, 15 Jul 2011 05:52:49 PDT

The correlation coefficient (CC) is a standard measure of the linear association between two random variables. The CC plays a significant role in many quantitative researches. In a bivariate normal distribution, there are many types of interval estimation for CC, such as z-transformation and maximum likelihood estimation based methods. However, when the underlying bivariate distribution is unknown, the construction of confidence intervals for the CC is still not well-developed. In this thesis, we discuss various interval estimation methods for the CC. We propose a generalized confidence interval and three empirical likelihood-based non-parametric intervals for the CC. We also conduct extensive simulation studies to compare the new intervals with existing intervals in terms of coverage probability and interval length. Finally, two real examples are used to demonstrate the application of the proposed methods.

Prevalence of Chronic Diseases and Risk Factors for Death among Elderly Americans

Guangming Han — Fri, 15 Jul 2011 05:39:03 PDT

The main aim of this study is to explore the effects of risk factors contributing to death in the elderly American population. To achieve this purpose, we constructed Cox proportional hazard regression models and logistic regression models with the complex survey dataset from the national Second Longitudinal Study of Aging (LSOA II) to calculate the hazard ratios (HR)/odds ratios (OR) and confidence interval (CI) of risk factors. Our results show that in addition to chronic disease conditions, many risk factors, such as demographic factors (gender and age), social factors (interaction with friends or relatives), personal health behaviors (smoking and exercise), and biomedical factors (Body mass index and emotional factors) have significant effects on death in the elderly American population. This will provide important information for elderly people to prolong lifespan regardless of whether they have chronic disease/diseases or not.