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.

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 Di 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.

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.

This paper presents numerical solutions to integration problems with bivariate integrands. Using equally spaced nodes in Adaptive Simpson's Rule as a base case, two ways of sampling the domain over which the integration will take place are examined. Drawing from Ouellette and Fiume, Voronoi sampling is used along both axes of integration and the corresponding points are used as nodes in an unequally spaced degree two Newton-Cotes method. Then the domain of integration is triangulated and used in the Triangular Prism Rules discussed by Limaye. Finally, both of these techniques are tested by running simulations over heavily oscillatory and monomial (up to degree five) functions over polygonal regions.

In biomedical research and lifetime data analysis, the comparison of two hazard functions usually plays an important role in practice. In this thesis, we consider the standard independent two-sample framework under right censoring. We construct efficient and useful confidence intervals for the ratio and difference of two hazard functions using smoothed empirical likelihood methods. The empirical log-likelihood ratio is derived and its asymptotic distribution is a chi-squared distribution. Furthermore, the proposed method can be applied to medical diagnosis research. Simulation studies show that the proposed EL confidence intervals have better performance in terms of coverage accuracy and average length than the traditional normal approximation method. Finally, our methods are illustrated with real clinical trial data. It is concluded that the empirical likelihood methods provide better inferential outcomes.

Strong evidence has linked dietary sodium intake to blood pressure, but the effects of sodium intake on cardiovascular diseases (CVD) outcomes remain elusive, especially for older population. We examined the association between estimated usual sodium intake and CVD and all-cause mortality in a nationally representative sample of 4068 US adults aged 51 and older surveyed in 1988-1994. After a mean follow-up of 12.9 years from 1988 to 2006, 1680 participants died: 734 from CVD; 392 from ischemic heart disease (IHD); and 144 from stroke. In the Cox proportional models adjusted for sociodemographic variables and CVD risk factors, sodium intake was not significantly associated with all-cause, CVD, IHD and stroke mortality. No significant interactions were observed between sodium intake and sex, race/ethnicity, hypertension status, body mass index or physical activity for any of the outcomes studied. However, among Mexican-Americans sodium intake was significantly and linearly associated with CVD mortality.

The estimation of low income proportions plays an important role in comparisons of poverty in different countries. In most countries, the stability of the society and the development of economics depend on the estimation of low income proportions. An accurate estimation of a low income proportion has a crucial role for the development of the natural economy and the improvement of people's living standards. In this thesis, the Jackknife empirical likelihood method is employed to construct confidence intervals for a low income proportion when the observed data had missing values. Comprehensive simulation studies are conducted to compare the relative performances of two Jackknife empirical likelihood based confidence intervals for low income proportions in terms of coverage probability. A real data example is used to illustrate the application of the proposed methods.

In the field of diagnostic test studies, the accuracy of a diagnostic test is essential in evaluating the performance of the test. The receiver operating characteristic (ROC) curve and the area under the curve (AUC) are widely used in such evaluation procedures. Meanwhile, the Youden index is also introduced into practice to measure the accuracy of the diagnostic test from another aspect. The Youden index maximizes the sum of sensitivity and specificity, assuring decent true positive and negative rates. It draws one's attention due to its merit of finding the optimal cut-off points of biomarkers. Similar to Partial ROC, a new index, called "Partial Youden index" can be defined as an extension of Youden's Index. It is more meaningful than regular Youden index since the regular one is just a special case of the Partial Youden Index. In this thesis, we focus on construction of generalized confidence intervals for the Partial Youden Index and its corresponding optimal cut-off points. Extensive simulation studies are conducted to evaluate the finite sample performances of the new intervals.

Receiver operating characteristic (ROC) curves have been widely used in evaluation of the goodness of the diagnostic method in many study fields, such as disease diagnosis in medicine. The area under the ROC curve (AUC) naturally became one of the most used variables in gauging the goodness of the diagnosis (Mossman, Somoza 1991). Since medical diagnosis often is not dichotomous, the ROC curve and AUC need to be generalized to a multi-dimensional case. The generalization of AUC to multi-class case has been studied by many researchers in the past decade. Most recently, Nakas & Yiannoutsos (2004) considered the ordered d classes ROC analysis by only considering the sensitivities of each class. Hence, their dimension is only d. Cha (2005) considered more types of mis-classification in the ordered multiple-class case, but reduced the dimension of Ferri, at.el. from d(d-1) to 2(d-1). In this dissertation we are trying to adjust and calculate the VUS for an ordered multipleclass with Cha’s 2(d-1)-dimension method. Our methodology of finding the VUS is introduced. We present the method of adjusting and calculating VUS and their statistical inferences for the 2(d-1)-dimension. Some simulation results are included and a real example will be presented. Intellectual outcomes in pediatric brain-tumor patients were investigated in a prospective longitudinal study. The Standard-Binet Intelligence Scale-Fourth Edition (SB-IV) Standard Age Score (SAS) and Composite intelligence quotient (IQ) score are examined as cognitive outcomes in pediatric brain-tumor patients. Treatment factors, patient factors and time since diagnosis are taken into account as the risk factors. Hierarchical linear/quadratic models and Gompertz based hierarchical nonlinear growth models were applied to build linear and nonlinear longitudinal curves. We use PRESS and Volume Under the Surface (VUS) as the criterions to compare these two methods. Some model interpretations are presented in this dissertation.

We look at three distinct topics in analysis. In the first we give a direct and easy proof that the usual Newton-Leibniz rule implies the fundamental theorem of algebra that any nonconstant complex polynomial of one complex variable has a complex root. Next, we look at the Riesz representation theorem and show that the Riesz representing measure often can be given in the form of mini sums just like in the case of the usual Lebesgue measure on a cube. Lastly, we look at the idea of holomorphic domination and use it to define a class of complex Banach manifolds that is similar in nature and definition to the class of Stein manifolds.