Youden index, a main summary index for the Receiver Operating Characteristic (ROC) curve, is a comprehensive measurement for the effectiveness of a diagnostic test. For a continuous-scale diagnostic test, the optimal cut-point for the positive of disease is the cut-point leading to the maximization of the sum of sensitivity and specificity. Finding the Youden index of the test is equivalent to maximize the sum of sensitivity and specificity for all the possible values of the cut-point. In this thesis, we propose a new non-parametric confidence interval for the Youden index. Extensive simulation studies are conducted to compare the relative performance of the new interval with the existing intervals for the index. Our simulation results indicate that the newly developed non-parametric method performs as well as the existing parametric method but it has better finite sample performance than the existing non-parametric methods. The new method is flexible and easy to implement in practice. A real example is also used to illustrate the application of the proposed interval.

This dissertation creates practical methods for Poincaré return mappings of individual and networked neuron models. Elliptic bursting models are found in numerous biological systems, including the external Globus Pallidus (GPe) section of the brain; the focus for studies of epileptic seizures and Parkinson's disease. However, the bifurcation structure for changes in dynamics remains incomplete. This dissertation develops computer-assisted Poincaré ́maps for mathematical and biologically relevant elliptic bursting neuron models and central pattern generators (CPGs). The first method, used for individual neurons, offers the advantage of an entire family of computationally smooth and complete mappings, which can explain all of the systems dynamical transitions. A complete bifurcation analysis was performed detailing the mechanisms for the transitions from tonic spiking to quiescence in elliptic bursters. A previously unknown, unstable torus bifurcation was found to give rise to small amplitude oscillations. The focus of the dissertation shifts from individual neuron models to small networks of neuron models, particularly 3-cell CPGs. A CPG is a small network which is able to produce specific phasic relationships between the cells. The output rhythms represent a number of biologically observable actions, i.e. walking or running gates. A 2-dimensional map is derived from the CPGs phase-lags. The cells are endogenously bursting neuron models mutually coupled with reciprocal inhibitory connections using the fast threshold synaptic paradigm. The mappings generate clear explanations for rhythmic outcomes, as well as basins of attraction for specific rhythms and possible mechanisms for switching between rhythms.

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.

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.

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.

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.

The receiver operating characteristic (ROC) curve methodology is the statistical methodology for assessment of the accuracy of diagnostics tests or bio-markers. Currently most widely used statistical methods for the inferences of ROC curves are complete-data based parametric, semi-parametric or nonparametric methods. However, these methods cannot be used in diagnostic applications with missing data. In practical situations, missing diagnostic data occur more commonly due to various reasons such as medical tests being too expensive, too time consuming or too invasive. This dissertation aims to develop new nonparametric statistical methods for evaluating the accuracy of diagnostic tests or biomarkers in the presence of missing data. Specifically, novel nonparametric statistical methods will be developed with different types of missing data for (i) the inference of the area under the ROC curve (AUC, which is a summary index for the diagnostic accuracy of the test) and (ii) the joint inference of the sensitivity and the specificity of a continuous-scale diagnostic test. In this dissertation, we will provide a general framework that combines the empirical likelihood and general estimation equations with nuisance parameters for the joint inferences of sensitivity and specificity with missing diagnostic data. The proposed methods will have sound theoretical properties. The theoretical development is challenging because the proposed profile log-empirical likelihood ratio statistics are not the standard sum of independent random variables. The new methods have the power of likelihood based approaches and jackknife method in ROC studies. Therefore, they are expected to be more robust, more accurate and less computationally intensive than existing methods in the evaluation of competing diagnostic tests.

In this dissertation, we investigate jackknife empirical likelihood methods motivated by recent statistics research and other related fields. Computational intensity of empirical likelihood can be significantly reduced by using jackknife empirical likelihood methods without losing computational accuracy and stability. We demonstrate that proposed jackknife empirical likelihood methods are able to handle several challenging and open problems in terms of elegant asymptotic properties and accurate simulation result in finite samples. These interesting problems include ROC curves with missing data, the difference of two ROC curves in two dimensional correlated data, a novel inference for the partial AUC and the difference of two quantiles with one or two samples. In addition, empirical likelihood methodology can be successfully applied to the linear transformation model using adjusted estimation equations. The comprehensive simulation studies on coverage probabilities and average lengths for those topics demonstrate the proposed jackknife empirical likelihood methods have a good performance in finite samples under various settings. Moreover, some related and attractive real problems are studied to support our conclusions. In the end, we provide an extensive discussion about some interesting and feasible ideas based on our jackknife EL procedures for future studies.

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.

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.