Ramsey Theorem, in the most simple form, states that if we are given a positive integer l, there exists a minimal integer r(l), called the Ramsey number, such any partition of the edges of K_r(l) into two sets, i.e. a 2-coloring, yields a copy of K_l contained entirely in one of the partitioned sets, i.e. a monochromatic copy of Kl. We prove an extension of Ramsey's Theorem, in the more general form, by replacing complete graphs by multipartite graphs in both senses, as the partitioned set and as the desired monochromatic graph. More formally, given integers l and k, there exists an integer p(m) such that any 2-coloring of the edges of the complete multipartite graph K_p(m);r(k) yields a monochromatic copy of K_m;k . The tools that are used to prove this result are the Szemeredi Regularity Lemma and the Blow Up Lemma. A full proof of the Regularity Lemma is given. The Blow-Up Lemma is merely stated, but other graph embedding results are given. It is also shown that certain embedding conditions on classes of graphs, namely (f , ?) -embeddability, provides a method to bound the order of the multipartite Ramsey numbers on the graphs. This provides a method to prove that a large class of graphs, including trees, graphs of bounded degree, and planar graphs, has a linear bound, in terms of the number of vertices, on the multipartite Ramsey number.

Diagnostic testing is essential to distinguish non-diseased individuals from diseased individuals. More accurate tests lead to improved treatment and thus reduce medical mistakes. The sensitivity and specificity are two important measurements for the diagnostic accuracy of a diagnostic test. When the test results are continuous, it is of interest to construct a confidence interval for the sensitivity at a fixed level of specificity for the test. In this thesis, we propose three empirical likelihood intervals for the sensitivity. Simulation studies are conducted to compare the empirical likelihood based confidence intervals with the existing normal approximation based confidence interval. Our studies show that the new intervals had better coverage probability than the normal approximation based interval in most simulation settings.

Compare the accuracy of two continuous-scale tests is increasing important when a new test is developed. The traditional approach that compares the entire areas under two Receiver Operating Characteristic (ROC) curves is not sensitive when two ROC curves cross each other. A better approach to compare the accuracy of two diagnostic tests is to compare the areas under two ROC curves (AUCs) in the interested specificity interval. In this thesis, we have proposed bootstrap and empirical likelihood (EL) approach for inference of the difference between two partial AUCs. The empirical likelihood ratio for the difference between two partial AUCs is defined and its limiting distribution is shown to be a scaled chi-square distribution. The EL based confidence intervals for the difference between two partial AUCs are obtained. Additionally we have conducted simulation studies to compare four proposed EL and bootstrap based intervals.

This study is performed to evaluate the factors associated with the recurrence of myoma after Myomectomy. Identifying the factors of myoma recurrence will assist the patient and her gynecologist in deciding the most appropriate method of treatment according to her specific social, medical and emotional needs. Multiple logistic regression is used to determine the factors affecting the recurrence. 'Age of Surgery', 'Tumor Size', 'Pelvic Pain' and the interaction between the 'Age of Surgery' and 'Tumor Size' are significant in the final model. Kaplan-Meier method is used to calculate the cumulative recurrence rate. The 5 year cumulative recurrence rate is 24.32% and the 10 year cumulative recurrence rate is 32.57%.

In this thesis we discuss how to find equivalent representations of polynomial functions over the ring of integers modulo a power of a prime. Specifically, we look for lower degree representations and representations with fewer variables for which important applications in electrical and computer engineering exist. We present several algorithms for finding these compact formulations.

We propose a resource allocation model for the management of the fund for the screening and treatment of women infected by Chlamydia trachomatis and Neisseria gonorrhoeae. The goal is to maximize the number of infected women cured of Chlamydia trachomatis and Neisseria gonorrhoeae infections. The population going for screening is divided into groups by ages and races. The group number is dynamic. Dierent groups have dierent infection rates. There are four possible test assays and four possible treatments. We employed a two-phase algorithm to solve the problem. The first phase is small so an exhaustive method is applied, while the second phase is transformed to a knapsack problem and a dynamic programming method is applied.

Diagnostic testing is essential to distinguish non-diseased individuals from diseased individuals. The sensitivity and specificity are two important indices for the diagnostic accuracy of continuous-scale diagnostic tests. If we want to compare the effectiveness of two tests, it is of interest to construct a confidence interval for the difference of the two sensitivities at a fixed level of specificity. In this thesis, we propose two empirical likelihood based confidence intervals (HBELI and HBELII) for the difference of two sensitivities at a predetermined specificity level. Simulation studies show that when correlation between the two test results exists, HBELI and HBELII intervals perform better than the existing bootstrap based BCa, BTI and BTII intervals due to shorter interval lengths. However, when there is no correlation, BCa, BTI and BTII intervals outperform HBELI and HBELII intervals due to better coverage probability in most simulation settings.

The analysis of longitudinal data has been a popular subject for the recent years. The growth of the Generalized Estimating Equation (GEE) Liang & Zeger, 1986) is one of the most influential recent developments in statistical practice for this practice. GEE methods are attractive both from a theoretical and a practical standpoint. In this paper, we are interested in the influence of different "working" correlation structures for modeling the longitudinal data. Furthermore, we propose a new AIC-like method for the model assessment which generalized AIC from the point of view of the data generating. By comparing the difference of the log-likelihood functions between different correlation models, we define the exact value to create an interval for our model selection. In this thesis, we combine the GEE method and a new generalized AIC Index for the longitudinal data with different correlation structures.

Model credibility index is defined to be a sample size under which the power of rejection equals 0.5. It applies goodness-of-fit testing thinking and uses a one-number summary statistic as an assessment tool in a false model world. The estimation of the model credibility index involves a bootstrap resampling technique. To assess the consistency of the estimator of model credibility index, we instead study the variance of the power achieved at a fixed sample size. An improved subsampling method is proposed to obtain an unbiased estimator of the variance of power. We present two examples to interpret the mechanics of building model credibility index and estimate its error in model selection. One example is two-way independent model by Pearson Chi-square test, and another example is multi-dimensional logistic regression model using likelihood ratio test.

The individual growth model is a relatively new statistical technique. It is now widely used to examine the trajectories of individuals and groups in repeated measures data. This study examines the association of the receptive vocabulary over time and characteristics of children who were treated for brain tumors. The children undertook different types of treatment from one to any combinations of surgery, radiation and chemotherapy. The individual growth model is used to analyze the longitudinal data and to address the issues behind the data. Results of this study present several factors' influences to the rate of change of PPVT scores. The conclusions of this thesis indicate that the decline in the PPVT scores is associated with gender, age at diagnosis, socioeconomic status, type of treatment and Neurological Predictor Scale.