In this thesis, we consider the problem of constructing an additive risk model based on the right censored survival data to predict the survival times of the cancer patients, especially when the dimension of the covariates is much larger than the sample size. For microarray Gene Expression data, the number of gene expression levels is far greater than the number of samples. Such ¡°small n, large p¡± problems have attracted researchers to investigate the association between cancer patient survival times and gene expression profiles for recent few years. We apply Partial Least Squares to reduce the dimension of the covariates and get the corresponding latent variables (components), and these components are used as new regressors to fit the extensional additive risk model. Also we employ the time dependent AUC curve (area under the Receiver Operating Characteristic (ROC) curve) to assess how well the model predicts the survival time. Finally, this approach is illustrated by re-analysis of the well known AML data set and breast cancer data set. The results show that the model fits both of the data sets very well.

This paper presents a survey of concepts in commutative algebra that have applications to topology and graph theory. The primary algebraic focus will be on Stanley-Reisner rings, classes of polynomial rings that can describe simplicial complexes. Stanley-Reisner rings are defined via square-free monomial ideals. The paper will present many aspects of the theory of these ideals and discuss how they relate to important constructions in commutative algebra, such as finite generation of ideals, graded rings and modules, localization and associated primes, primary decomposition of ideals and Hilbert series. In particular, the primary decomposition and Hilbert series for certain types of monomial ideals will be analyzed through explicit examples of simplicial complexes and graphs.

The case-control studies are primary tools for the study of risk factors (exposures) related to the disease interested. The case-control studies using longitudinal data are cost and time efficient when the disease is rare and assessing the exposure level of risk factors is difficult. Instead of GEE method, the method of using a prospective logistic model for analyzing case-control longitudinal data was proposed and the semiparametric inference procedure was explored by Park and Kim (2004). In this thesis, we apply an empirical likelihood ratio method to derive limiting distribution of the empirical likelihood ratio and find one likelihood-ratio based confidence region for the unknown regression parameters. Our approach does not require estimating the covariance matrices of the parameters. Moreover, the proposed confidence region is adapted to the data set and not necessarily symmetric. Thus, it reflects the nature of the underlying data and hence gives a more representative way to make inferences about the parameter of interest. We compare empirical likelihood method with normal approximation based method, simulation results show that the proposed empirical likelihood ratio method performs well in terms of coverage probability.

Cancer is a dangerous disease causing the most deaths in the world today and around 550,000 deaths in America per year (American Cancer Society). Larynx cancer data was recorded by Kardaun (1983). The data was collected at a Dutch hospital during 1970 – 1978. Ninety male adults with cancer of larynx were involved into the study. Each patient was divided into one of four groups depending on his or her illness condition. The data also recorded their age, lifetime, and year of entering the research. These are common factors as factors of other cancer data. The purpose of this thesis is to apply proportional hazard regression model, additive hazard regression model, censored quantiles regression model, and censored linear regression model to analyze the above larynx cancer data and find the best regression model of data by using each method. Comparison and suggestion for which method should be used in specific situation are also made. Some related topics are also mentioned so we can have resource for future study. Key words: right censoring, proportional risk model, additive risk model, quantiles regression model, linear regression model.

The mean residual life (MRL) function is an important function in survival analysis, actuarial science, economics and other social sciences and reliability for characterizing lifetime. Different methods have been proposed for doing inference on the MRL but their coverage probabilities for small sample sizes are not good enough. In this thesis we apply the empirical likelihood method and carry out a simulation study of the MRL function using different statistical distributions. The simulation study does a comparison of the empirical likelihood method and the normal approximation method. The comparisons are based on the average lengths of confidence intervals and coverage probabilities. We also did comparisons based on median lengths of confidence intervals for the MRL. We found that the empirical likelihood method gives better coverage probability and shorter confidence intervals than the normal approximation method for almost all the distributions that we considered. Applying the two methods to real data we also found that the empirical likelihood method gives thinner pointwise confidence bands.

The Differential Operator D is a linear operator from C1[0,1] onto C[0,1]. Its domain C1[0,1] is thoroughly studied as a meager subspace of C[0,1]. This is analogous to the status of the set of all rational numbers Q in the set of the real numbers R. On the polynomial vector space Pn the Differential Operator D is a nilpotent operator. Using the invariant subspace and reducing subspace technique an appropriate basis for the underlying vector space can be found so that the nilpotent operator admits its Jordan Canonical form. The study of D on Pn is completely carried out. Finally, the solution space V of the nth order differential equation with leading coefficient one is studied. The behavior of D on V is explored using some notions from linear algebra and linear operators. NOTE- Due to the limitation of the above being in "text only form" , further details of this abstract can be viewed in the pdf file.

Bioequivalence determines if two drugs are alike. The three kinds of bioequivalence are Average, Population, and Individual Bioequivalence. These Bioequivalence criteria can be evaluated using aggregate and disaggregate methods. Considerable work assessing bioequivalence in a frequentist method exists, but the advantages of Bayesian methods for Bioequivalence have been recently explored. Variance parameters are essential to any of theses existing Bayesian Bioequivalence metrics. Usually, the prior distributions for model parameters use either informative priors or vague priors. The Bioequivalence inference may be sensitive to the prior distribution on the variances. Recently, there have been questions about the routine use of inverse gamma priors for variance parameters. In this paper we examine the effect that changing the prior distribution of the variance parameters has on Bayesian models for assessing Bioequivalence and the carry-over effect. We explore our method with some real data sets from the FDA.

We will propose a random changepoint model for the analysis of longitudinal CD4 and CD8 T-cell counts, as well as viral RNA loads, for HIV infected subjects following highly active antiretroviral treatment. The data was taken from two studies, one of the Aids Clinical Group Trial 398 and one performed by the Terry Beirn Community Programs for Clinical Research on AIDS. Models were created with the changepoint following both exponential and truncated normal distributions. The estimation of the changepoints was performed in a Bayesian analysis, with implementation in the WinBUGS software using Markov Chain Monte Carlo methods. For model selection, we used the deviance information criterion (DIC), a two term measure of model adequacy and complexity. DIC indicates that the data support a random changepoint model with the changepoint following an exponential distribution. Visual analyses of the posterior densities of the parameters also support these conclusions.

A sign pattern (matrix) is a matrix whose entries are from the set {+,–, 0}. A sign pattern matrix A is a spectrally arbitrary pattern if for every monic real polynomial p(x) of degree n there exists a real matrix B whose entries agree in sign with A such that the characteristic polynomial of B is p(x). All 3 × 3 SAP's, as well as tree sign patterns with star graphs that are SAP's, have already been characterized. We investigate tridiagonal sign patterns of order 4. All irreducible tridiagonal SAP's are identified. Necessary and sufficient conditions for an irreducible tridiagonal pattern to be an SAP are found. Some new techniques, such as innovative applications of Gröbner bases for demonstrating that a sign pattern is not potentially nilpotent, are introduced. Some properties of sign patterns that allow every possible inertia are established. Keywords: Sign pattern matrix, Spectrally arbitrary pattern (SAP), Inertially arbitrary pattern (IAP), Tree sign pattern (tsp), Potentially nilpotent pattern, Gröbner basis, Potentially stable pattern, Sign nonsingular, Sign singular

The Association of Hypertension Diagnosis with Smoking Cessation: Application of Multiple Logistic Regression Using Biostatistical and Epidemiological Methods by LaTonia A. Clay Under the Direction of Yu-Sheng Hsu, PhD. ABSTRACT Hypertension and smoking are two major issues threatening the nation’s health. Previous studies examining their relationship have resulted in conflicting reports. The aim of this study is to determine if a relationship exists between smoking cessation and hypertension diagnosis. Data from the Third National Health and Nutrition Examination Survey (NHANES III) were used in this investigation. Physical examination measurements of blood pressure and self-reported diagnosis and smoking behavior were used to define hypertension and smoking status. The odds of prior hypertension diagnosis associated with smoking cessation was estimated from a multivariate logistic regression model, adjusting for gender, age, ethnicity, BMI, physical activity, HDL cholesterol, and alcohol use. Results of the investigation found that unsuccessful smoking cessation was associated with a decreased odds of prior hypertension diagnosis, adjusting for the presence of confounders (OR=0.816, p<.001). Thus, hypertension diagnosis may indeed lead to the decision to quit smoking. Future studies on this finding are encouraged. INDEX WORDS: hypertension, smoking, cessation, diagnosis, logistic regression, NHANES