Document Type

Dissertation

Degree

Ph.D.

Degree Granting Department

Mathematics and Statistics

Major Professor

Chris P. Tsokos

Keywords

Bayesian Statistics, Cancer Mortality, Functional Data Analysis, Global Warming, Joinpoint Regression

Abstract

The present study is divided into two parts: the first is on developing the statistical analysis and modeling of mortality (or incidence) trends using Bayesian joinpoint regression and the second is on fitting differential equations from time series data to derive the rate of change of carbon dioxide in the atmosphere.

Joinpoint regression model identifies significant changes in the trends of the incidence, mortality, and survival of a specific disease in a given population. Bayesian approach of joinpoint regression is widely used in modeling statistical data to identify the points in the trend where the significant changes occur. The purpose of the present study is to develop an age-stratified Bayesian joinpoint regression model to describe mortality trends assuming that the observed counts are probabilistically characterized by the Poisson distribution. The proposed model is based on Bayesian model selection criteria with the smallest number of joinpoints that are sufficient to explain the Annual Percentage Change (APC). The prior probability distributions are chosen in such a way that they are automatically derived from the model index contained in the model space. The proposed model and methodology estimates the age-adjusted mortality rates in different epidemiological studies to compare the trends by accounting the confounding effects of age. The future mortality rates are predicted using the Bayesian Model Averaging (BMA) approach.

As an application of the Bayesian joinpoint regression, first we study the childhood brain cancer mortality rates (non age-adjusted rates) and their Annual Percentage Change (APC) per year using the existing Bayesian joinpoint regression models in the literature. We use annual observed mortality counts of children ages 0-19 from 1969-2009 obtained from Surveillance Epidemiology and End Results (SEER) database of the National Cancer Institute (NCI). The predictive distributions are used to predict the future mortality rates. We also compare this result with the mortality trend obtained using joinpoint software of NCI, and to fit the age-stratified model, we use the cancer mortality counts of adult lung and bronchus cancer (25-85+ years), and brain and other Central Nervous System (CNS) cancer (25-85+ years) patients obtained from the Surveillance Epidemiology and End Results (SEER) data base of the National Cancer Institute (NCI).

The second part of this study is the statistical analysis and modeling of noisy data using functional data analysis approach. Carbon dioxide is one of the major contributors to Global Warming. In this study, we develop a system of differential equations using time series data of the major sources of the significant contributable variables of carbon dioxide in the atmosphere. We define the differential operator as data smoother and use the penalized least square fitting criteria to smooth the data. Finally, we optimize the profile error sum of squares to estimate the necessary differential operator. The proposed models will give us an estimate of the rate of change of carbon dioxide in the atmosphere at a particular time. We apply the model to fit emission of carbon dioxide data in the continental United States. The data set is obtained from the Carbon Dioxide Information Analysis Center (CDIAC), the primary climate-change data and information analysis center of the United States Department of Energy.

The first four chapters of this dissertation contribute to the development and application of joinpiont and the last chapter discusses the statistical modeling and application of differential equations through data using functional data analysis approach.

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