In the course Discrete Deterministic Chaos, Dr. Mark Daniels introduces students to Chaos Theory and explores many topics within the field. Students prove many of the key results that are discussed in class and work through examples of each topic. Connections to the secondary mathematics curriculum are made throughout the course, and students discuss how the topics in the course could be implemented in the classroom. This paper will provide an overview of the topics covered in the course, Discrete Deterministic Chaos, and provide additional discussion on various related topics. / text

This study focused on identifying factors that significantly affect the potentially insolvent status of life insurers. The potentially insolvent status is indicated based on insurer’s Risk-based capital ratio (RBC ratio) reported in the National Association of Insurance Commissioners (NAIC) database of life insurers’ annual statements. A logistic regression analysis is performed to explore the relationship between the RBC insolvent indicator and a set of explanatory variables including insurer’s size, capital, governance structure, membership in a group of affiliated companies, and various risk measures during the 2006-2008 period. The results suggest that the probability of potential insolvency for an individual insurer is significantly affected by its size, capital-to-asset ratio, returns on capital, health product risk and proportion of products reinsured. It could be also possibly affected by the insurer’s regulatory asset risk. However, the results indicate that the probability is not significant related to the insurer’s annuity product risk, opportunity asset risk, governance structure and its membership in a group of affiliated companies. On average, by holding all other explanatory variables constant, every 1% increase in total assets will result in a decrease of 0.19 to 0.36% on the odds of potentially insolvent rates; every 0.01 unit increase in capital-to-asset ratio will result in a decrease of a multiplicative factor of 0.951 to 0.956 on the odds; every 0.01 unit increase in return on capital will result in a decrease of a multiplicative factor of 0.984 to 0.985 on the odds; every 0.01 unit increase in health product risk will result in an increase of a multiplicative factor of 1.021 to 1.031 on the odds; and every 0.01 unit increase in proportion of products reinsured will result in an increase of a multiplicative factor of 1.015 to 1.026 on the odds. The assumptions of independency and absence of harmful multicolliearity are both valid for this logistic model, suggesting that the model is adequate and the conclusion is warranted. Although the potentially insolvent indicator, instead of the real insolvent indicator is used, this model could still be useful to identify the significant factors which affect life insurers’ potentially insolvent status. / text

Μελέτη μη γραμμικών μοντέλων παλινδρόμησης(λογιστικό, εκθετικό, Poisson,γενικευμένα γραμμικά μοντέλα) όσον αφορά διαστημα εμπιστοσύνης, έλεγχο υποθέσεων και καλή προσαρμογή. / Analysis of nonlinear regression models (logistic, exponential, Poisson, generalized linear models) regarding confidence interval estimation, tests and good fit.

The purpose of this study was to investigate the relationship between cigarette smoking and socio-demographic variables based on the empirical literature and the primitive theories in the field. Two regression approaches, logistic regression and linear multiple regression, were conducted on the two most recent Arizona Adult Tobacco Surveys to test the hypothesized models. The results showed that cigarette smokers in Arizona are mainly residents who have not completed a four-year college degree, who are unemployed, White, non-Hispanic, or young to middle-aged adults. Among the socio-demographic predictors of interest, education is the most important variable in identifying cigarette smokers, even though the predictive power of these socio-demographic variables is small. Practical and methodological implications of these findings are discussed.

Diversity in the population along with the diversity of testing usage has resulted in smaller identified groups of test takers. In addition, computer adaptive testing sometimes results in a relatively small number of items being used for a particular assessment. The need and use for statistical techniques that are able to effectively detect differential item functioning (DIF) when the population is small and or the assessment is short is necessary. Identification of empirically biased items is a crucial step in creating equitable and construct-valid assessments. Parshall and Miller (1995) compared the conventional asymptotic Mantel-Haenszel (MH) with the exact test (ET) for the detection of DIF with small sample sizes. Several studies have since compared the performance of MH to logistic regression (LR) under a variety of conditions. Both Swaminathan and Rogers (1990), and Hildalgo and López-Pina (2004) demonstrated that MH and LR were comparable in their detection of items with DIF. This study followed by comparing the performance of the MH, the ET, and LR performance when both the sample size is small and test length is short. The purpose of this Monte Carlo simulation study was to expand on the research done by Parshall and Miller (1995) by examining power and power with effect size measures for each of the three DIF detection procedures. The following variables were manipulated in this study: focal group sample size, percent of items with DIF, and magnitude of DIF. For each condition, a small reference group size of 200 was utilized as well as a short, 10-item test. The results demonstrated that in general, LR was slightly more powerful in detecting items with DIF. In most conditions, however, power was well below the acceptable rate of 80%. As the size of the focal group and the magnitude of DIF increased, the three procedures were more likely to reach acceptable power. Also, all three procedures demonstrated the highest power for the most discriminating item. Collectively, the results from this research provide information in the area of small sample size and DIF detection.

Most quantitative research is conducted by randomly selecting members of a population on which to conduct a study. When statistics are run on a sample, and not the entire population of interest, they are subject to a certain amount of error. Many factors can impact the amount of error, or bias, in statistical estimates. One important factor is sample size; larger samples are more likely to minimize bias than smaller samples. Therefore, determining the necessary sample size to obtain accurate statistical estimates is a critical component of designing a quantitative study. Much research has been conducted on the impact of sample size on simple statistical techniques such as group mean comparisons and ordinary least squares regression. Less sample size research, however, has been conducted on complex techniques such as hierarchical linear modeling (HLM). HLM, also known as multilevel modeling, is used to explain and predict an outcome based on knowledge of other variables in nested populations. Ordinal logistic HLM (OLHLM) is used when the outcome variable has three or more ordered categories. While there is a growing body of research on sample size for two-level HLM utilizing a continuous outcome, there is no existing research exploring sample size for OLHLM. The purpose of this study was to determine the impact of sample size on statistical estimates for ordinal logistic hierarchical linear modeling. A Monte Carlo simulation study was used to investigate this research query. Four variables were manipulated: level-one sample size, level-two sample size, sample outcome category allocation, and predictor-criterion correlation. Statistical estimates explored include bias in level-one and level-two parameters, power, and prediction accuracy. Results indicate that, in general, holding other conditions constant, bias decreases as level-one sample size increases. However, bias increases or remains unchanged as level-two sample size increases, holding other conditions constant. Power to detect the independent variable coefficients increased as both level-one and level-two sample size increased, holding other conditions constant. Overall, prediction accuracy is extremely poor. The overall prediction accuracy rate across conditions was 47.7%, with little variance across conditions. Furthermore, there is a strong tendency to over-predict the middle outcome category.

This study is an analysis of adolescent risk behaviors contributing to an increased rate of suicidal ideation for 12 to 18 year olds. The Youth Risk Behavior Surveillance System Survey (YRBSS) is an epidemiologic survey designed to monitor the prevalence of risky behaviors of adolescents in middle and high school1. The YRBSS is a complex sample survey with a three-stage cluster design. Multiple logistic regression is used to analyze the data, including methods of analysis to address issues in complex survey design. Results of this study indicate several different risk factors that influence the rate of suicidal ideation among adolescents, including alcohol and drug use, sexual risky behaviors, unhealthy weight loss methods, depressed mood, sex and race/ethnicity. The conclusions of this study indicate that many risk factors associated with suicidal ideation are behaviors that could be addressed with early intervention strategies to reduce the risk of suicidal ideation.

To significantly improve the early detection of cervical precancers and cancers, LightTouch™ is under development by SpectRx Inc.. LightTouch™ identifies cancers and precancers quickly by using a spectrometer to analyze light reflected from the cervix. Data from the spectrometer is then used to create an image of the cervix that highlights the location and severity of disease. Our research is conducted to find the appropriate models that can be used to generate map-like image showing disease tissue from normal and further diagnose the cervical cancerous conditions. Through large work of explanatory variable search and reduction, logistic regression and Partial Least Square Regression successfully applied to our modeling process. These models were validated by 60/40 cross validation and 10 folder cross validation. Further examination of model performance, such as AUC, sensitivity and specificity, threshold had been conducted.

Substance abuse is the overindulgence in and dependence on a drug or chemical leading to detrimental effects on the individual’s health and the welfare of those surrounding him or her. Logistic regression analysis is an important tool used in the analysis of the relationship between various explanatory variables and nominal response variables. The objective of this study is to use this statistical method to determine the factors which are considered to be significant contributors to the use or abuse of substances in school-aged children and also determine what measures can be implemented to minimize their effect. The logistic regression model was used to build models for the three main types of substances used in this study; Tobacco, Alcohol and Drugs and this facilitated the identification of the significant factors which seem to influence their use in children.

Most American universities and colleges require students to provide faculty evaluation at end of each academic term, as a way of measuring faculty teaching performance. Although some analysts think that this kind of evaluation does not necessarily provide a good measurement of teaching effectiveness, there is a growing agreement in the academic world about its reliability. This study attempts to find any strong statistical evidence supporting faculty evaluation by students as a measure of faculty teaching effectiveness. Emphasis will be on analyzing relationships between instructor ratings by students and corresponding students’ grades. Various statistical methods are applied to analyze a sample of real data and derive conclusions. Methods considered include multivariate statistical analysis, principal component analysis, Pearson's correlation coefficient, Spearman's and Kendall’s rank correlation coefficients, linear and logistic regression analysis.