The purpose of this paper is to present a discussion on the basic fundamentals of the theory of sets. Primarily, the discussion will be confined to the study of cardinal and ordinal numbers. The concepts of sets, classes of sets, and families of sets will be undefined quantities, and the concept of the class of all sets will be avoided.

Thesis (Honors)--College of William and Mary, 2009. / Includes bibliographical references (leaf 34). Also available via the World Wide Web.

Many promising Phase II trials have been carried out in head injury however to date there has been no successful translation of the positive results from these explanatory trials into improved patient outcomes in Phase III trials. Many reasons have been hypothesised for this failure. Outcomes in head injury trials are usually measured using the five point Glasgow Outcome Scale. Traditionally the ordinality of this scale is disregarded and it is dichotomised into two groups, favourable and unfavourable outcome. This thesis explores whether suboptimal statistical analysis techniques, including the dichotomisation of outcomes could have contributed to the reasons why Phase III trials have been unsuccessful. Based on eleven completed head injury studies, simulation modelling is used to compare outcome as assessed by the conventional dichotomy with both modelling that takes into account the ordered nature of the outcome (proportional odds modelling) and modelling which individualises a patient’s risk of a good or poor outcome ( the ‘sliding dichotomy’). The results of this modelling show that both analyses which use the full outcome scale and those which individualise risk show great efficiency gains (as measured by reduction in required sample sizes) over the conventional analysis of the binary outcome. These results are consistent both when the simulated treatment effects followed a proportional odds model and when they did not. Consistent results were also observed when targeting or restricting improvement to groups of subjects based on clinical characteristics or prognosis. Although proportional odds modelling shows consistently greater sample size reductions the choice of whether to use proportional odds modelling or the sliding dichotomy depends on the question of interest.

Neste trabalho, apresentamos algumas metodologias para analisar dados que possuem variável resposta categórica ordinal. Descrevemos os principais Modelos de Regressão conhecidos atualmente que consideram a ordenação das categorias de resposta, entre eles: Modelos Cumulativos e Modelos Sequenciais. Discutimos também o problema de discriminação e classificação de elementos em grupos ordinais, comentando sobre os preditores mais comuns para dados desse tipo. Apresentamos ainda a técnica de Análise Discriminante Ótima e sua versão aprimorada, baseada na utilização de métodos bootstrap. Por fim, aplicamos algumas das técnicas descritas a dados reais da área financeira, com o intuito de classificar possíveis clientes, no momento da aquisição de um cartão de crédito, como futuros bons, médios ou maus pagadores. Para essa aplicação, discutimos as vantagens e desvantagens dos modelos utilizados em termos de qualidade da classificação. / In this work, some methods to analyse data with ordinal categorical response are presented. We describe the most important and widely used Regression Models which consider the ordering of response categories like: Cumulative Models and Sequential Models. We also discuss the problem of how to discriminate and classify elements in ordinal groups, commenting on the most common predictors to this kind of data. Also we present the technique known as optimal discriminant analysis and its improved version, based on the use of bootstrap methods. Finally, we apply some of the described techniques to real financial data, intending to classify possible consumers, on acquistion of a credit card, as high, medium and low risk customers. With this application, we discuss the advantages and disadvantages of the models used in terms of quality of classification.

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Microfiche copy available in Archives and Science. / Vita. / Bibliography: leaves 123-125. / by Sy D. Friedman. / Ph.D.

Neste trabalho, apresentamos algumas metodologias para analisar dados que possuem variável resposta categórica ordinal. Descrevemos os principais Modelos de Regressão conhecidos atualmente que consideram a ordenação das categorias de resposta, entre eles: Modelos Cumulativos e Modelos Sequenciais. Discutimos também o problema de discriminação e classificação de elementos em grupos ordinais, comentando sobre os preditores mais comuns para dados desse tipo. Apresentamos ainda a técnica de Análise Discriminante Ótima e sua versão aprimorada, baseada na utilização de métodos bootstrap. Por fim, aplicamos algumas das técnicas descritas a dados reais da área financeira, com o intuito de classificar possíveis clientes, no momento da aquisição de um cartão de crédito, como futuros bons, médios ou maus pagadores. Para essa aplicação, discutimos as vantagens e desvantagens dos modelos utilizados em termos de qualidade da classificação. / In this work, some methods to analyse data with ordinal categorical response are presented. We describe the most important and widely used Regression Models which consider the ordering of response categories like: Cumulative Models and Sequential Models. We also discuss the problem of how to discriminate and classify elements in ordinal groups, commenting on the most common predictors to this kind of data. Also we present the technique known as optimal discriminant analysis and its improved version, based on the use of bootstrap methods. Finally, we apply some of the described techniques to real financial data, intending to classify possible consumers, on acquistion of a credit card, as high, medium and low risk customers. With this application, we discuss the advantages and disadvantages of the models used in terms of quality of classification.

Sample size calculation methods for ordinal logistic regression are proposed to test statistical hypotheses. The author was motivated to do this work by the need for statistical analysis of the red imported fire ants data. The proposed methods use the concept of approximation by the moment-generating function. Some correction methods are also suggested. When a prior data set is available, an empirical method is explored. Application of the proposed methodology to the fire ant mating flight data is demonstrated. The proposed sample size and power calculation methods are applied in the hypothesis testing problems. Simulation studies are also conducted to illustrate their performance and to compare them with existing methods.

Sample size calculation methods for ordinal logistic regression are proposed to test statistical hypotheses. The author was motivated to do this work by the need for statistical analysis of the red imported fire ants data. The proposed methods use the concept of approximation by the moment-generating function. Some correction methods are also suggested. When a prior data set is available, an empirical method is explored. Application of the proposed methodology to the fire ant mating flight data is demonstrated. The proposed sample size and power calculation methods are applied in the hypothesis testing problems. Simulation studies are also conducted to illustrate their performance and to compare them with existing methods.

University students often complete evaluations of their courses and instructors. The evaluation tool typically contains questions about the course and the instructor on an ordinal Likert scale. We assess instructor effectiveness while adjusting for known confounders. We present a probit regression model with a latent variable to measure the instructor effectiveness accounting for student specific covariates, such as student grade in the course, high school and university GPA, and ACT score.

I denna studie undersöks förklaringsfaktorer till lärares trivsel med sin skolledning. Studien använder data från Skolverkets enkätundersökning Attityder till skolan 2015 samt 2014/2015 års version av Lärarregistret. Bland annat betraktas förutsättningar för undervisningen, inflytande på arbetsplatsen, stressnivå samt bakgrundsvariabler för lärare, rektorer och skolor. För att undersöka effekterna används ordinal logistisk regression. De variabler och faktorer som uppvisar en positiv signifikant effekt på trivsel med skolledningen är lärarens inflytande över resursfördelning, lärarens upplevelse av skolledningens mottaglighet för kritik, tillgången till teknisk IT-support samt stöd för undervisningen. Lärarens stressnivå har en negativ signifikant effekt.