In this paper we propose two simple algorithms to estimate parameters £] and baseline survival function in Cox proportional hazard model with application to Massachusetts Health Care Panel Study (MHCPS) (Chappell, 1991) data which is a left truncated and interval censored data. We find that, in the estimation of £] and baseline survival function, Kaplan and Meier algorithm is uniformly better than the Empirical algorithm. Also, Kaplan and Meier algorithm is uniformly more powerful than the Empirical algorithm in testing whether two groups of survival functions are the same. We also define a distance measure D and compare the performance of these two algorithms through £] and D.

Emanating from classical Weibull mixture models we propose a framework for clustering survival data with various proportionality restrictions imposed. By introducing mixtures of Weibull proportional hazards models on a multivariate data set a parametric cluster approach based on the EM-algorithm is carried out. The problem of non-response in the data is considered. The application example is a real life data set stemming from the analysis of a world-wide operating eCommerce application. Sessions are clustered due to the dwell times a user spends on certain page-areas. The solution allows for the interpretation of the navigation behavior in terms of survival and hazard functions. A software implementation by means of an R package is provided. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Thesis (Ph.D.)--University of Alabama at Birmingham, 2010. / Title from PDF title page (viewed on June 28, 2010). Includes bibliographical references.

Thesis (M.A.)--University of South Florida, 2006. / Title from PDF of title page. Document formatted into pages; contains 71 pages. Includes bibliographical references.

O Atendimento Pré Hospitalar (APH) é um importante recurso no atendimento à vítimas de trauma. No entanto, há muitas dificuldades para demonstrar o efeito benéfico das intervenções do APH na sobrevivência das vítimas, sobretudo as de suporte avançado à vida (SAV). A proposta deste estudo é caracterizar as vítimas de acidentes trânsito, com Revised Trauma Score (RTS) <11, atendidas pelo SAV municipal e encaminhadas a hospitais terciários em São Paulo, além de identificar as variáveis da fase pré-hospitalar associadas à sobrevivência e avaliar o valor predeterminante dessas variáveis sobre o resultado obtido pelas vítimas. As variáveis avaliadas foram: sexo, idade, mecanismos do acidente, procedimentos de suporte básico e SAV realizados, repercussão fisiológica do trauma na cena do acidente, (considerando o RTS , seus parâmetros e flutuações), o tempo consumido no APH, gravidade do trauma segundo o Injury Severity Score (ISS),a Maximum Abbreviated Injury Scale (MAIS) e número de lesões para cada segmento corporal. Os resultados obtidos por 175 vítimas entre 12 e 65 anos, foram submetidos a ”Análise de Sobrevivência de Kaplan Meier” e ao “Modelo de Riscos Proporcionais de Cox”. A variável dependente foi o tempo de sobrevivência após o acidente, considerando os intervalos até 6h,12h, 24h, 48h, até 7 dias e até o término da internação. Os homens (86,9%) e a faixa etária de 20 a 29 anos (36,0%) foram as mais freqüentes. Os atropelamentos (45,1%) e o envolvimento de motocicletas e seus ocupantes (30,9%) foram os destaques dentre os mecanismos de trauma. A média do RTS na cena e do ISS, foram respectivamente 8,8 e 19,4.Os segmentos corpóreos mais atingidos foram: cabeça (58,8%), membros inferiores (45,1%) e superfície externa (40%). A média de tempo consumido na fase de APH foi 41min (tempo de cena 20,2min). Ocorreram 36% de óbitos, (metade em até 6 horas). A análise estatística revelou 24 fatores associados à sobrevivência, dentre eles, os procedimentos respiratórios avançados e os circulatórios básicos, as variáveis relativas ao RTS e a gravidade (ISS, MAIS e o número de lesões). No modelo final de Cox, ter sido submetido a procedimentos respiratórios avançados, compressões torácicas, apresentar lesão abdominal e ISS>25, foi associado a maior risco para o óbito até 48h após o trauma. Até 7 dias, a compressão torácica não se manteve no modelo final e a PAS de zero a 75mmHg apresentou associação com a morte após o acidente. Até a alta hospitalar, a ausência de PAS na avaliação inicial permaneceu no modelo. A reposição de volume foi o único fator com valor protetor para o risco de óbito presente em todos os momentos / The prehospital care (PH) is an important resource to trauma victims’ care. Nevertheless, there is great difficulty in demonstrating the PH intervention’s positive effect in victim’s survival, especially when concerning the advanced life support (ALS). The aim of this study is to characterize motor vehicle crash victims with Revised Trauma Score (RTS) <11 cared by municipal ALS and moved to tertiary hospitals in São Paulo in addition to identifying the prehospital variables associated to survival, and to evaluate their values as victim survival outcome determinant. The variables evaluated were: sex, age, trauma mechanism, basic life support and ALS procedures, physiological measures in the accident scene (considering the RTS, its parameters and fluctuations), the time consumed in PH phase, trauma severity by Injury Severity Score (ISS), the Maximum Abbreviated Injury Scale (MAIS) and number of lesions in each body region. The main results obtained by 175 victims between 12 e 65 years of age were submitted to the Kaplan Meier Survival Analysis and to Cox Proportional hazards Regression Analysis. The dependent variable was the survival time after the motor vehicle accident considering the intervals up to 6,12,24 and 48hs , up to 7 days and until the time of hospital discharge. Men (86,9%) and the 20 to 29 aged group (36%) were the most frequent. The pedestrians struck by car (45,1%) and the motorcycles (and their riders) (30,9%)were the highlight in trauma mechanisms. The RTS and the ISS average were 8,8 and 19,4 respectively. The more damaged body regions were head (58,8%), lower limbs (45,1%) and external surface (40%).The prehospital time average was 41 min (scene time 20,2min).Death rate was 36% (half of which up to 6hs).The statistical analysis revealed 24 survival associated factors. The ALS and the circulatory basic procedures, the RTS variables and the trauma severity (ISS,MAIS and number of lesions) were within them. In the final Cox Model were associated to higher risk of death up to 48hs after trauma: the submission to ALS respiratory procedures, chest compressions, the presence of abdominal injuries and ISS>25 .Until the 7th day the chest compression was not sustained in a final model and the systolic blood pressure (SBP) from zero to 75mmHg revealed statistical association with death after trauma. Until hospital discharge the SBP absence in scene evaluation remained in the model. The prehospital intravenous fluid refilling was the only factor of protector value to death risk in all moments

O objetivo deste trabalho foi comparar o desempenho do modelo de riscos proporcionais de Cox convencional, modelo de Cox modificado quando os riscos não são proporcionais e o modelo de análise de sobrevida baseado na teoria de processos de contagem. Para tanto utilizou-se uma coorte de 648 pacientes portadores de câncer de pênis, atendidos no Departamento de Cirurgia Pélvica do Hospital A. C. Camargo, no período de 1953 a 1985. Dessa coorte foram selecionadas três amostras com o objetivo de validar internamente os resultados da análise de sobrevida do banco de dados original. Os resultados do modelo de riscos proporcionais de Cox, no banco de dados original, foram confirmados por uma das amostras desse conjunto de dados. Apenas o estadiamento N foi confirmado como fator prognóstico também nas outras duas amostras. O modelo de riscos proporcionais de Cox e o modelo de análise de sobrevida baseado na teoria de processos de contagem apresentaram resultados semelhantes, na definição dos fatores prognósticos dessa coorte de pacientes com câncer de pênis. O modelo utilizando processos de contagem é mais sofisticado, do ponto de vista matemático. Porém o modelo de Cox está disponível em grande número de pacotes estatísticos e a interpretação de seus coeficientes se faz com maior facilidade. Por isso, talvez, continue a ser a técnica estatística mais utilizada quando o objetivo do estudo é definir fatores prognósticos e grupos de risco. Os fatores prognósticos para a sobrevida de pacientes com câncer de pênis foram os estadiamentos T e N e o grau de diferenciação do tumor. Esses resultados foram ajustados pelo ano de início de tratamento no Hospital A.C. Camargo. Os pacientes com prognóstico favorável foram os que apresentaram tumor pequeno, sem presença de linfonodos clinicamente positivos, e tumor bem diferenciado. / The aim of this study was to compare the performance of the Cox proportional hazards model, the Cox model with time-dependent covariates and the survival model using the counting process theory. These methods were applied in a cohort of 648 patients with penile cancer treated at the Department of Pelvic Surgery, Hospital A.C. Camargo (São Paulo-Brazil), between 1953 and 1985. Three samples were selected from the total database in order to check the internal validity. The prognostic factors selected using the Cox proportional hazards model were the same in one sample. The only prognostic factor selected in all samples was the N stage. The T and N stages, and the grade of differentiation were independent prognostic factors of survival using both the Cox proportional hazards model and the survival,model using the counting process theory. The statistical significance was the same and even the values of estimation of the coefficients were very close. The survival model using the counting process is more sophisticated from the mathematical point of view, but the Cox model is more available in statistical software, and, probably because of this, is more applied in survival analysis than the model using the counting processo Patients with small tumors, clinically negatives nodes and well differentiated tumors showed a favorable prognosis. These results were adjusted by year of the beginning in the study.

Nous nous intéressons à des processus empiriques particuliers pour l'inférence dans le modèle à risques non proportionnels. Ce modèle permet au coefficient de régression de varier avec le temps et généralise le modèle à risques proportionnels très utilisé pour modéliser des données de survie. Le processus du score standardisé que nous étudions est une somme séquentielle des résidus standardisés du modèle. Le processus est considéré en présence d'une covariable dans le modèle, avant d'être étendu au cas de multiples covariables pouvant être corrélées. Le plan du manuscrit se décompose en trois parties. Dans un premier temps, nous établissons les propriétés limites du processus sous le modèle et sous un modèle mal spécifié. Dans une deuxième partie, nous utilisons les résultats de convergence du processus pour dériver des tests de la valeur du paramètre du modèle. Nous montrons qu'un des tests proposés est asymptotiquement équivalent au test de référence du log-rank pour comparer les fonctions de survie de plusieurs groupes de patients. Nous construisons des tests plus puissants que le test du log-rank sous certaines alternatives. Enfin, dans la dernière partie, nous étudions comment lier prédiction et adéquation dans le modèle à risques non proportionnels. Nous proposons une méthode de construction d'un modèle bien ajusté en maximisant sa capacité prédictive. Aussi, nous introduisons un test d'adéquation du modèle à risques proportionnels. Les performances des méthodes proposées, qu'il s'agisse des tests sur le paramètre ou de l'adéquation du modèle, sont comparées à des méthodes de référence par des simulations. Les méthodes sont illustrées sur des données réelles. / In this thesis, we focus on particular empirical processes on which we can base inference in the non-proportional hazards model. This time-varying coefficient model generalizes the widely used proportional hazards model in the field of survival analysis. Our focus is on the standardized score process that is a sequential sum of standardized model-based residuals. We consider first the process with one covariate in the model, before looking at its extension for multiple and possibly correlated covariates. The outline of the manuscript is composed of three parts. In the first part, we establish the limit properties of the process under the model as well as under a misspecified model. In the second part, we use these convergence results to derive tests for the value of the model parameter. We show that one proposed test is asymptotically equivalent to the log-rank test, which is a benchmark for comparing survival experience of two or more groups. We construct more powerful tests under some alternatives. Finally, in the last part, we propose a methodology linking prediction and goodness of fit in order to construct models. The resulting models will have a good fit and will optimize predictive ability. We also introduce a goodness-of-fit test of the proportional hazards model. The performances of our methods, either tests for the parameter value or goodness-of-fit tests, are compared to standard methods via simulations. The methods are illustrated on real life datasets.

For both randomized clinical trials and prospective cohort studies, the Cox regression model is a powerful tool for evaluating the effect of a treatment or an explanatory variable on time-to-event outcome. This method assumes proportional hazards over time. Systematic approaches to efficiently evaluate non-proportionality and to model data in the presence of non-proportionality are investigated. Six graphical methods are assessed to verify the proportional hazards assumption based on characteristics of the survival function, cumulative hazard, or the feature of residuals. Their performances are empirically evaluated with simulations by checking their ability to be consistent and sensitive in detecting proportionality or non-proportionality. Two-sample data are generated in three scenarios of proportional hazards and five types of alternatives (that is, non-proportionality). The usefulness of these graphical assessment methods depends on the event rate and type of non-proportionality. Three numerical (statistical testing) methods are compared via simulation studies to investigate the proportional hazards assumption. In evaluating data for proportionality versus non-proportionality, the goal is to test a non-zero slope in a regression of the variable or its residuals on a specific function of time, or a Kolmogorov-type supremum test. Our simulation results show that statistical test performance is affected by the number of events, event rate, and degree of divergence of non-proportionality for a given hazards scenario. Determining which test will be used in practice depends on the specific situation under investigation. Both graphical and numerical approaches have benefits and costs, but they are complementary to each other. Several approaches to model and summarize non-proportionality data are presented, including non-parametric measurements and testing, semi-parametric models, and a parametric approach. Some illustrative examples using simulated data and real data are also presented. In summary, we present a systemic approach using both graphical and numerical methods to identify non-proportionality, and to provide numerous modeling strategies when proportionality is violated in time-to-event data.

Premarital fertility, that is, childbearing before first marriage, is an important yet under researched demographic topic in sub-Saharan Africa. In Zimbabwe, the distinction by marital status in fertility research is hardly drawn. Hence, a gap exists in the knowledge of premarital fertility levels. This research aims to investigate levels of, and factors associated with, premarital fertility since the onset of Zimbabwe's fertility transition in the mid-1980s. The research employed direct fertility estimation techniques to effectively compare premarital, marital, and overall fertility trends between 1988 and 2015. Cox proportional-hazards regression and forest plot analyses were then used to explain changes in factors associated with the timing of premarital first births over the same period. Data quality assessments were carried out using the method of cohortperiod fertility rates to provide explanations for any erratic results. The results showed that premarital fertility was constant and moderate, with an average of 0.7 children per woman, between 1988 and 2015. While most premarital first births consistently occurred to younger women, from 2005 onwards, they increased among women aged above 24 years and decreased among adolescents. An increase in age, commencing sexual activity after adolescence, and improved socio-economic status including level of education decreased the relative risk of having a premarital first birth. However, delaying marriage past young womanhood, history of contraceptive use, Ndebele ethnicity, and residence in regions other than Manicaland and Masvingo, especially Ndebele dominated regions, increased the same risk by 465.0%, 45.5%, 136.0% and up to 135.0% respectively. The stagnation of premarital fertility between 1988 and 2015 while both marital and overall fertility first declined and then stalled indicates that there is insufficient evidence to suggest that premarital fertility had contributed to the stall of fertility decline in Zimbabwe from the mid-1990s. The timing of premarital first births since the start of the fertility transition in the 1980s has had a strong ethnic and cultural bias. Due to evidence of the effect of migrancy and tourism on premarital fertility in border and tourism towns, an extension into the theory of migrant premarital sexual behaviour to detail the risk of premarital fertility among border town residents who interact with but are neither migrants nor tourists is recommended.

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