Master of Science / Department of Statistics / Paul I. Nelson / Quantiles are useful in describing distributions of component lifetimes. Data, consisting of the lifetimes of sample units, used to estimate quantiles are often censored. Right censoring, the setting investigated here, occurs, for example, when some test units may still be functioning when the experiment is terminated. This study investigated and compared the performance of parametric and nonparametric estimators of quantiles from right censored data generated from Weibull and Lognormal distributions, models which are commonly used in analyzing lifetime data. Parametric quantile estimators based on these assumed models were compared via simulation to each other and to quantile estimators obtained from the nonparametric Kaplan- Meier Estimator of the survival function. Various combinations of quantiles, censoring proportion, sample size, and distributions were considered. Our simulation show that the larger the sample size and the lower the censoring rate the better the performance of the estimates of the 5th percentile of Weibull data. The lognormal data are very sensitive to the censoring rate and we observed that for higher censoring rates the incorrect parametric estimates perform the best. If you do not know the underlying distribution of the data, it is risky to use parametric estimates of quantiles close to one. A limitation in using the nonparametric estimator of large quantiles is their instability when the censoring rate is high and the largest observations are censored. Key Words: Quantiles, Right Censoring, Kaplan-Meier estimator

M.Sc. (Mathematical Statistics) / Comparison of two distributions via use of the quantile comparison function is carried out specifically from possibly censored data. A semi-parametric method which assumes linearity of the quantile comparison function is examined thoroughly for non-censored data and then extended to incorporate censored data. A fully nonparametric method to construct confidence bands for the quantile comparison function is set out. The performance of all methods examined is tested using Monte Carlo Simulation.

The measurement of human immunodeficiency virus ribonucleic acid levels over time leads to censored longitudinal data. Suitable models for dynamic modelling of these levels need to take this data characteristic into account. If groups of patients with different developments of the levels over time are suspected the model class of finite mixtures of mixed effects models with censored data is required.We describe the model specification and derive the estimation with a suitable expectation-maximization algorithm.We propose a convenient implementation using closed form formulae for the expected mean and variance of the truncated multivariate distribution. Only efficient evaluation of the cumulative multivariate normal distribution function is required. Model selection as well as methods for inference are discussed. The application is demonstrated on the clinical trial ACTG 315 data.

Neste trabalho estudou-se a existência de viés de gênero nas escolhas familiares em gastar recursos na educação dos filhos e filhas. Foi obtida uma amostra de 11386 famílias da Pesquisa de Orçamentos Familiares - POF 2002-2003 do Instituto Brasileiro de Geografia e Estatística (IBGE). As famílias tinham filhos e filhas entre 7 e 20 anos de idade, com pelo menos um deles matriculado e sem que outro membro da família, pais ou outros parentes, também estivesse matriculado. Curvas de Engel para as despesas com educação e para gastos com grupos de itens de despesas educacionais foram estimadas. Entre as variáveis explicativas estão o número de filhos e número de filhas matriculados e o número de filhos e o número de filhas não matriculados segundo faixas etárias. Outras variáveis socioeconômicas e demográficas controlaram as demais características familiares. Como 2066 famílias não apresentaram despesas com educação, as curvas de Engel foram estimadas com a utilização de modelos Tobit. As curvas de Engel também foram estimadas por Mínimos Quadrados e os resultados comparados com os obtidos pelos modelos Tobit. A igualdade entre os parâmetros estimados do número de filhos e filhas matriculados em cada faixa etária foi verificada com testes de Wald, isto para os dois procedimentos de estimação utilizados. Os resultados econométricos foram consistentes com a hipótese do trade-off quantidade qualidade existente nas decisões familiares de alocar recursos para a prole, pois o número total de filhos e filhas pressionou as despesas com educação a taxas decrescentes. As análises das despesas com educação, realizadas com os dois procedimentos de estimação, indicaram diferenças significativas no impacto causado por filhos e filhas matriculados nas faixas etárias de 7 a 10 anos e de 15 a 20 anos, com viés pró-feminino. A análise dos gastos com mensalidades escolares com modelos Tobit indicou diferenças significativas para duas faixas etárias, de 11 a 14 anos e de 15 a 20 anos, com viés pró-feminino na última faixa. Com Mínimos Quadrados houve viés pró-feminino na faixa de 15 a 20 anos, nas demais faixas não houve diferenças significativas. Quando foram analisados os gastos com cursos não regulares com um modelo Tobit, foram observadas diferenças significativas de 7 a 10 anos e de 11 a 14 anos, as duas pró-femininas. Ao analisar os gastos com cursos não regulares por Mínimos Quadrados, detectou-se viés pró-masculino na faixa etária de 11 a 14 anos e viés pró-feminino nas demais faixas. Entre as demais estimativas não houve diferenças significativas ou elas foram pró-femininas, independente do procedimento de estimação. Os resultados indicam que a formação das jovens não sofreu discriminação no que tange à disposição das famílias de gastarem com a formação de sua prole, salvo as duas exceções mencionadas. Estes resultados mostraram-se consistentes com o cenário favorável à escolarização feminina no Brasil já apontado em outros estudos. / This study investigated the gender bias presence within the family choices about spending resourses with their children\'s education. A sample of 11386 families was obtained from Household Budget Survey - POF 2002-2003, of the Brazilian Institute of Geography and Statistic - IBGE. The families had children between 7 and 20 years old, with at least one of them enrolled, considering that no other member of the family, parents or relatives, was enrolled. Engel curves for the overall expenses and for expenditures with educational item groups were estimated. The number of sons and daughters enrolled and the number of sons and daughters not enrolled according to age level are among the explanatory variables. Other socioeconomics and demographic variables controlled the rest of the family characteristics. As 2066 families did not showed expenses with education, the Engel curves were estimated using Tobit models. The Engel curves were also estimated by the method of Least Squares and the results were compared to the ones obtained by Tobit model. The equity between the estimated parameters of the number of sons and daughters enrolled in each age level was verified with Wald test for both the estimation procedures used. The econometric results were consistent with the hypothesis of quantity-quality trade-off which exists within the family decisions when allocating resourses for their children because the total number of children pressed the educational expenditures at decreasing rates. The analysis of total expenses with education, accomplished by both the estimation procedures, indicated significant differences in the impact caused by sons and daughters enrolled at the age level from 7 to 10 and from 15 to 20 with pro-female bias. The analysis of the expenditures with school monthly fees with Tobit models indicated significant differences for both age levels, from 11 to 14 and from 15 to 20, with a pro-female bias in the latter level. With the Minimum Square method there was a pro-female bias in the level of age from 15 to 20 years old, in the other levels there were no significant differences. When the expenditures with non regular courses were analyzed with Tobit model, significant differences were observed from 7 to 10 and from 11 to 14 years old, both pro-female. When the expenditures with non regular courses were analyzed with the Minimum Square method a pro-male bias was detected in the age level from 11 to 14 and a pro-female bias in the other levels. Among the other estimations there were no significant differences or they were pro-female, despite the estimation procedure. The results show that the young girls\' education did not suffer any discrimination related to the family disposition of spending with their children\'s education, apart from the two exceptions mentioned. These results are consistent with the favorable scenario to the female education in Brazil already pointed out in other studies.

Neste trabalho considera-se uma reparametrização no modelo log-gama generalizado para a inclusão de dados com sobreviventes de longa duração. Os modelos tentam estimar separadamente os efeitos das covariáveis na aceleração ou desaceleração no tempo e na fração de sobreviventes que é a proporção da população para o qual o evento não ocorre. A função logística é usada para o modelo de regressão com fração de cura. Os parâmetros do modelo, serão estimados através do método de máxima verossimilhança. Alguns métodos de influência, como a influência local e a influência local total de um indivíduo, serão introduzidos, calculados, analisados e discutidos. Finalmente, um conjunto de dados médicos será analisado sob o modelo log-gama generalizado com fração de cura. Uma análise de resíduos será executada para verificar a qualidade de ajuste do modelo. / In this work the generalized log-gama model is modified for possibility that long-term survivors are present in the data . The models attempt to estimate separately the effects of covariates on the accelaration/decelaration of the timing of a given event and surviving fraction; that is, the proportion of the population for which the event never occurs. The logistic function is used for the regression model of the surviving fraction. Inference for the model parameters is considered via maximum likelihood. Some influence methods, such as the local influence, total local influence of an individual are derived, analyzed and discussed. Finally, a data set from the medical area is analyzed under log-gama generalized mixture model. A residual analysis is performed in order to select an appropriate model.

Neste trabalho, é apresentado vários métodos de diagnóstico para modelos de riscos múltiplos. A vantagem desse modelo é sua flexibilidade em relação aos modelos de risco simples, como, os modelos Weibull e log-logístico, pois acomoda uma grande classe de funções de risco, função de risco não-monótona, por exemplo, forma de "banheira" e curvas multimodal. Alguns métodos de influência, assim como, a influência local, influência local total de um indivíduo são calculadas, analizadas e discutidas. Uma discussão computacional do método do afastamento da verossimilhança, bem como da curvatura normal em influência local são apresentados. Finalmente, um conjunto de dados reais é usado para ilustrar a teoria estudada. Uma análise de resíduo é aplicada para a seleção do modelo apropriado. / In this paperwork is present various diagnostic methods for polyhazard models. Polyhazard models are a flexible family for fitting lifetime data. Their main advantage over the single hazard models, such as the Weibull and the log-logistic models, is to include a large amount of nonmonotone hazard shapes, as bathtub and multimodal curves. Some influence methods, such as the local influence, total local influence of an individual are derived, analyzed and discussed. A discussion of the computation of the likelihood displacement as well as the normal curvature in the local influence method are presented. Finally, an example with real data is given for illustration. A residual analysis is performed in order to select an appropriate model.

In this dissertation we solve the nonidentifiability problem of Archimedean copula models based on dependent censored data (see [Wang, 2012]). We give a set of identifiability conditions for a special class of bivariate frailty models. Our simulation results show that our proposed model is identifiable under our proposed conditions. We use EM algorithm to estimate unknown parameters and the proposed estimation approach can be applied to fit dependent censored data when the dependence is of research interest. The marginal survival functions can be estimated using the copula-graphic estimator (see [Zheng and Klein, 1995] and [Rivest and Wells, 2001]) or the estimator proposed by [Wang, 2014]. We also propose two model selection procedures for Archimedean copula models, one for uncensored data and the other one for right censored bivariate data. Our simulation results are similar to that of [Wang and Wells, 2000] and suggest that both procedures work quite well. The idea of our proposed model selection procedure originates from the model selection procedure for Archimedean copula models proposed by [Wang and Wells, 2000] for right censored bivariate data using the L2 norm corresponding to the Kendall distribution function. A suitable bootstrap procedure is yet to be suggested for our method. We further propose a new parameter estimator and a simple goodness-of-fit test for Archimedean copula models when the bivariate data is under fixed left truncation. Our simulation results suggest that our procedure needs to be improved so that it can be more powerful, reliable and efficient. In our strategy, to obtain estimates for the unknown parameters, we heavily exploit the concept of truncated tau (a measure of association established by [Manatunga and Oakes, 1996] for left truncated data). The idea of our goodness of fit test originates from the goodness-of-fit test for Archimedean copula models proposed by [Wang, 2010] for right censored bivariate data.

En recherche médicale, les cohortes permettent de mieux comprendre l'évolution d'une pathologie et d'améliorer la prise en charge des patients. La mise en évidence de liens de causalité entre certains facteurs de risque et l'évolution de l'état de santé des patients est possible grâce à des études étiologiques. L'analyse de cohortes permet aussi d'identifier des marqueurs pronostiques de l'évolution d'un état de santé. Cependant, les facteurs de confusion constituent souvent une source de biais importante dans l'interprétation des résultats des études étiologiques ou pronostiques. Dans ce manuscrit, nous présentons deux travaux de recherche en Biostatistique dans la thématique des scores de propension. Dans le premier travail, nous comparons les performances de différents modèles permettant d'évaluer la causalité d'une exposition sur l'incidence d'un événement en présence de données censurées à droite. Dans le second travail, nous proposons un estimateur de courbes ROC dépendantes du temps standardisées et pondérées permettant d'estimer la capacité prédictive d'un marqueur en prenant en compte les facteurs de confusion potentiels.En cohérence avec l'objectif de fournir des outils statistiques adaptés, nous présentons également dans ce manuscrit une application nommée Plug-Stat®. En lien direct avec la base de données, elle permet de réaliser des analyses statistiques adaptées à la pathologie afin de faciliter la recherche épidémiologique et de mieux valoriser les données de cohortes observationnelles. / In medical research, cohorts help to better understandthe evolution of a pathology and improve the care ofpatients. Causal associations between risk factors andoutcomes are regularly studied through etiological studies. Cohorts analysis also allow the identification of new markers for the prediction of the patient evolution.However, confounding factors are often source of bias in the interpretation of the results of etiologic or prognostic studies.In this manuscript, we presented two research works in Biostatistics, the common topic being propensity scores.In the first work, we compared the performances of different models allowing to evaluate the causality of an exposure on an outcome in the presence of rightc ensored data. In the second work, we proposed anestimator of standardized and weighted time-dependentROC curves. This estimator provides a measure of theprognostic capacities of a marker by taking into accountthe possible confounding factors. Consistent with our objective to provide adapted statistical tools, we also present in this manuscript an application, so-calledPlug-Stat®. Directly linked with the database, it allows toperform statistical analyses adapted to the pathology in order to facilitate epidemiological studies and improve the valorization of data from observational cohorts.

Hierarchical or asymmetrical power relationships among states have long been a focus of scholarly attention (e.g., asymmetrical alliances, trade dependencies). While the "power to hurt" is one expression of power, an alternative approach is to gain and exercise authority, or "rightful rule." One of the major impediments to the study of social concepts such as authority or legitimacy, however, is in their informal or intangible nature. This dissertation uses game theoretic and latent variable approaches to capture informal, social authority relationships, or social hierarchies, among international states and explores the effects of these hierarchies on security and economic behavior. I posit that states adopt one of two social roles--that of a dominant or a subordinate. Each subordinate chooses a degree of autonomy that it is willing to cede to the dominant in exchange for a corresponding degree of protection. Ranging from complete autonomy to complete control, these dyadic bargains make up a social international hierarchy. This hierarchy affects the relationships between each subordinate and the dominant, as well as the relationships among subordinates. In the security realm, the probability of conflict initiation is inversely related to the degree of subordination. When conflict does occur, dominants are more likely to intervene when the target is located at a higher position in the dominant's social hierarchy than the aggressor state. Economically, the probability that a state enacts illiberal policies is inversely related to its degree of subordination. Moreover, more subordinated states face a lower risk of economic sanction than states located lower in the hierarchy, even for similar illiberal actions. Empirical analysis of states within the US hierarchy (1950-2000) and UK hierarchy (1870-1913) using strategic probit models supports these theoretical predictions.

L'anàlisi de la supervivència s'utilitza en diversos àmbits per tal d'analitzar dades que mesuren el temps transcorregut entre dos successos. També s'anomena anàlisi de la història dels esdeveniments, anàlisi de temps de vida, anàlisi de fiabilitat o anàlisi del temps fins a l'esdeveniment. Una de les dificultats que té aquesta àrea de l'estadística és la presència de dades censurades. El temps de vida d'un individu és censurat quan només és possible mesurar-lo de manera parcial o inexacta. Hi ha diverses circumstàncies que donen lloc a diversos tipus de censura. La censura en un interval fa referència a una situació on el succés d'interès no es pot observar directament i només tenim coneixement que ha tingut lloc en un interval de temps aleatori. Aquest tipus de censura ha generat molta recerca en els darrers anys i usualment té lloc en estudis on els individus són inspeccionats o observats de manera intermitent. En aquesta situació només tenim coneixement que el temps de vida de l'individu es troba entre dos temps d'inspecció consecutius.Aquesta tesi doctoral es divideix en dues parts que tracten dues qüestions importants que fan referència a dades amb censura en un interval. La primera part la formen els capítols 2 i 3 els quals tracten sobre condicions formals que asseguren que la versemblança simplificada pot ser utilitzada en l'estimació de la distribució del temps de vida. La segona part la formen els capítols 4 i 5 que es dediquen a l'estudi de procediments estadístics pel problema de k mostres. El treball que reproduïm conté diversos materials que ja s'han publicat o ja s'han presentat per ser considerats com objecte de publicació.En el capítol 1 introduïm la notació bàsica que s'utilitza en la tesi doctoral. També fem una descripció de l'enfocament no paramètric en l'estimació de la funció de distribució del temps de vida. Peto (1973) i Turnbull (1976) van ser els primers autors que van proposar un mètode d'estimació basat en la versió simplificada de la funció de versemblança. Altres autors han estudiat la unicitat de la solució obtinguda en aquest mètode (Gentleman i Geyer, 1994) o han millorat el mètode amb noves propostes (Wellner i Zhan, 1997).El capítol 2 reprodueix l'article d'Oller et al. (2004). Demostrem l'equivalència entre les diferents caracteritzacions de censura no informativa que podem trobar a la bibliografia i definim una condició de suma constant anàloga a l'obtinguda en el context de censura per la dreta. També demostrem que si la condició de no informació o la condició de suma constant són certes, la versemblança simplificada es pot utilitzar per obtenir l'estimador de màxima versemblança no paramètric (NPMLE) de la funció de distribució del temps de vida. Finalment, caracteritzem la propietat de suma constant d'acord amb diversos tipus de censura. En el capítol 3 estudiem quina relació té la propietat de suma constant en la identificació de la distribució del temps de vida. Demostrem que la distribució del temps de vida no és identificable fora de la classe dels models de suma constant. També demostrem que la probabilitat del temps de vida en cadascun dels intervals observables és identificable dins la classe dels models de suma constant. Tots aquests conceptes elsil·lustrem amb diversos exemples.El capítol 4 s'ha publicat parcialment en l'article de revisió metodològica de Gómez et al. (2004). Proporciona una visió general d'aquelles tècniques que s'han aplicat en el problema no paramètric de comparació de dues o més mostres amb dades censurades en un interval. També hem desenvolupat algunes rutines amb S-Plus que implementen la versió permutacional del tests de Wilcoxon, Logrank i de la t de Student per a dades censurades en un interval (Fay and Shih, 1998). Aquesta part de la tesi doctoral es complementa en el capítol 5 amb diverses propostes d'extensió del test de Jonckeere. Amb l'objectiu de provar una tendència en el problema de k mostres, Abel (1986) va realitzar una de les poques generalitzacions del test de Jonckheere per a dades censurades en un interval. Nosaltres proposem altres generalitzacions d'acord amb els resultats presentats en el capítol 4. Utilitzem enfocaments permutacionals i de Monte Carlo. Proporcionem programes informàtics per a cada proposta i realitzem un estudi de simulació per tal de comparar la potència de cada proposta sota diferents models paramètrics i supòsits de tendència. Com a motivació de la metodologia, en els dos capítols s'analitza un conjunt de dades d'un estudi sobre els beneficis de la zidovudina en pacients en els primers estadis de la infecció del virus VIH (Volberding et al., 1995).Finalment, el capítol 6 resumeix els resultats i destaca aquells aspectes que s'han de completar en el futur. / Survival analysis is used in various fields for analyzing data involving the duration between two events. It is also known as event history analysis, lifetime data analysis, reliability analysis or time to event analysis. One of the difficulties which arise in this area is the presence of censored data. The lifetime of an individual is censored when it cannot be exactly measured but partial information is available. Different circumstances can produce different types of censoring. Interval censoring refers to the situation when the event of interest cannot be directly observed and it is only known to have occurred during a random interval of time. This kind of censoring has produced a lot of work in the last years and typically occurs for individuals in a study being inspected or observed intermittently, so that an individual's lifetime is known only to lie between two successive observation times.This PhD thesis is divided into two parts which handle two important issues of interval censored data. The first part is composed by Chapter 2 and Chapter 3 and it is about formal conditions which allow estimation of the lifetime distribution to be based on a well known simplified likelihood. The second part is composed by Chapter 4 and Chapter 5 and it is devoted to the study of test procedures for the k-sample problem. The present work reproduces several material which has already been published or has been already submitted.In Chapter 1 we give the basic notation used in this PhD thesis. We also describe the nonparametric approach to estimate the distribution function of the lifetime variable. Peto (1973) and Turnbull (1976) were the first authors to propose an estimation method which is based on a simplified version of the likelihood function. Other authors have studied the uniqueness of the solution given by this method (Gentleman and Geyer, 1994) or have improved it with new proposals (Wellner and Zhan, 1997).Chapter 2 reproduces the paper of Oller et al. (2004). We prove the equivalence between different characterizations of noninformative censoring appeared in the literature and we define an analogous constant-sum condition to the one derived in the context of right censoring. We prove as well that when the noninformative condition or the constant-sum condition holds, the simplified likelihood can be used to obtain the nonparametric maximum likelihood estimator (NPMLE) of the failure time distribution function. Finally, we characterize the constant-sum property according to different types of censoring. In Chapter 3 we study the relevance of the constant-sum property in the identifiability of the lifetime distribution. We show that the lifetime distribution is not identifiable outside the class of constant-sum models. We also show that the lifetime probabilities assigned to the observable intervals are identifiable inside the class of constant-sum models. We illustrate all these notions with several examples.Chapter 4 has partially been published in the survey paper of Gómez et al. (2004). It gives a general view of those procedures which have been applied in the nonparametric problem of the comparison of two or more interval-censored samples. We also develop some S-Plus routines which implement the permutational version of the Wilcoxon test, the Logrank test and the t-test for interval censored data (Fay and Shih, 1998). This part of the PhD thesis is completed in Chapter 5 by different proposals of extension of the Jonckeere's test. In order to test for an increasing trend in the k-sample problem, Abel (1986) gives one of the few generalizations of the Jonckheree's test for interval-censored data. We also suggest different Jonckheere-type tests according to the tests presented in Chapter 4. We use permutational and Monte Carlo approaches. We give computer programs for each proposal and perform a simulation study in order compare the power of each proposal under different parametric assumptions and different alternatives. We motivate both chapters with the analysis of a set of data from a study of the benefits of zidovudine in patients in the early stages of the HIV infection (Volberding et al., 1995).Finally, Chapter 6 summarizes results and address those aspects which remain to be completed.