A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, December 2014. / Industrial units cannot operate without failure forever. When the operation of a unit deviates from industrial standards, it is considered to have failed. The time from the moment a unit enters service until it fails is its lifetime. Within reliability and often in life data analysis in general, lifetime is the event of interest. For highly reliable units, accelerated life testing is required to obtain lifetime data quickly. Accelerated tests where failure is not instantaneous, but the end point of an underlying degradation process are considered. Failure during testing occurs when the performance of the unit falls to some specified threshold value such that the unit fails to meet industrial specifications though it has some residual functionality (degraded failure) or decreases to a critical failure level so that the unit cannot perform its function to any degree (critical failure). This problem formulation satisfies the random signs property, a notable competing risks formulation originally developed in maintenance studies but extended to accelerated testing here. Since degraded and critical failures are linked through the degradation process, the open problem of modelling dependent competing risks is discussed. A copula model is assumed and expert opinion is used to estimate the copula. Observed occurrences of degraded and critical failure times are interpreted as times when the degradation process first crosses failure thresholds and are therefore postulated to be distributed as inverse Gaussian. Based on the estimated copula, a use-level unit lifetime distribution is extrapolated from test data. Reliability metrics from the extrapolated use-level unit lifetime distribution are found to differ slightly with respect to different degrees of stochastic dependence between the risks. Consequently, a degree of dependence between the risks that is believed to be realistic to admit is considered an important factor when estimating the use-level unit lifetime distribution from test data. Keywords: Lifetime; Accelerated testing; Competing risks; Copula; First passage time.

Currently, there are an estimated 2.8 million breast cancer survivors in the United States. Due to modern screening practices and raised awareness, the majority of these cases will be diagnosed in the early stages of disease where highly effective treatment options are available, leading a large proportion of these patients to fail from causes other than breast cancer. The primary cause of death in the United States today is cardiovascular disease, which can be delayed or prevented with interventions such as lifestyle modifications or medications. In order to identify individuals who may be at high risk for a cardiovascular event or cardiovascular mortality, a number of prognostic models have been developed. The majority of these models were developed on populations free of comorbid conditions, utilizing statistical methods that did not account for the competing risks of death from other causes, therefore it is unclear whether they will be generalizable to a cancer population remaining at an increased risk of death from cancer and other causes. Consequently, the purpose of this work is multi-fold. We will first summarize the major statistical methods available for analyzing competing risk data and include a simulation study comparing them. This will be used to inform the interpretation of the real data analysis, which will be conducted on a large, contemporary cohort of breast cancer survivors. For these women, we will categorize the major causes of death, hypothesizing that it will include cardiovascular failure. Next, we will evaluate the existing cardiovascular disease risk models in our population of cancer survivors, and then propose a new model to simultaneously predict a survivor's risk of death due to her breast cancer or due to cardiovascular disease, while accounting for additional competing causes of death. Lastly, model predicted outcomes will be calculated for the cohort, and evaluation methods will be applied to determine the clinical utility of such a model.

<b>Background:</b> 'Competing risks' are events that either preclude or alter the probability of experiencing the primary study outcome(s). Many standard survival models fail to account for competing risks, introducing an unknown level of bias in their measures of absolute and relative risk. Individuals with type 2 diabetes mellitus (T2DM) and albuminuria are at increased risk of multiple competing causes of mortality, including cardiovascular disease (CVD), cancer and renal disease, yet studies to date have not implemented competing risks methodology. <b>Aim:</b> Using albuminuria in T2DM as a case study, this Thesis set out to quantify differences between standard- and competing-risks-adjusted survival analysis estimates of absolute and relative risk for the outcomes of cardiovascular and cancer mortality. <b>Methods:</b> 86,962 patients aged &ge;35 years with T2DM present on or before 2005 were identified in the Clinical Practice Research Datalink. To quantify differences in measures of absolute risk, cumulative risk estimates for cardiovascular and cancer mortality from standard survival analysis methods (Kaplan-Meier estimator) were compared to those from competing-risks-adjusted methods (cumulative incidence competing risk estimator). Cumulative risk estimates were stratified by patient albuminuria level (normoalbuminuria vs albuminuria). To quantify differences in measures of relative risk, estimates for the effect of albuminuria on the relative hazards of cardiovascular and cancer mortality were compared between standard cause-specific hazard (CSH) models (Cox-proportional-hazards regression), competing risk CSH models (unstratified Lunn-McNeil model), and competing risk subdistribution hazard (SDH) models (Fine-Gray model). <b>Results:</b> Patients with albuminuria, compared to those with normoalbuminuria, were older (p&LT;0.001), had higher systolic blood pressure (p&LT;0.001), had worse glycaemic control (p&LT;0.001), and were more likely to be current or ex-smokers (p&LT;0.001). Over the course of nine years of follow-up 22,512 patients died; 8,800 from CVD, 5,239 from cancer, and 8,473 from other causes. Median follow-up was 7.7 years. In patients with normoalbuminuria, nine-year standard and competing-risks-adjusted cumulative risk estimates for cardiovascular mortality were 11.1% (95% confidence interval (CI): 10.8-11.5%) and 10.2% (95% CI: 9.9-10.5%), respectively. For cancer mortality, these figures were 8.0% (95% CI: 7.7-8.3%) and 7.2% (95% CI: 6.9-7.5%). In patients with albuminuria, standard and competing-risks-adjusted estimates for cardiovascular mortality were 21.8% (95% CI: 20.9-22.7%) and 18.5% (95% CI: 17.8-19.3%), respectively. For cancer mortality, these figures were 10.7% (95% CI: 10.0-11.5%) and 8.6% (8.1-9.2%). For the effect of albuminuria on cardiovascular mortality, hazard ratios from multivariable standard CSH, competing risks CSH, and subdistribution hazard ratios from competing risks SDH models were 1.75 (95% CI: 1.63-1.87), 1.75 (95% CI: 1.64-1.87), and 1.58 (95% CI: 1.48-1.69), respectively. For the effect of albuminuria on cancer mortality, these values were 1.27 (95% CI: 1.16-1.39), 1.28 (95% CI: 1.17-1.40), and 1.11 (95% CI: 1.01-1.21). <b>Conclusions:</b> When evaluating measures of absolute risk, differences between standard and competing-risks-adjusted methods were small in absolute terms, but large in relative terms. For the investigation of epidemiological relationships using relative hazards models, standard survival analysis methods produced near-identical risk estimates to the CSH competing risks methods for the clinical associations evaluated in this Thesis. For the evaluation of risk prediction using relative hazards models, CSH models produced consistently higher risk estimates than SDH models, and their use may lead to over-estimation of the predictive effect of albuminuria on either outcome. Where outcomes are less common (like cancer) CSH models provide poor estimates of risk prediction, and SDH models should be used. This research demonstrates that differences can be present between risk estimates derived using CSH and SDH methods, and that the two are not necessarily interchangeable. Moreover, such differences may be present in other clinical areas.

Orientador: Luiz Koodi Hotta / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-21T13:55:30Z (GMT). No. of bitstreams: 1 Tsai_Rodrigo_D.pdf: 3859687 bytes, checksum: 1064b1fa05b98307d97763bb79e95de4 (MD5) Previous issue date: 2012 / Resumo: Nesse trabalho discutimos aplicações de cópulas a modelos de riscos múltiplos com dependência e modelos de misturas de distribuições. Numa primeira parte analisamos a inclusão de dependência entre os fatores de risco do modelo de riscos múltiplos. Os modelos de riscos múltiplos são uma família de modelos flexíveis para representar dados de tempos de vida. Suas maiores vantagens sobre os modelos de risco simples incluem a habilidade de representar funções de taxa de falha com formas não usuais e a facilidade de incluir covariáveis. O objetivo principal dessa parte é modelar a dependência existente entre as causas latentes de falha do modelo de riscos múltiplos por meio de funções de cópulas. A escolha da função de cópulas bem como das funções de distribuição dos tempos latentes de falha resultam numa classe flexível de distribuições de sobrevivência que é capaz de representar funções de taxa de falha de formas multimodais, forma de banheira e contendo efeitos locais dados pela concorrência dos riscos. A identificação e estimação do modelo proposto também são discutidas. Ao eliminar a restrição de suporte positivo para as variáveis latentes, o método pode ser utilizado para gerar uma família rica de distribuições univariadas contendo assimetrias e múltiplas modas. Na segunda parte propomos um modelo de mistura de distribuições generalizado utilizando cópulas. O parâmetro da cópula é útil para definir formas de assimetria e ponderar com maior ou menor peso determinadas regiões do suporte das distribuições componentes para compor a mistura. pesos das distribuições componentes variam no suporte da distribuição e não são restritos à soma unitária. A modelagem resultante acrescenta uma maior flexibilidade aos modelos de misturas na representação de dados com densidades de várias formas multimodais e assimétricas. O modelo tem como casos particulares o modelo de mistura tradicional, o modelo de riscos múltiplos e o modelo de fração de cura. Os modelos são aplicados a dados simulados e reais da literatura. Foram utilizados os métodos de estimação de máxima verossimilhança e os critérios de ajuste de Akaike e Bayesiano para a seleção dos modelos. Os modelos representaram bem os conjuntos de dados analisados em comparação com metodologias propostas na literatura / Abstract: In this work, we discuss the application of copula to polyhazard and mixture models. First we analyse the inclusion of dependence among failure causes in the polyhazard models. The polyhazard models constitute a family of flexible models to represent lifetime data. Their main advantages over single hazard models include the ability to represent hazard rate functions with unusual shapes and the ease of including covariates. The main purpose in this first part is to model the dependence that exists among the latent causes of failure in the polyhazard model by copula functions. The choice of the copula function as well as the latent failure distributions produces a flexible class of survival distributions that is able to model hazard functions with unusual shapes such as bathtub or multimodal curves, while also modelling local effects given by the competing risks. The model identification and estimation are also discussed. Dropping the restriction of positive support for the latent variables, the method can be used to generate a rich family of univariate distributions with asymmetries and multiple modes. In the second part a generalized mixture model using copula functions is proposed. To assemble the mixture model, the parameter of the copula function is used to define asymmetry shapes and to attribute more or less weight to chosen regions of the component distributions. The weights of the component distributions vary on the support of the distribution and are not restricted to the unitary sum. The resulting model increases the flexibility of the mixture models to represent data with densities with several multimodal and asymmetric shapes. Special cases of the model are the traditional mixture models, the polyhazard model, and the cure fraction model. Simulated and empirical data from the literature are analysed by the proposed models. The estimation was done by maximum likelihood methods and the selection of the models used the Akaike and Bayesian criteria. The proposed models exhibited very good fit to the data sets in comparison to other methodologies presented in the literature / Doutorado / Estatistica / Doutor em Estatística

Orientador: Djalma de Carvalho Moreira Filho / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Ciências Médicas / Made available in DSpace on 2018-08-26T22:55:22Z (GMT). No. of bitstreams: 1 Ferraz_RosemeiredeOlanda_D.pdf: 2711370 bytes, checksum: b4966f4c4ea3b88daffa54c0576bd307 (MD5) Previous issue date: 2015 / Resumo: O objetivo deste estudo é identificar os fatores associados ao tempo de sobrevida do câncer de mama, como idade, estadiamento e extensão do tumor, utilizando modelos de riscos proporcionais de Cox e de riscos competitivos de Fine-Gray. E também propor um modelo de regressão paramétrico para ajustar o tempo de sobrevida na presença dos riscos competitivos. É um estudo de coorte retrospectivo de base-populacional referente a 524 mulheres diagnosticadas com câncer de mama no período de 1993 a 1995, acompanhadas até 2011, residentes no município de Campinas/SP. Um ponto de corte para a variável contínua da idade foi escolhido utilizando-se modelos de Cox. Nos ajustes de modelos simples e múltiplo de Fine-Gray e de Cox, a idade não foi significativa quando o óbito por câncer de mama foi o evento de interesse. As curvas de sobrevivências estimadas por Kaplan-Meier evidenciaram diferenças expressivas nas probabilidades comparando-se os óbitos por câncer de mama e por riscos competitivos. As curvas de sobrevida por câncer de mama não apresentaram diferenças significativas quando comparadas as categorias de idades, segundo teste de log rank. Os modelos de Fine-Gray e Cox identificaram praticamente as mesmas covariáveis influenciando no tempo de sobrevida para ambos eventos de interesse, óbitos por câncer de mama e óbitos por riscos competitivos. Foram comparados os modelos exponencial, de Weibull e lognormal com o modelo gama generalizada e conclui-se que o modelo de regressão de Weibull foi o mais adequado para ajustar o tempo de sobrevida na presença dos riscos competitivos, conforme resultados dos testes de razões de verossimilhanças / Abstract: The aim of this study is to identify associated factors to time failure survival of breast cancer such as age, stage and extent of the tumor using Cox's proportional hazards and Fine-Gray competing risks models. It is a retrospective cohort study of population-based concerning to 524 women diagnosed with breast cancer in the period 1993-1995, followed until 2011, living in the city of Campinas, São Paulo State, Brazil. The cutoff age variable has been defined using Cox models. In the settings of simple and multiple models of Fine-Gray and Cox age was not significant when the death from breast cancer was the outcome of interest. The survival curves estimated by Kaplan-Meier showed significant differences in the odds comparing the deaths from breast cancer and competing risks. The survival curves for breast cancer showed no significant differences when comparing age groups, according to the logrank test. The Fine-Gray and Cox models identified the same covariates influencing the survival time for both events of interest: deaths from breast cancer and deaths from competing risks. The exponential, Weibull and lognormal regression models were compared with generalized gamma model and it is concluded that the Weibull regression model was the most appropriate to adjust the survival time in the presence of competing risks, according to results of the ratio likelihood tests / Doutorado / Epidemiologia / Doutora em Saúde Coletiva

Ce travail a eu pour objectif de proposer des méthodes statistiques pour évaluer et comparer les capacités prédictives de divers outils pronostiques. Le Brier score et principalement les courbes ROC dépendant du temps ont été étudiés. Tous deux dépendent d'un temps t, représentant un horizon de prédiction. Motivé par les applications à la prédiction de la démence et des données de cohortes de personnes âgées, ce travail s'est spécifiquement intéressé à des procédures d'inférence en présence de données censurées et de risques concurrents. Le risque concurrent de décès sans démence est en effet important lorsque l'on s'intéresse à prédire une démence chez des sujets âgés. Pour obtenir des estimateurs consistants, nous avons utilisé une méthode appelée “Inverse Probability of Censoring Weighting” (IPCW). Dans un premier travail, nous montrons qu'elle permet d'étendre simplement les estimateurs pour données non censurées et de prendre en compte une censure éventuellement dépendante de l'outil pronostique étudié. Dans un second travail, nous proposons des adaptations pour les situations de risques concurrents. Quelques résultats asymptotiques sont donnés et permettent de dériver des régions de confiance et des tests de comparaison d'outils pronostiques. Enfin, un troisième travail s'intéresse à la comparaison d'outils pronostiques dynamiques, basés sur des marqueurs longitudinaux. Les mesures de capacités pronostiques dépendent ici à la fois du temps s auquel on fait la prédiction et de l'horizon de prédiction t. Des courbes de capacités pronostiques selon s sont proposées pour leur évaluation et quelques procédures d'inférence sont développées, permettant de construire des régions de confiance et des tests de comparaison de ces courbes. L'application des méthodes proposées a permis de montrer que des outils prédictifs de la démence basés sur des tests cognitifs ou des mesures répétées de ces tests ont de bonnes capacités pronostiques. / The objective of this work is to develop statistical methods that can be used to evaluate and compare the prognostic ability of different prognostic tools. To measure prognostic ability, mainly the time-dependent ROC curve is studied and also the Brier score for a prediction horizon t. Motivated by applications where the aim is to predict the risk of dementia in cohort data of elderly people, this work focuses on inference procedures in the presence of right censoring and competing risks. In elderly populations death is a highly prevalent competing risk. To define consistent estimators of the prediction ability measures, we use the inverse probability of censoring weighting (IPCW) approach. In our first work, we show that the IPCW approach provides consistent estimators of prediction ability based on right censored data, even when the censoring distribution is marker-dependent. In our second work, we adapt the estimators to settings with competing risks. Asymptotic results are provided and we derive confidence regions and tests for comparing different prognostic tools. Finally, in a third work we focus on comparing dynamic prognostic tools which use information from repeated marker measurements to predict future events. The prognostic ability measures now depend on both the time s at which predictions are made and on the prediction horizon t. Curves of the prognostic ability as a function of s are developed for the evaluation of dynamic risk predictions. Inference procedures are adapted and so are confidence regions and tests to compare the curves. The applications of the proposed methods to cohort data show that the prognostic tools that use cognitive tests, or repeated measurements of cognitive tests, have high prognostic abilities.

Les infections nosocomiales demeurent un véritable défi de santé publique en dépit des progrès techniques considérables enregistrés. Inhérentes à la prise en charge de soins, se sont les services de réanimation qui comptabilisent les plus fort taux de prévalence. En effet, quelle que soit leur orientation (chirurgicale, médicale ou mixte), ces services, ont pour mission la prise en charge des patients dont le processus vital est menacé par la survenue brutale d’une ou de plusieurs défaillances organiques nécessitant un arsenal diagnostic et/ou thérapeutique souvent invasif.Parmi les conséquences induites par ces infections ; i) l’allongement de la durée de séjour, ii) le surcoût, iii) une augmentation de la mortalité, et iv) la résistance bactérienne.Pouvoir anticiper en amont et en aval cette problématique aux motifs complexes et aux conséquences parfois fatales serait un atout majeur au profit des patients et un outil stratégique pour les équipes soignantes.Organisée autour de trois étapes charnières, la présente étude s’est d’abord attelée à la phasede l’identification des facteurs de risque de l’évènement nosocomial et de mortalité au service de réanimation ou s’est passé l’étude –prise en compte du case-mix du service de réanimation CHU la TIMONE-. Réalisée grâce à deux méthodes statistiques différentes à savoir la régression logistique et la méthode des risques compétitifs. L’étape suivante a consisté dans un premier temps à comparer les capacités prédictives des scores APACHE II, LOD, SOFA et SAPS II chez ces patients -hospitalisés en réanimation-ayant développé un épisode nosocomial. Dans un second temps de déterminer si la variation des scores LOD, SOFA, APACHEII et SAPS II est un facteur pronostique du risque nosocomial. Les résultats obtenus révèlent que la meilleure performance prédictive est objectivée au profit du SOFA et que seule la variation de ce même score entre le premier jour d’hospitalisation et celui du diagnostic de l’infection nosocomiale mesurée grâce à l’AUC est prédictive du risque nosocomial.À l’issue de ces étapes et au moyen des résultats obtenus une construction d’un score prédictif est réalisée grâce à la méthode de régression logistique. L’objectif de ce score est d’éclairer voire d’influencer le prescripteur lors de ses prises de décisions ou d’éventuelle démarche d’ajustement de ses conduites thérapeutiques. / Limiting nosocomial infections is still a health challenge although the technical development has improved. They are inherent in medical care and the health care services have the highest prevalence. Indeed, whatever the service (surgical, medical or both), the patients life-giving process is under attack because of the emergence of one or several organ faillures;This generates a diagnostic and therapeutic arsenal which is often invasive.Among the consequences resulting from these infections we will take into account :i) a longer stay in hospitalii) an extra costiii) a higher mortality rateiv) bacterial resistance .If we could anticipate upstream and downstream this issue with complex origins and sometimes fatal consequences, it would be a major asset for patients and a strategic tool for medical teams.The present study is organized in three parts, and first focusses onto the identification of the nosocomial event and death risk factors in intensive care where the study took place. We took into account the the case-mix of the intensive care unit in the TIMONE University Hospital. The study was made with two different statistic methods that is logistic regression and the competitive risks method.The next step first consisted in comparing the predictive capacities of the APACHE II, LOD, SOFA and SAPS II scores in nosocomial patients hospitalized in intensive care . Then it tried to determine if the variation of the LOD, SOFA, APACHEII and SAPS II scores was a prognostic risk factor.Results showed that the best predictive performance was objectively measured by the SOFA and that only the variation of this score between the first day in hospital and the day of the diagnosis of a nosocomial infection, calculated thanks to the AUC, could be predictive of a nosocomal risk. After these steps, and with the results calculated , the construction of a predictive score could be established thanks to the logistic regression method. The objective of this score is to help, or even influence the prescribing doctors when they take decisions or when they try to adjust their therapeutic practices.

In this thesis, the main objective is to study certain aspects of modeling failure time data of repairable systems under a competing risks framework. We consider two different models and propose more efficient Bayesian methods for estimating the parameters. In the first model, we discuss inferential procedures based on an objective Bayesian approach for analyzing failures from a single repairable system under independent competing risks. We examined the scenario where a minimal repair is performed at each failure, thereby resulting in that each failure mode appropriately follows a power-law intensity. Besides, it is proposed that the power-law intensity is reparametrized in terms of orthogonal parameters. Then, we derived two objective priors known as the Jeffreys prior and reference prior. Moreover, posterior distributions based on these priors will be obtained in order to find properties which may be optimal in the sense that, for some cases, we prove that these posterior distributions are proper and are also matching priors. In addition, in some cases, unbiased Bayesian estimators of simple closed-form expressions are derived. In the second model, we analyze data from multiple repairable systems under the presence of dependent competing risks. In order to model this dependence structure, we adopted the well-known shared frailty model. This model provides a suitable theoretical basis for generating dependence between the components failure times in the dependent competing risks model. It is known that the dependence effect in this scenario influences the estimates of the model parameters. Hence, under the assumption that the cause-specific intensities follow a PLP, we propose a frailty-induced dependence approach to incorporate the dependence among the cause-specific recurrent processes. Moreover, the misspecification of the frailty distribution may lead to errors when estimating the parameters of interest. Because of this, we considered a Bayesian nonparametric approach to model the frailty density in order to offer more flexibility and to provide consistent estimates for the PLP model, as well as insights about heterogeneity among the systems. Both simulation studies and real case studies are provided to illustrate the proposed approaches and demonstrate their validity. / Nesta tese, o objetivo principal é estudar certos aspectos da modelagem de dados de tempo de falha de sistemas reparáveis sob uma estrutura de riscos competitivos. Consideramos dois modelos diferentes e propomos métodos Bayesianos mais eficientes para estimar os parâmetros. No primeiro modelo, discutimos procedimentos inferenciais baseados em uma abordagem Bayesiana objetiva para analisar falhas de um único sistema reparável sob riscos competitivos independentes. Examinamos o cenário em que um reparo mínimo é realizado em cada falha, resultando em que cada modo de falha segue adequadamente uma intensidade de lei de potência. Além disso, propõe-se que a intensidade da lei de potência seja reparametrizada em termos de parâmetros ortogonais. Então, derivamos duas prioris objetivas conhecidas como priori de Jeffreys e priori de referência. Além disso, distribuições posteriores baseadas nessas prioris serão obtidas a fim de encontrar propriedades que podem ser ótimas no sentido de que, em alguns casos, provamos que essas distribuições posteriores são próprias e que também são matching priors. Além disso, em alguns casos, estimadores Bayesianos não-viesados de forma fechada são derivados. No segundo modelo, analisamos dados de múltiplos sistemas reparáveis sob a presença de riscos competitivos dependentes. Para modelar essa estrutura de dependência, adotamos o conhecido modelo de fragilidade compartilhada. Esse modelo fornece uma base teórica adequada para gerar dependência entre os tempos de falha dos componentes no modelo de riscos competitivos dependentes. Sabe-se que o efeito de dependência neste cenário influencia as estimativas dos parâmetros do modelo. Assim, sob o pressuposto de que as intensidades específicas de causa seguem um PLP, propomos uma abordagem de dependência induzida pela fragilidade para incorporar a dependência entre os processos recorrentes específicos da causa. Além disso, a especificação incorreta da distribuição de fragilidade pode levar a erros na estimativa dos parâmetros de interesse. Por isso, consideramos uma abordagem Bayesiana não paramétrica para modelar a densidade da fragilidade, a fim de oferecer mais flexibilidade e fornecer estimativas consistentes para o modelo PLP, bem como insights sobre a heterogeneidade entre os sistemas. São fornecidos estudos de simulação e estudos de casos reais para ilustrar as abordagens propostas e demonstrar sua validade.