Em estudos cujo interesse é avaliar o efeito de fatores prognósticos sobre a sobrevida ou algum outro evento de interesse, é comum o uso de modelos de regressão que relacionam tempos de sobrevivência e covariáveis. Quando covariáveis que apresentam dados omissos são incluídas nos modelos de regressão, os programas estatísticos usuais automaticamente excluem aqueles indivíduos que apresentam omissão em pelo menos uma das covariáveis. Com isso, muitos pesquisadores utilizam apenas as observações completas, descartando grande parte da informação disponível. Está comprovado que a análise baseada apenas nos dados completos pode levar a estimadores altamente viesados e ineficientes. Para lidar com este problema, alguns métodos foram propostos na literatura. O objetivo deste trabalho é estender métodos que lidam com dados de sobrevivência e omissão nas covariáveis para a situação em que existe uma proporção de pacientes na população que não são suscetíveis ao evento de interesse. A idéia principal é utilizar modelos com fração de cura incluindo ponderações para compensar possíveis desproporcionalidades na subamostra de casos completos, levando-se em conta uma possível relação entre omissão e pior prognóstico. Foi considerado um modelo de mistura no qual os tempos de falha foram modelados através da família Weibull ou do modelo semiparamétrico de Cox e as probabilidade de cura foram especificadas por um modelo logístico. Os métodos propostos foram aplicados a dados reais, em que a omissão foi simulada em 10\\%, 30\\% e 50\\% das observações. / Survival regression models are considered to evaluate the effect of prognostic factors for survival or some other event of interest. The standard statistical packages automatically exclude cases with at least one missing covariate value. Thus, many researchers use only the complete cases, discarding substantial part of the available information. It is known that this complete case analysis provides biased and inefficient estimates. The aim of this work is to extend survival models with missing covariate values to situations where some individuals are not susceptible to the event of interest. The main idea is to use cure rate models introducing individual weights to incorporate possible bias in the sample with complete cases, taking a possible relation between missingness and worse prognosis into account. Mixture models in which Weibull and Cox models are used to represent the failure times and logistic models to model the cure probabilities are considered. The performance of the procedure was evaluated via a simulation study. The proposed methods were applied to real data where the missingness was simulated in 10\\%, 30\\% and 50\\% of the observations.

Em estudos cujo interesse é avaliar o efeito de fatores prognósticos sobre a sobrevida ou algum outro evento de interesse, é comum o uso de modelos de regressão que relacionam tempos de sobrevivência e covariáveis. Quando covariáveis que apresentam dados omissos são incluídas nos modelos de regressão, os programas estatísticos usuais automaticamente excluem aqueles indivíduos que apresentam omissão em pelo menos uma das covariáveis. Com isso, muitos pesquisadores utilizam apenas as observações completas, descartando grande parte da informação disponível. Está comprovado que a análise baseada apenas nos dados completos pode levar a estimadores altamente viesados e ineficientes. Para lidar com este problema, alguns métodos foram propostos na literatura. O objetivo deste trabalho é estender métodos que lidam com dados de sobrevivência e omissão nas covariáveis para a situação em que existe uma proporção de pacientes na população que não são suscetíveis ao evento de interesse. A idéia principal é utilizar modelos com fração de cura incluindo ponderações para compensar possíveis desproporcionalidades na subamostra de casos completos, levando-se em conta uma possível relação entre omissão e pior prognóstico. Foi considerado um modelo de mistura no qual os tempos de falha foram modelados através da família Weibull ou do modelo semiparamétrico de Cox e as probabilidade de cura foram especificadas por um modelo logístico. Os métodos propostos foram aplicados a dados reais, em que a omissão foi simulada em 10\\%, 30\\% e 50\\% das observações. / Survival regression models are considered to evaluate the effect of prognostic factors for survival or some other event of interest. The standard statistical packages automatically exclude cases with at least one missing covariate value. Thus, many researchers use only the complete cases, discarding substantial part of the available information. It is known that this complete case analysis provides biased and inefficient estimates. The aim of this work is to extend survival models with missing covariate values to situations where some individuals are not susceptible to the event of interest. The main idea is to use cure rate models introducing individual weights to incorporate possible bias in the sample with complete cases, taking a possible relation between missingness and worse prognosis into account. Mixture models in which Weibull and Cox models are used to represent the failure times and logistic models to model the cure probabilities are considered. The performance of the procedure was evaluated via a simulation study. The proposed methods were applied to real data where the missingness was simulated in 10\\%, 30\\% and 50\\% of the observations.

<p>Recently, the Conway-Maxwell Poisson (COM-Poisson) cure rate model has been proposed which includes as special cases some of the well-known cure rate models discussed in the literature. Data obtained from cancer clinical trials are often right censored and the expectation maximization (EM) algorithm can be efficiently used for the determination of the maximum likelihood estimates (MLEs) of the model parameters based on right censored data.</p> <p>By assuming the lifetime distribution to be exponential, lognormal, Weibull, and gamma, the necessary steps of the EM algorithm are developed for the COM-Poisson cure rate model and some of its special cases. The inferential method is examined by means of an extensive simulation study. Model discrimination within the COM-Poisson family is carried out by likelihood ratio test as well as by information-based criteria. Finally, the proposed method is illustrated with a cutaneous melanoma data on cancer recurrence. As the lifetime distributions considered are not nested, it is not possible to carry out a formal statistical test to determine which among these provides an adequate fit to the data. For this reason, the wider class of generalized gamma distributions is considered which contains all of the above mentioned lifetime distributions as special cases. The steps of the EM algorithm are then developed for this general class of distributions and a simulation study is carried out to evaluate the performance of the proposed estimation method. Model discrimination within the generalized gamma family is carried out by likelihood ratio test and information-based criteria. Finally, for the considered cutaneous melanoma data, the two-way flexibility of the COM-Poisson family and the generalized gamma family is utilized to carry out a two-way model discrimination to select a parsimonious competing cause distribution along with a suitable choice of a lifetime distribution that provides the best fit to the data.</p> / Doctor of Philosophy (PhD)

Afin de mieux connaitre puis d'optimiser les trajectoires suivies par les patients arrivés au stade terminal de leur insuffisance rénale chronique, il a été nécessaire de mettre au point des outils permettant de modéliser ces trajectoires complexes. Les différentes modalités de traitement n'ont pas été comparées une à une mais une approche globale a été privilégiée tenant compte d'une vision intégrée où les modalités de traitement sont considérées comme complémentaires et non concurrentielles. Ce travail de modélisation a utilisé des modèles à compartiments avec prise en compte de risque concurrents et un modèle de mélange pour données de survie avec fraction non à risque. Les paramètres des modèles ont été estimés à partir des données du registre du Réseau Épidémiologie et Information en Néphrologie (REIN). L'outil de prédiction développé a également pu être alimenté par les données de remboursement de l'assurance maladie (SNIIRAM) sur l'année 2009. Cette première version de l'outil a permis d'évaluer les conséquences en termes d'espérance de vie restreinte à 15 ans et de coût moyen par mois de différentes stratégies simulées de prise en charge des patients en IRCT dans le cadre d'une analyse médico-économique, en partenariat avec la Haute Autorité de Santé. L'objectif final de ce travail sera de proposer des outils d'aide à la décision reposant sur des stratégies de prise en charge les mieux adaptées aux besoins des patients. A terme, les outils développés lors de ce travail pourraient également servir de base à une plateforme de simulation afin d'accompagner les décideurs publics lors de la réflexion sur les schémas d'organisation sanitaire / In order to better understand and then optimize the trajectories followed by end-stage renal disease patients, it was necessary to develop tools to model these complex trajectories. The different treatment modalities were not compared but a comprehensive approach was preferred taking into account an integrated vision where treatment modalities are considered complementary and non-competitive. We used compartments models which took into account competitive risk and a mixture model for survival with fraction not at risk. The model parameters were estimated from the data from the Renal Epidemiology and Information Network registry. Reimbursement data from the national health insurance (SNIIRAM) were also used. The prediction tool developed was used to evaluate the consequences in terms of expected 15- years restricted lifetime and average cost per month for different strategies in a medicoeconomic analysis, in partnership with the Haute Autorité de Santé. The final aim of this work is to offer decision support tools based on strategies best adapted to patients’ needs. The tools developed in this work could also serve as a basis for a simulation platform to accompany public decision-makers in their reflection on health organization

Cure rate models, introduced by Boag (1949), are very commonly used while modelling lifetime data involving long time survivors. Applications of cure rate models can be seen in biomedical science, industrial reliability, finance, manufacturing, demography and criminology. In this thesis, cure rate models are discussed under a competing cause scenario, with the assumption of proportional odds (PO) lifetime distributions for the susceptibles, and statistical inferential methods are then developed based on right-censored data. In Chapter 2, a flexible cure rate model is discussed by assuming the number of competing causes for the event of interest following the Conway-Maxwell (COM) Poisson distribution, and their corresponding lifetimes of non-cured or susceptible individuals can be described by PO model. This provides a natural extension of the work of Gu et al. (2011) who had considered a geometric number of competing causes. Under right censoring, maximum likelihood estimators (MLEs) are obtained by the use of expectation-maximization (EM) algorithm. An extensive Monte Carlo simulation study is carried out for various scenarios, and model discrimination between some well-known cure models like geometric, Poisson and Bernoulli is also examined. The goodness-of-fit and model diagnostics of the model are also discussed. A cutaneous melanoma dataset example is used to illustrate the models as well as the inferential methods. Next, in Chapter 3, the destructive cure rate models, introduced by Rodrigues et al. (2011), are discussed under the PO assumption. Here, the initial number of competing causes is modelled by a weighted Poisson distribution with special focus on exponentially weighted Poisson, length-biased Poisson and negative binomial distributions. Then, a damage distribution is introduced for the number of initial causes which do not get destroyed. An EM-type algorithm for computing the MLEs is developed. An extensive simulation study is carried out for various scenarios, and model discrimination between the three weighted Poisson distributions is also examined. All the models and methods of estimation are evaluated through a simulation study. A cutaneous melanoma dataset example is used to illustrate the models as well as the inferential methods. In Chapter 4, frailty cure rate models are discussed under a gamma frailty wherein the initial number of competing causes is described by a Conway-Maxwell (COM) Poisson distribution in which the lifetimes of non-cured individuals can be described by PO model. The detailed steps of the EM algorithm are then developed for this model and an extensive simulation study is carried out to evaluate the performance of the proposed model and the estimation method. A cutaneous melanoma dataset as well as a simulated data are used for illustrative purposes. Finally, Chapter 5 outlines the work carried out in the thesis and also suggests some problems of further research interest. / Thesis / Doctor of Philosophy (PhD)

當造成失敗的原因不只一種時，若各對象同一時間最多只經歷一種失敗原因，則這些失敗原因稱為競爭風險。然而，有些個體不會失敗或者經過治療之後已痊癒，我們稱這部分的群體為治癒群。本文考慮同時處理競爭風險及治癒率的混合模式，即競爭風險的治癒率模式，亦將解釋變數結合到治癒率、競爭風險的條件失敗機率，或未治癒下競爭風險的條件存活函數中，並以建立在完整資料上之擴充的概似函數為貝氏分析的架構。對於右設限對象則以插補方式決定是否會治癒或會因何種風險而失敗，並推導各參數的完全條件後驗分配及其性質。由於邊際後驗分配的數學形式無法明確呈現，再加上需對右設限者判斷其狀態，所以採用屬於馬可夫鏈蒙地卡羅法的Gibbs抽樣法及適應性拒絕抽樣法(adaptive rejection sampling) ，執行參數之模擬抽樣及設算右設限者之治癒或失敗狀態。實證部分，我們分析Klein and Moeschberger (1997)書中骨髓移植後的血癌病患的資料，並用不同模式之下的參數模擬值計算各對象之條件預測指標(CPO)，換算成各模式的對數擬邊際概似函數值(LPML)，比較不同模式的優劣。 / In case that there are more than one possible failure types, if each subject experiences at most one failure type at one time, then these failure types are called competing risks. Moreover, some subjects have been cured or are immune so they never fail, then they are called the cured ones. This dissertation discusses several mixture models containing competing risks and cure rate. Furthermore, covariates are associated with cure rate, conditional failure rate of each risk, or conditional survival function of each risk, and we propose the Bayesian procedure based on the augmented likelihood function of complete data. For right censored subjects, we make use of imputation to determine whether they were cured or failed by which risk and derive full conditional posterior distributions. Since all marginal posterior distributions don’t have closed forms and right censored subjects need to be identified their statuses, we take Gibbs sampling and adaptive rejection sampling of Markov chain Monte Carlo method to simulate parameter values. We illustrate how to conduct Bayesian analysis by using the bone marrow transplant data from the book written by Klein and Moeschberger (1997). To do model selection, we compute the conditional predictive ordinate(CPO) for every subject under each model, then the goodness is determined by the comparing the value of log of pseudo marginal likelihood (LMPL) of each model.

當母體中有部份對象因被治癒或免疫而不會失敗時，需考慮這群對象所佔的比率，即存活分率。本文主要在探討如何以貝氏方法對含存活分率之治癒率模式進行分析，並特別針對兩種含存活分率的迴歸模式，分別是Weibull迴歸模式以及對數邏輯斯迴歸模式，導出概似函數與各參數之完全條件後驗分配及其性質。由於聯合後驗分配相當複雜，各參數之邊際後驗分配之解析形式很難表達出。所以，我們採用了馬可夫鏈蒙地卡羅方法（MCMC）中的Gibbs抽樣法及Metropolis法，模擬產生參數值，以進行貝氏分析。實證部份，我們分析了黑色素皮膚癌的資料，這是由美國Eastern Cooperative Oncology Group所進行的第三階段臨床試驗研究。有關模式選取的部份，我們先分別求出各對象在每個模式之下的條件預測指標（CPO），再據以算出各模式的對數擬邊際概似函數值（LPML），以比較各模式之適合性。 / When we face the problem that part of subjects have been cured or are immune so they never fail, we need to consider the fraction of this group among the whole population, which is the so called survival fraction. This article discuss that how to analyze cure rate models containing survival fraction based on Bayesian method. Two cure rate models containing survival fraction are focused; one is based on the Weibull regression model and the other is based on the log-logistic regression model. Then, we derive likelihood functions and full conditional posterior distributions under these two models. Since joint posterior distributions are both complicated, and marginal posterior distributions don’t have closed form, we take Gibbs sampling and Metropolis sampling of Markov Monte Carlo chain method to simulate parameter values. We illustrate how to conduct Bayesian analysis by using the data from a melanoma clinical trial in the third stage conducted by Eastern Cooperative Oncology Group. To do model selection, we compute the conditional predictive ordinate (CPO) for every subject under each model, then the goodness is determined by the comparing the value of log of pseudomarginal likelihood (LPML) of each model.