Frequentemente eventos intermediários fornecem informações mais detalhadas sobre o processo da doença ou recuperação, por exemplo, e permitem uma maior precisão na previsão do prognóstico de pacientes. Tais eventos não fatais durante o curso da doença podem ser vistos como transições de um estado para outro. A ideia básica dos modelos multiestado é que o indivíduo se move através de uma serie de estados em tempo contínuo, sendo possível estimar as probabilidades e intensidades de transição entre eles e o efeito das covariáveis associadas a cada transição. Muitos estudos incluem o agrupamento dos tempos de sobrevivência como, por exemplo, em estudos multicêntricos, e também é de interesse estudar a evolução dos pacientes ao longo do tempo, caracterizando assim dados multiestado agrupados. Devido ao fato de os dados virem de diferentes centros/grupos, os tempos de falha desses indivíduos estarem agrupados e a fatores de risco comuns não observados, é interessante considerar o uso de fragilidades para que possamos capturar a heterogeneidade entre os grupos no risco para os diferentes tipos de transição, além de considerar a estrutura de dependência entre transições dos indivíduos de um mesmo grupo. Neste trabalho apresentamos a metodologia dos modelos multiestado, dos modelos de fragilidade e, em seguida, a integração dos modelos multiestado com modelos de fragilidade, tratando do seu processo de estimação paramétrica e semiparamétrica. O estudo de simulação realizado mostrou a importância de considerarmos fragilidade sem modelos multiestado agrupados, pois sem considerá-las, as estimativas tornam-se viesadas. Além disso, verificamos as propriedades frequentistas dos estimadores do modelo multiestado com fragilidades aninhadas. Por fim, como um exemplo de aplicação a um conjunto de dados reais, utilizamos o processo de recuperação de transplante de medula óssea de pacientes tratados em quatro hospitais. Fizemos uma comparação de modelos por meio das medidas de qualidade do ajuste AIC e BIC, chegando à conclusão de que o modelo que considera dois efeitos aleatórios (uma para o hospital e outro para a interação transição-hospital) ajusta-se melhor aos dados. Além de considerar a heterogeneidade entre os hospitais, tal modelo também considera a heterogeneidade entre os hospitais em cada transição. Sendo assim, os valores das fragilidades estimadas da interação transição-hospital revelam o quão frágeis os pacientes de cada hospital são para experimentarem determinado tipo de evento/transição. / Often intermediate events provide more detailed information about the disease process or recovery, for example, and allow greater accuracy in predicting the prognosis of patients. Such non-fatal events during the course of the disease can be seen as transitions from one state to another. The basic idea of a multistate models is that the person moves through a series of states in continuous time, it is possible to estimate the transition probabilities and intensities between them and the effect of covariates associated with each transition. Many studies include the grouping of survival times, for example, in multi-center studies, and is also of interest to study the evolution of patients over time,characterizing grouped multistate data. Because the data coming from different centers/groups, the failure times these individuals are grouped and the common risk factors not observed, it is interesting to consider the use of frailty so that we can capture the heterogeneity between the groups at risk for different types of transition, in addition to considering the dependence structure between transitions of individuals of the same group. In this work we present the methodology of multistate models, frailty models and then the integration of models with multi-state fragility models, dealing with the process of parametric and semi-parametric estimation. The conducted simulation study showed the importance of considering frailty in grouped multistate models, because without conside- ring them, the estimates become biased. Furthermore, we find the frequentist properties of estimators of multistate model with nested frailty. Finally, as an application example to a set of real data, we use the process of bone marrow transplantation recovery of patients in four hospitals. We did a comparison of models through quality measures setting AIC and BIC, coming to the conclusion that the model considers two random effects (one for the hospital and another for interaction transition-hospital) fits the data better. In addition to considering the heterogeneity between hospitals, such a model also considers the heterogeneity between hospitals in each transition. Thus,the values of the frailty estimated interaction transition-hospital reveal how fragile patients from each hospital are to experience certain type of event/transition.

Os métodos de estimação para modelos de fragilidade vêm sendo bastante discutidos na literatura estatística devido a sua grande utilização em estudos de Análise de Sobrevivência. Vários métodos de estimação de parâmetros dos modelos foram desenvolvidos: procedimentos de estimação baseados no algoritmo EM, cadeias de Markov de Monte Carlo, processos de estimação usando verossimilhança parcial, verossimilhança penalizada, quasi-verossimilhança, entro outros. Uma alternativa que vem sendo utilizada atualmente é a utilização da verossimilhança hierárquica. O objetivo principal deste trabalho foi estudar as vantagens e desvantagens da verossimilhança hierárquica para a inferência em modelos de fragilidade em relação a verossimilhança penalizada, método atualmente mais utilizado. Nós aplicamos as duas metodologias a um banco de dados real, utilizando os pacotes estatísticos disponíveis no software R, e fizemos um estudo de simulação, visando comparar o viés e o erro quadrático médio das estimativas de cada abordagem. Pelos resultados encontrados, as duas metodologias apresentaram estimativas muito próximas, principalmente para os termos fixos. Do ponto de vista prático, a maior diferença encontrada foi o tempo de execução do algoritmo de estimação, muito maior na abordagem hierárquica. / Estimation procedures for frailty models have been widely discussed in the statistical literature due its widespread use in survival studies. Several estimation methods were developed: procedures based on the EM algorithm, Monte Carlo Markov chains, estimation processes based on parcial likelihood, penalized likelihood and quasi-likelihood etc. An alternative currently used is the hierarchical likelihood. The main objective of this work was to study the hierarchical likelihood advantages and disadvantages for inference in frailty models when compared with the penalized likelihood method, which is the most used one. We applied both approaches to a real data set, using R packages available. Besides, we performed a simulation study in order to compare the methods through out the bias and the mean square error of the estimators. Both methodologies presented very similar estimates, mainly for the fixed effects. In practice, the great difference was the computational cost, much higher in the hierarchical approach.

L'analyse du risque de survenue d'événements récurrents est une motivation majeure dans de nombreuses études de recherche clinique ou épidémiologique. En cancérologie, certaines stratégies thérapeutiques doivent être évaluées au cours d'essais randomisés où l'efficacité est mesurée à partir de la survenue d'événements successifs marquant la progression de la maladie. L'état de santé de patients infectés par le VIH évolue en plusieurs étapes qui ont pu être définies par la survenue d'événements cliniques successifs.Ce travail de thèse porte sur les modèles de régression du risque pour l'analyse de la survenue d'événements successifs. En pratique, la présence de corrélations entre les temps d'attente séparant les événements successifs est une hypothèse qui peut rarement être écartée d'emblée. L'objectif de la thèse porte sur le développement de modèles de régression permettant d'évaluer une telle corrélation. Dans ce cadre, la méthode le plus souvent utilisée suppose que la corrélation entre les délais successifs a pour origine une hétérogénéité aléatoire, non observée, entre sujets. Le modèle correspondant définit le risque instantané individuel en fonction d'un terme aléatoire, ou « fragilité », de distribution gamma et dont la variance quantifie l'hétérogénéité entre sujets et donc la corrélation entre délais d'un même sujet. Cependant, l'utilisation de ce modèle pour évaluer l'ampleur des corrélations présente l'inconvénient de conduire à une estimation biaisée de la variance de la fragilité.Une première approche a été définie pour deux événements successifs dans une échelle de temps « par intervalles », c'est-à-dire où le risque est exprimé en fonction du temps écoulé depuis l'événement précédent. L'approche mise au point a été obtenue à partir d'une approximation du risque de second événement conditionnellement au premier délai dans un modèle à fragilité pour plusieurs distributions de fragilité. Une seconde approche a été définie en échelle de temps « calendaire », où le risque est exprimé en fonction du temps écoulé depuis le début du suivi du sujet. L'approche retenue a été obtenue à partir d'une approximation de l'intensité conditionnelle au passé dans un modèle à fragilité. Dans les deux échelles de temps, l'approche mise au point consiste à introduire une covariable interne, calculée sur le passé du processus, qui correspond à la différence entre le nombre d'événements observés pour le sujet sur la période passée, et le nombre attendu d'événements pour ce sujet sur la même période compte tenu de ses covariables externes. Une revue de la littérature des études de simulations a montré que le cas d'une hétérogénéité dans la population face au risque d'événement était souvent envisagé par les auteurs. En revanche, dans beaucoup d'études de simulations, le cas d'un risque dépendant du temps, ou d'une dépendance entre événements, n'étaient pas considérés. Des études de simulations ont permis de montrer dans les deux échelles de temps considérées un gain de puissance du test mis au point par rapport au test d'homogénéité correspondant au modèle à fragilité gamma. Ce gain est plus marqué en échelle de temps par intervalles. Par ailleurs, dans cette échelle de temps, le modèle proposé permet une amélioration de l'estimation de la variance de la fragilité dans le cas d'une hétérogénéité faible ou modérée, plus particulièrement pour de petits échantillons.L'approche développée en échelle de temps par intervalles a été utilisée pour analyser les données d'une cohorte de patients infectés par le VIH, montrant une corrélation négative entre le délai entre infection et première manifestation mineure d'immunodéficience et le délai entre première manifestation mineure d'immunodéficience et stade SIDA déclaré. / The risk analysis for the occurrence of recurrent events is a major concern in many clinical research studies or epidemiological studies. In the field of oncology, therapeutic strategies are evaluated in randomised clinical trials in which efficacy is assessed through the occurrence of sequential events that define the progression of the disease. In HIV-infected patients, the infection evolves in several stages that have been defined by the occurrence of successive clinical events. The frame of this work is the regression models for the risk of multiple successive events. In practice, the hypothesis of existing correlations between the inter-event times cannot be a priori discarded. The aim of this work is to develop a regression model that would assess such correlations. In this setting, the most common method is to assume that correlations between inter-event times are induced by a random, unobserved heterogeneity across individuals. The corresponding model defines the individual hazard as a function of a random variable, or " frailty ", assumed to be gamma-distributed with a variance that quantifies the heterogeneity across individuals and incidentally the correlations between inter-event times. However, the use of this model when evaluating the correlations has the drawback that it tends to underestimate the variance of the frailty.A first approach was proposed for two sequential events in a "gap-timescale", in which the risk is defined as a function of the time elapsed since the previous event. The proposed method was derived from an approximation of the risk of second event given the first time-to-event in a frailty model for various frailty distributions. Another approach was defined in "calendar-time", in which the risk is expressed as a function of the time elapsed since the beginning of the subject's follow-up. The proposed method was derived from an approximation of the intensity conditional on the past in a frailty model. In both timescales, the method that was developed consists in including in the model an internal covariate, that is calculated on the history of the process, and that corresponds to the difference between the observed number of events and the expected number of events in the past period given the individual's other covariates.A review of the literature involving simulation studies showed that when defining the generation processes, most authors considered the case of heterogeneity in the population. However, in many simulation studies, only constant hazards are considered, and no event-dependence is introduced. Simulations studies showed that in both timescales, the test of the effect of the internal covariate in the proposed model proved more powerful that the usual test of homogeneity in the gamma frailty model. This gain of power is more noticeable in gap-time. Additionally, in this timescale, the proposed model provides a better estimation of the variance of the frailty when heterogeneity is low or moderate, more particularly in small samples.The method developed in gap-time was used to analyse data from a cohort of HIV-infected patients. It showed a negative correlation between the time from infection to first minor manifestation of immunodeficiency and the time from first minor manifestation of immunodeficiency to AIDS. The method developed in calendar-time was used to study the occurrence of repeated progressions and severe toxicities in a clinical trial for patients with advanced colorectal cancer. In this example, the method corroborated the results obtained with a gamma frailty model which showed a significant heterogeneity.

Avaliamos a incidência e o clareamento para o vírus do papiloma humano (HPV) dos tipos oncogênicos e não-oncogênicos em uma coorte aberta com 202 mulheres portadoras do vírus da imunodeficiência humana (HIV), e identificamos alguns fatores de risco e proteção associados a cada desfecho utilizando modelos de fragilidade Gama. No modelo de incidência, foram estudados os tempos até incidência de HPV oncogênicos e não-oncogênicos para cada mulher; no modelo de clareamento, foram estudados os correspondentes tempos até clareamento. Comparamos os erros-padrões estimados pela matriz de informação observada com os erros-padrões bootstrap para ambos os modelos e verificamos que a proposta de Verweij & Houwelingen (1994) para a matriz de variâncias e covariâncias dos parâmetros é a mais apropriada. Para a incidência de HPV oncogênicos, identificamos como fator de risco o uso de drogas em que a taxa de incidência para as pacientes que usam drogas é 1.88 (IC 90%, 1.01; 3.5) vezes aquela correspondente a mulheres que não usam e como fator de proteção a renda em que a taxa de incidência de pacientes com renda igual ou superior a 3 salários mínimos é 0.62 (IC 90%, 0.38; 1.00) vezes a taxa referente àquelas com renda menor que 3 salários mínimos. Para a incidência de HPV não-oncogênicos identificamos como fatores de risco a escolaridade e o total de gestações, em que, para a última, a taxa de incidência para as mulheres que tiveram mais do que uma gestação é 1.76 (IC 90%, 1.09; 2.86) vezes a taxa referente àquelas que tiveram uma ou nenhuma. Para o clareamento de HPV oncogênicos identificamos como fatores que indicam um clareamento mais rápido a renda, a idade e o tratamento antirretroviral (ARV), em que, para a última, supondo mulheres com fragilidades iguais, a taxa de clareamento para as pacientes que eram tratadas com o esquema inibidor de protease (IP) é 1.79 (IC 90%, 1.1; 2.9) vezes aquela correspondente a mulheres que não foram tratadas com nenhum tratamento ARV e como fator que indicam um clareamento mais lento o número de parceiros sexuais no último ano, em que, as pacientes com mais de um parceiro tiveram taxa de clareamento 0.39 (IC 90%, 0.16; 0.98) vezes a taxa de clareamento referente à uma mulher que teve um parceiro ou menos. Para o clareamento de HPV não-oncogênicos tivemos como fator que indica um clareamento mais lento o hábito tabagista em que, supondo fragilidades iguais, pacientes fumantes tem a taxa de clareamento 0.53 (IC 90%, 0.32; 0.87) vezes a taxa referente à uma mulher que não fuma. / We evaluated the incidence and clearance for oncogenic and non-oncogenic human papilloma virus (HPV) in an open cohort of 202 women infected with human immunodeficiency virus (HIV), and we identified some risk factors and protective factors for each outcome using Gamma frailty models. In the incidence model, we studied the incidence of stroke by oncogenic and non-oncogenic HPV for each woman; in the clearance model, the corresponding times to clearance were studied. We compared the standard errors estimated by the observed information matrix with bootstrap standard errors for both models and found that the variance and covariance matrix of the parameters proposed by Verweij & Houwelingen (1994) is more appropriate. For the incidence of oncogenic HPV, identified as a risk factor drug use and the incidence rate for patients who use drugs is 1.88 (90% CI, 1.01; 3.5) times the rate for those who do not use and as a protective factor income where the incidence rate is 0.62 (90% CI, 0.38; 1.00) times the rate for those earning less than 3 minimum wages. For the incidence of non-oncogenic HPV identified as risk factors schooling and total pregnancies, in which, for the latter, the incidence rate for women who had more than one pregnancy is 1.76 (90% CI, 1.09; 2.86) times the rate for those which have one or none. For clearance of oncogenic HPV identified as factors that indicate a faster clearance income, age and antiretroviral therapy (ART), in which, to the last, with women assuming equal frailties, the rate of clearance for patients who were treated with the protease inhibitor (IP) regimen is 1.79 (90% CI, 1.1; 2.9) times the rate for those who were not treated with any antiretroviral regimen and as a factor that indicates slower clearance the number of sexual partners in the last year, and for patients with more than one partner the clearance rate 0.39 (IC 90%, 0.16; 0.98) times the rate referring to a woman who had up to a partner. For the clearance of non-oncogenic HPV had a factor which indicates a slower clearance smoking habit, assuming equal frailties, smokers have the clearance rate 0.53 (90% CI, 0.32; 0.87) times the rate referring to a woman who does not smoke.

This thesis focuses on two fundamental topics, specifically in medical statistics: the modelling of correlated survival datasets and the variable selection of the significant covariates and random effects. In particular, two types of survival data are considered: the classical survival datasets, where subjects are likely to experience only one type of event and the competing risks datasets, where subjects are likely to experience one of several types of event. In Chapter 2, among other topics, we highlight the importance of adding frailty terms on the proposed models in order to account for the association between the survival time and characteristics of subjects/groups. The main novelty of this thesis is to simultaneously select fixed effects and frailty terms through the proposed statistical models for each survival dataset. Chapter 3 covers the analysis of the classical survival dataset through the proposed Cox Proportional Hazard (PH) model. Utilizing a Cox PH frailty model, may increase the dimension of variable components and estimation of the unknown coefficients becomes very challenging. The method proposed for the analysis of classical survival datasets involves simultaneous variable selection on both fixed effects and frailty terms through penalty functions. The benefit of penalty functions is that they identify the non-significant parameters and set them to have a zero effect in the model. Hence, the idea is to 'doubly-penalize' the partial likelihood of the Cox PH frailty model; one penalty for each term. Estimation and selection implemented through Newton-Raphson algorithms, whereas closed iterative forms for the estimation and selection of fixed effects and prediction of frailty terms were obtained. For the selection of frailty terms, penalties imposed on their variances since frailties are random effects. Based on the same idea, we further extend the simultaneous variable selection in the competing risks datasets in Chapter 4, using extended cause-specific frailty models. Two different scenarios are considered for frailty terms; in the first case we consider that frailty terms vary among different types of events (similar to the fixed effects) whereas in the second case we consider shared frailties over all the types of events. Moreover, our 'individual penalization' approach allows for one covariate to be significant for some types of events, in contrast to the frequently used 'group-penalization' where a covariate is entirely removed when it is not significant over all the events. For both proposed methods, simulation studies were conduced and showed that the proposed procedure followed for each analysis works well in simultaneously selecting and estimating significant fixed effects and frailty terms. The proposed methods are also applied to real datasets analysis; Kidney catheter infections, Diabetes Type 2 and Breast Cancer datasets. Association of the survival times and unmeasured characteristics of the subjects was studied as well as a variable selection for fixed effects and frailties implemented successfully.

Os métodos de estimação para modelos de fragilidade vêm sendo bastante discutidos na literatura estatística devido a sua grande utilização em estudos de Análise de Sobrevivência. Vários métodos de estimação de parâmetros dos modelos foram desenvolvidos: procedimentos de estimação baseados no algoritmo EM, cadeias de Markov de Monte Carlo, processos de estimação usando verossimilhança parcial, verossimilhança penalizada, quasi-verossimilhança, entro outros. Uma alternativa que vem sendo utilizada atualmente é a utilização da verossimilhança hierárquica. O objetivo principal deste trabalho foi estudar as vantagens e desvantagens da verossimilhança hierárquica para a inferência em modelos de fragilidade em relação a verossimilhança penalizada, método atualmente mais utilizado. Nós aplicamos as duas metodologias a um banco de dados real, utilizando os pacotes estatísticos disponíveis no software R, e fizemos um estudo de simulação, visando comparar o viés e o erro quadrático médio das estimativas de cada abordagem. Pelos resultados encontrados, as duas metodologias apresentaram estimativas muito próximas, principalmente para os termos fixos. Do ponto de vista prático, a maior diferença encontrada foi o tempo de execução do algoritmo de estimação, muito maior na abordagem hierárquica. / Estimation procedures for frailty models have been widely discussed in the statistical literature due its widespread use in survival studies. Several estimation methods were developed: procedures based on the EM algorithm, Monte Carlo Markov chains, estimation processes based on parcial likelihood, penalized likelihood and quasi-likelihood etc. An alternative currently used is the hierarchical likelihood. The main objective of this work was to study the hierarchical likelihood advantages and disadvantages for inference in frailty models when compared with the penalized likelihood method, which is the most used one. We applied both approaches to a real data set, using R packages available. Besides, we performed a simulation study in order to compare the methods through out the bias and the mean square error of the estimators. Both methodologies presented very similar estimates, mainly for the fixed effects. In practice, the great difference was the computational cost, much higher in the hierarchical approach.

Submitted by Luciana Sebin (lusebin@ufscar.br) on 2016-09-23T13:03:28Z No. of bitstreams: 1 DissRSC.pdf: 1649931 bytes, checksum: c3449a4367ea7de9e327fa7dc9110861 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-27T19:22:02Z (GMT) No. of bitstreams: 1 DissRSC.pdf: 1649931 bytes, checksum: c3449a4367ea7de9e327fa7dc9110861 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-27T19:22:09Z (GMT) No. of bitstreams: 1 DissRSC.pdf: 1649931 bytes, checksum: c3449a4367ea7de9e327fa7dc9110861 (MD5) / Made available in DSpace on 2016-09-27T19:22:16Z (GMT). No. of bitstreams: 1 DissRSC.pdf: 1649931 bytes, checksum: c3449a4367ea7de9e327fa7dc9110861 (MD5) Previous issue date: 2016-03-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Often intermediate events provide more detailed information about the disease process or recovery, for example, and allow greater accuracy in predicting the prognosis of patients. Such non-fatal events during the course of the disease can be seen as transitions from one state to another. The basic idea of a multistate models is that the person moves through a series of states in continuous time, it is possible to estimate the transition probabilities and intensities between them and the effect of covariates associated with each transition. Many studies include the grouping of survival times, for example, in multi-center studies, and is also of interest to study the evolution of patients over time, characterizing grouped multistate data. Because the data coming from different centers/groups, the failure times these individuals are grouped and the common risk factors not observed, it is interesting to consider the use of frailty so that we can capture the heterogeneity between the groups at risk for different types of transition, in addition to considering the dependence structure between transitions of individuals of the same group. In this work we present the methodology of multistate models, frailty models and then the integration of models with multi-state fragility models, dealing with the process of parametric and semi-parametric estimation. The conducted simulation study showed the importance of considering frailty in grouped multistate models, because without considering them, the estimates become biased. Furthermore, we find the frequentist properties of estimators of multistate model with nested frailty. Finally, as an application example to a set of real data, we use the process of bone marrow transplantation recovery of patients in four hospitals.We did a comparison of models through quality teasures setting AIC and BIC, coming to the conclusion that the model considers two random effects (one for the hospital and another for interaction transition-hospital) fits the data better. In addition to considering the heterogeneity between hospitals, such a model also considers the heterogeneity between hospitals in each transition. Thus, the values of the frailty estimated interaction transition-hospital reveal how fragile patients from each hospital are to experience certain type of event/transition. / Frequentemente eventos intermediários fornecem informações mais detalhadas sobre o processo da doença ou recuperação, por exemplo, e permitem uma maior precisão na previsão do prognóstico de pacientes. Tais eventos não fatais durante o curso da doença podem ser vistos como transições de um estado para outro. A ideia básica dos modelos multiestado é que o indivíduo se move através de uma série de estados em tempo contínuo, sendo possível estimar as probabilidades e intensidades de transição entre eles e o efeito das coivaráveis associadas a cada transição. Muitos estudos incluem o agrupamento dos tempos de sobrevivência como, por exemplo, em estudos multicêntricos, e também é de interesse estudar a evolução dos pacientes ao longo do tempo, caracterizando assim dados multiestado agrupados. Devido ao fato de os dados virem de diferentes centros/grupos, os tempos de falha desses indivíduos estarem agrupados e a fatores de risco comuns não observados, é interessante considerar o uso de fragilidades para que possamos capturar a heterogeneidade entre os grupos no risco para os diferentes tipos de transição, além de considerar a estrutura de dependência entre transições dos indivíduos de um mesmo grupo. Neste trabalho apresentamos a metodologia dos modelos multiestado, dos modelos de fragilidade e, em seguida, a integração dos modelos multiestado com modelos de fragilidade, tratando do seu processo de estimação paramétrica e semiparamétrica. O estudo de simulação realizado mostrou a importância de considerarmos fragilidades em modelos multiestado agrupados, pois sem consider´a-las, as estimativas tornam-se viesadas. Al´em disso, verificamos as propriedades frequentistas dos estimadores do modelo multiestado com fragilidades aninhadas. Por fim, como um exemplo de aplicação a um conjunto de dados reais, utilizamos o processo de recuperação de transplante de medula óssea de pacientes tratados em quatro hospitais. Fizemos uma comparação de modelos por meio das medidas de qualidade do ajuste AIC e BIC, chegando `a conclusão de que o modelo que considera dois efeitos aleatórios (uma para o hospital e outro para a interação transição-hospital) ajusta-se melhor aos dados. Além de considerar a heterogeneidade entre os hospitais, tal modelo também considera a heterogeneidade entre os hospitais em cada transição. Sendo assim, os valores das fragilidades estimadas da interação transição-hospital revelam o quão frágeis os pacientes de cada hospital são para experimentarem determinado tipo de evento/transição.

Avaliamos a incidência e o clareamento para o vírus do papiloma humano (HPV) dos tipos oncogênicos e não-oncogênicos em uma coorte aberta com 202 mulheres portadoras do vírus da imunodeficiência humana (HIV), e identificamos alguns fatores de risco e proteção associados a cada desfecho utilizando modelos de fragilidade Gama. No modelo de incidência, foram estudados os tempos até incidência de HPV oncogênicos e não-oncogênicos para cada mulher; no modelo de clareamento, foram estudados os correspondentes tempos até clareamento. Comparamos os erros-padrões estimados pela matriz de informação observada com os erros-padrões bootstrap para ambos os modelos e verificamos que a proposta de Verweij & Houwelingen (1994) para a matriz de variâncias e covariâncias dos parâmetros é a mais apropriada. Para a incidência de HPV oncogênicos, identificamos como fator de risco o uso de drogas em que a taxa de incidência para as pacientes que usam drogas é 1.88 (IC 90%, 1.01; 3.5) vezes aquela correspondente a mulheres que não usam e como fator de proteção a renda em que a taxa de incidência de pacientes com renda igual ou superior a 3 salários mínimos é 0.62 (IC 90%, 0.38; 1.00) vezes a taxa referente àquelas com renda menor que 3 salários mínimos. Para a incidência de HPV não-oncogênicos identificamos como fatores de risco a escolaridade e o total de gestações, em que, para a última, a taxa de incidência para as mulheres que tiveram mais do que uma gestação é 1.76 (IC 90%, 1.09; 2.86) vezes a taxa referente àquelas que tiveram uma ou nenhuma. Para o clareamento de HPV oncogênicos identificamos como fatores que indicam um clareamento mais rápido a renda, a idade e o tratamento antirretroviral (ARV), em que, para a última, supondo mulheres com fragilidades iguais, a taxa de clareamento para as pacientes que eram tratadas com o esquema inibidor de protease (IP) é 1.79 (IC 90%, 1.1; 2.9) vezes aquela correspondente a mulheres que não foram tratadas com nenhum tratamento ARV e como fator que indicam um clareamento mais lento o número de parceiros sexuais no último ano, em que, as pacientes com mais de um parceiro tiveram taxa de clareamento 0.39 (IC 90%, 0.16; 0.98) vezes a taxa de clareamento referente à uma mulher que teve um parceiro ou menos. Para o clareamento de HPV não-oncogênicos tivemos como fator que indica um clareamento mais lento o hábito tabagista em que, supondo fragilidades iguais, pacientes fumantes tem a taxa de clareamento 0.53 (IC 90%, 0.32; 0.87) vezes a taxa referente à uma mulher que não fuma. / We evaluated the incidence and clearance for oncogenic and non-oncogenic human papilloma virus (HPV) in an open cohort of 202 women infected with human immunodeficiency virus (HIV), and we identified some risk factors and protective factors for each outcome using Gamma frailty models. In the incidence model, we studied the incidence of stroke by oncogenic and non-oncogenic HPV for each woman; in the clearance model, the corresponding times to clearance were studied. We compared the standard errors estimated by the observed information matrix with bootstrap standard errors for both models and found that the variance and covariance matrix of the parameters proposed by Verweij & Houwelingen (1994) is more appropriate. For the incidence of oncogenic HPV, identified as a risk factor drug use and the incidence rate for patients who use drugs is 1.88 (90% CI, 1.01; 3.5) times the rate for those who do not use and as a protective factor income where the incidence rate is 0.62 (90% CI, 0.38; 1.00) times the rate for those earning less than 3 minimum wages. For the incidence of non-oncogenic HPV identified as risk factors schooling and total pregnancies, in which, for the latter, the incidence rate for women who had more than one pregnancy is 1.76 (90% CI, 1.09; 2.86) times the rate for those which have one or none. For clearance of oncogenic HPV identified as factors that indicate a faster clearance income, age and antiretroviral therapy (ART), in which, to the last, with women assuming equal frailties, the rate of clearance for patients who were treated with the protease inhibitor (IP) regimen is 1.79 (90% CI, 1.1; 2.9) times the rate for those who were not treated with any antiretroviral regimen and as a factor that indicates slower clearance the number of sexual partners in the last year, and for patients with more than one partner the clearance rate 0.39 (IC 90%, 0.16; 0.98) times the rate referring to a woman who had up to a partner. For the clearance of non-oncogenic HPV had a factor which indicates a slower clearance smoking habit, assuming equal frailties, smokers have the clearance rate 0.53 (90% CI, 0.32; 0.87) times the rate referring to a woman who does not smoke.

Ce travail a eu pour objectif de proposer des modèles conjoints d'intensités de processus d'événements récurrents et d'un événement terminal dépendant. Nous montrons que l'analyse séparée de ces événements conduit à des biais d'estimation importants. C'est pourquoi il est nécessaire de prendre en compte les dépendances entre les différents événements d'intérêt. Nous avons choisi de modéliser ces dépendances en introduisant des effets aléatoires (ou fragilités) et de travailler sur la structure de dépendance. Ces effets aléatoires prennent en compte les dépendances entre événements, les dépendances inter-récurrences et l'hétérogénéité non-observée. Nous avons, en premier lieu, développé un modèle conjoint à fragilités pour un type d'événement récurrent et un événement terminal dépendant en introduisant deux effets aléatoires indépendants pour prendre en compte et distinguer la dépendance inter-récurrences et celle entre les risques d'événements récurrents et terminal. Ce modèle a été ajusté pour des données de patients atteints de lymphome folliculaire où les événements d'intérêt sont les rechutes et le décès. Le second modèle développé permet de modéliser conjointement deux types d'événements récurrents et un événement terminal dépendant en introduisant deux effets aléatoires corrélés et deux paramètres de flexibilités. Ce modèle s'avère adapté pour l'analyse des risques de récidives locorégionales, de récidives métastatiques et de décès chez des patientes atteintes de cancer du sein. Nous confirmons ainsi que le décès est lié aux récidives métastatiques mais pas aux récidives locorégionales tandis que les deux types de récidives sont liés. Cependant ces approches font l'hypothèse de proportionnalité des intensités conditionnellement aux fragilités, que nous allons tenter d'assouplir. Dans un troisième travail, nous proposons de modéliser un effet potentiellement dépendant du temps des covariables en utilisant des fonctions B-Splines. / This work aimed to propose joint models for recurrent events and a dependent terminal event. We show how separate analyses of these events could lead to important biases. That is why it seems necessary to take into account the dependencies between events of interest. We choose to model these dependencies through random effects (or frailties) and work on the dependence structure. These random effects account for dependencies between events, inter-dependence recurrences and unobserved heterogeneity. We first have developed a joint frailty model for one type of recurrent events and a dependent terminal event with two independent random effects to take into account and distinguish the inter-recurrence dependence and between recurrent events and terminal event. This model was applied to follicular lymphoma patient’s data where events of interest are relapses and death. The second proposed model is used to model jointly two types of recurrent events and a dependent terminal event by introducing two correlated random effects and two flexible parameters. This model is suitable for analysis of locoregional recurrences, metastatic recurrences and death for breast cancer patients. It confirms that the death is related to metastatic recurrence but not locoregional recurrence while both types of recurrences are related. However, these approaches do the assumption of proportional intensities conditionally on frailties, which we want to relax. In a third study, we propose to model potentially time-dependent regression coefficient using B-splines functions.

Thesis (Ph.D. (Statistics)) -- University of Limpopo, 2022 / This thesis is based on application of frailty and copula models to under five child mortality data set in South Africa. The main purpose of the study was to apply sample splitting techniques in a survival analysis setting and compare clustered survival models considering left truncation to the under five child mortality data set in South Africa. The major contributions of this thesis is in the application of the shared frailty model and a class of Archimedean copulas in particular, Clayton-Oakes copula with completely monotone generator, and introduction of sample splitting techniques in a survival analysis setting. The findings based on shared frailty model show that clustering effect was sig nificant for modelling the determinants of time to death of under five children, and revealed the importance of accounting for clustering effect. The conclusion based on Clayton-Oakes model showed association between survival times of children from the same mother. It was found that the parameter estimates for the shared frailty and the Clayton-Oakes models were quite different and that the two models cannot be comparable. Gender, province, year, birth order and whether a child is part of twin or not were found to be significant factors affect ing under five child mortality in South Africa. / NRF-TDG Flemish Interuniversity Council Institutional corporation (VLIR-IUC) VLIR-IUC Programme of the University of Limpopo