Interval censoring arises frequently in life history data, as individuals are often only observed at a sequence of assessment times. This leads to a situation where we do not know when an event of interest occurs, only that it occurred somewhere between two assessment times. Here, the focus will be on methods of estimation for recurrent event data, current status data, and multistate data, subject to interval censoring. With recurrent event data, the focus is often on estimating the rate and mean functions. Nonparametric estimates are readily available, but are not smooth. Methods based on local likelihood and the assumption of a Poisson process are developed to obtain smooth estimates of the rate and mean functions without specifying a parametric form. Covariates and extra-Poisson variation are accommodated by using a pseudo-profile local likelihood. The methods are assessed by simulations and applied to a number of datasets, including data from a psoriatic arthritis clinic. Current status data is an extreme form of interval censoring that occurs when each individual is observed at only one assessment time. If current status data arise in clusters, this must be taken into account in order to obtain valid conclusions. Copulas offer a convenient framework for modelling the association separately from the margins. Estimating equations are developed for estimating marginal parameters as well as association parameters. Efficiency and robustness to the choice of copula are examined for first and second order estimating equations. The methods are applied to data from an orthopedic surgery study as well as data on joint damage in psoriatic arthritis. Multistate models can be used to characterize the progression of a disease as individuals move through different states. Considerable attention is given to a three-state model to characterize the development of a back condition known as spondylitis in psoriatic arthritis, along with the associated risk of mortality. Robust estimates of the state occupancy probabilities are derived based on a difference in distribution functions of the entry times. A five-state model which differentiates between left-side and right-side spondylitis is also considered, which allows us to characterize what effect spondylitis on one side of the body has on the development of spondylitis on the other side. Covariate effects are considered through multiplicative time homogeneous Markov models. The robust state occupancy probabilities are also applied to data on CMV infection in patients with HIV.

Interval censoring arises frequently in life history data, as individuals are often only observed at a sequence of assessment times. This leads to a situation where we do not know when an event of interest occurs, only that it occurred somewhere between two assessment times. Here, the focus will be on methods of estimation for recurrent event data, current status data, and multistate data, subject to interval censoring. With recurrent event data, the focus is often on estimating the rate and mean functions. Nonparametric estimates are readily available, but are not smooth. Methods based on local likelihood and the assumption of a Poisson process are developed to obtain smooth estimates of the rate and mean functions without specifying a parametric form. Covariates and extra-Poisson variation are accommodated by using a pseudo-profile local likelihood. The methods are assessed by simulations and applied to a number of datasets, including data from a psoriatic arthritis clinic. Current status data is an extreme form of interval censoring that occurs when each individual is observed at only one assessment time. If current status data arise in clusters, this must be taken into account in order to obtain valid conclusions. Copulas offer a convenient framework for modelling the association separately from the margins. Estimating equations are developed for estimating marginal parameters as well as association parameters. Efficiency and robustness to the choice of copula are examined for first and second order estimating equations. The methods are applied to data from an orthopedic surgery study as well as data on joint damage in psoriatic arthritis. Multistate models can be used to characterize the progression of a disease as individuals move through different states. Considerable attention is given to a three-state model to characterize the development of a back condition known as spondylitis in psoriatic arthritis, along with the associated risk of mortality. Robust estimates of the state occupancy probabilities are derived based on a difference in distribution functions of the entry times. A five-state model which differentiates between left-side and right-side spondylitis is also considered, which allows us to characterize what effect spondylitis on one side of the body has on the development of spondylitis on the other side. Covariate effects are considered through multiplicative time homogeneous Markov models. The robust state occupancy probabilities are also applied to data on CMV infection in patients with HIV.

In this dissertation, we are interested in studying processes which generate events repeatedly over the follow-up time of a given subject. Such processes are called recurrent event processes and the data they provide are referred to as recurrent event data. Examples include the cancer recurrences, recurrent infections or disease episodes, hospital readmissions, the filing of warranty claims, and insurance claims for policy holders. In particular, we focus on the multi-type recurrent event times which usually arise when two or more different kinds of events may occur repeatedly over a period of observation. Our main objectives are to describe features of each marginal process simultaneously and study the dependence among different types of events. We present applications to a real dataset collected from the Nutritional Prevention of Cancer Trial. The objective of the clinical trial was to evaluate the efficacy of Selenium in preventing the recurrence of several types of skin cancer among 1312 residents of the Eastern United States. Four chapters are involved in this dissertation. Chapter 1 introduces a brief background to the statistical techniques used to develop the proposed methodology. We cover some concepts and useful functions related to survival data analysis and present a short introduction to frailty distributions. The Monte Carlo expectation maximization (MCEM) algorithm and copula functions for the multivariate variables are also presented in this chapter. Chapter 2 develops a multi-type recurrent events model with multivariate Gaussian random effects (frailties) for the intensity functions. In this chapter, we present nonparametric baseline intensity functions and a multivariate Gaussian distribution for the multivariate correlated random effects. An MCEM algorithm with MCMC routines in the E-step is adopted for the partial likelihood to estimate model parameters. Equations for the variances of the estimates are derived and variances of estimates are computed by Louis' formula. Predictions of the individual random effects are obtained because in some applications the magnitude of the random effects is of interest for a better understanding and interpretation of the variability in the data. The performance of the proposed methodology is evaluated by simulation studies, and the developed model is applied to the skin cancer dataset. Chapter 3 presents copula-based semiparametric multivariate frailty models for multi-type recurrent event data with applications to the skin cancer data. In this chapter, we generalize the multivariate Gaussian assumption of the frailty terms and allow the frailty distributions to have more features than the symmetric, unimodal properties of the Gaussian density. More flexible approaches to modeling the correlated frailty, referred to as copula functions, are introduced. Copula functions provide tremendous flexibility especially in allowing taking the advantages of a variety of choices for the marginal distributions and correlation structures. Semiparametric intensity models for multi-type recurrent events based on a combination of the MCEM with MCMC sampling methods and copula functions are introduced. The combination of the MCEM approach and copula function is flexible and is a generally applicable approach for obtaining inferences of the unknown parameters for high dimension frailty models. Estimation procedures for fixed effects, nonparametric baseline intensity functions, copula parameters, and predictions for the subject-specific multivariate frailties and random effects are obtained. Louis' formula for variance estimates are derived and calculated. We investigate the impact of the specification of the frailty and random effect models on the inference of covariate effects, cumulative baseline intensity functions, prediction of random effects and frailties, and the estimation of the variance-covariance components. Performances of proposed models are evaluated by simulation studies. Applications are illustrated through the dataset collected from the clinical trial of patients with skin cancer. Conclusions and some remarks for future work are presented in Chapter 4. / Ph. D.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-21T19:45:33Z No. of bitstreams: 1 TeseMACM.pdf: 1006551 bytes, checksum: 49a419a1f18e05827f34631564fe431b (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-09-28T18:19:58Z (GMT) No. of bitstreams: 1 TeseMACM.pdf: 1006551 bytes, checksum: 49a419a1f18e05827f34631564fe431b (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-09-28T18:20:07Z (GMT) No. of bitstreams: 1 TeseMACM.pdf: 1006551 bytes, checksum: 49a419a1f18e05827f34631564fe431b (MD5) / Made available in DSpace on 2016-09-28T18:26:37Z (GMT). No. of bitstreams: 1 TeseMACM.pdf: 1006551 bytes, checksum: 49a419a1f18e05827f34631564fe431b (MD5) Previous issue date: 2015-09-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / In this thesis it is proposed alternative methodologies and extensions on models for recurrent event data. Speci cally, we propose a model in which the distribution of the gap time is easily derived from the marginal rate function providing more direct practical interpretation besides to consider the relation between successive gap times for each individual. Another model that extends the frailty models for recurrent event data to allow a Bernoulli, Geometric, Poisson, Discrete Weibull, Negative Binomial or other discrete distribution of the frailty variable has also been proposed. The parameter estimation procedure for both models was conducted considering maximum likelihood methods. Simulation studies were performed in order to examine some frequentist properties of the estimation method and evaluate the maximum likelihood estimates quality. Real data applications demonstrated the use of the proposed models. Overall, the proposed models were suitable for analyzing recurrent event data. / Neste trabalho propomos metodologias alternativas e extensões em modelos para dados de eventos recorrentes. Especificamente, propomos um modelo em que a distribuição condicional do tempo entre sucessivas ocorrências de um evento recorrente e derivada facilmente da função de taxa marginal, proporcionando interpretações praticas mais diretas, além de considerar a relação entre as sucessivas ocorrências para cada indivíduo. O outro modelo, que estende os modelos de fragilidade para dados de eventos recorrentes permitindo o uso de distribuições como Bernoulli, Geométrica, Poisson, Weibull Discreta, Binomial Negativa ou outra distribuição discreta para a variável de fragilidade, também foi proposto. O procedimento de estimação dos parâmetros para ambos modelos foi realizado considerando-se o método de máxima verossimilhança. Estudos de simulação foram realizados com o objetivo de analisar algumas propriedades frequentistas do método de estimação e avaliar a qualidade das estimativas de máxima verossimilhança. Aplicações a conjuntos de dados reais mostraram a aplicabilidade dos modelos propostos. De modo geral, os modelos propostos mostraram-se adequados para a modelagem de dados de eventos recorrentes.

Dans le contexte de la modélisation aléatoire des événements récurrents, un modèle statistique particulier est exploré. Ce modèle est fondé sur la théorie des processus de comptage et est construit dans le cadre d'analyse de défaillances dans les réseaux d'eau. Dans ce domaine nous disposons de données sur de nombreux systèmes observés durant une certaine période de temps. Les systèmes étant posés à des instants différents, leur âge est utilisé en tant qu'échelle temporelle dans la modélisation. Le modèle tient compte de l'historique incomplet d'événements, du vieillissement des systèmes, de l'impact négatif des défaillances précédentes sur l'état des systèmes et des covariables. Le modèle est positionné parmi d'autres approches visant à l'analyse d'événements récurrents utilisées en biostatistique et en fiabilité. Les paramètres du modèle sont estimés par la méthode du Maximum de Vraisemblance (MV). Une covariable dépendante du temps est intégrée au modèle. Il est supposé qu'elle est extérieure au processus de défaillance et constante par morceaux. Des méthodes heuristiques sont proposées afin de tenir compte de cette covariable lorsqu'elle n'est pas observée. Des méthodes de simulation de données artificielles et des estimations en présence de la covariable temporelle sont proposées. Les propriétés de l'estimateur (la normalité, le biais, la variance) sont étudiées empiriquement par la méthode de Monte Carlo. L'accent est mis sur la présence de deux directions asymptotiques : asymptotique en nombre de systèmes n et asymptotique en durée d'observation T. Le comportement asymptotique de l'estimateur MV constaté empiriquement est conforme aux résultats théoriques classiques. Il s'agit de l'asymptotique en n. Le comportement T-asymptotique constaté empiriquement n'est pas classique. L'analyse montre également que les deux directions asymptotiques n et T peuvent être combinées en une unique direction : le nombre d'événements observés. Cela concerne les paramètres classiques du modèle (les coefficients associés aux covariables fixes et le paramètre caractérisant le vieillissement des systèmes). Ce n'est en revanche pas le cas pour le coefficient associé à la covariable temporelle et pour le paramètre caractérisant l'impact négatif des défaillances précédentes sur le comportement futur du système. La méthodologie développée est appliquée à l'analyse des défaillances des réseaux d'eau. L'influence des variations climatiques sur l'intensité de défaillance est prise en compte par une covariable dépendante du temps. Les résultats montrent globalement une amélioration des prédictions du comportement futur du processus lorsque la covariable temporelle est incluse dans le modèle. / In the context of stochastic modeling of recurrent events, a particular model is explored. This model is based on the counting process theory and is built to analyze failures in water distribution networks. In this domain the data on a large number of systems observed during a certain time period are available. Since the systems are installed at different dates, their age is used as a time scale in modeling. The model accounts for incomplete event history, aging of systems, negative impact of previous failures on the state of systems and for covariates.The model is situated among other approaches to analyze the recurrent events, used in biostatistics and in reliability. The model parameters are estimated by the Maximum Likelihood method (ML). A method to integrate a time-dependent covariate into the model is developed. The time-dependent covariate is assumed to be external to the failure process and to be piecewise constant. Heuristic methods are proposed to account for influence of this covariate when it is not observed. Methods for data simulation and for estimations in presence of the time-dependent covariate are proposed. A Monte Carlo study is carried out to empirically assess the ML estimator's properties (normality, bias, variance). The study is focused on the doubly-asymptotic nature of data: asymptotic in terms of the number of systems n and in terms of the duration of observation T. The asymptotic behavior of the ML estimator, assessed empirically agrees with the classical theoretical results for n-asymptotic behavior. The T-asymptotics appears to be less typical. It is also revealed that the two asymptotic directions, n and T can be combined into one unique direction: the number of observed events. This concerns the classical model parameters (the coefficients associated to fixed covariates, the parameter characterizing aging of systems). The presence of one unique asymptotic direction is not obvious for the time-dependent covariate coefficient and for a parameter characterizing the negative impact of previous events on the future behavior of a system.The developed methodology is applied to the analysis of failures of water networks. The influence of climatic variations on failure intensity is assessed by a time-dependent covariate. The results show a global improvement in predictions of future behavior of the process when the time-dependent covariate is included into the model.

可轉換公司債為複合式證券，除了具有債券性質外，並給予持有者於債券流通期間內行使轉換為股票之權利。以存活分析方法探討可轉債之研究尚屬少見，本論文乃以台灣上市櫃公司發行之5年期可轉債為研究資料，先整理出與公司經營有關的變數，再分別以Cox模式與再發事件之兩種邊際模型(marginal model)：A-G (Anderson-Gill) 模式、PWP-TT (Prentice-Williams-Petersen)模式為研究分析方法，探討可轉債之流通時間及大量交易時間的問題。本論文並將可轉債分類為債券類型、混合類型和權益類型，且由於不同類型可轉債之流通時間有所差異，因此以其為分層條件加入模式中進行分析。研究結果發現，資產總額、總負債率、TCRI評等及董監持股率等變數，具有顯著解釋可轉債流通時間的能力，可見公司財務負債狀況與穩定性與流通期間有關；而最高差價(當月最高股價與轉換價之相對差價)、長期負債率、總負債率及股價報酬率等變數，則可顯著解釋大量交易的發生時間，表示公司財務負債狀況與股價利潤差與大量交易發生之快慢有關，其中資產總額、最高差價、TCRI評等及股價報酬率之係數均顯著為正，長期負債率、總負債率及董監持股率之係數則顯著為負。由於平均表現之存活曲線與經驗存活曲線相當接近，以Kolmogorov-Smirnov檢定多無顯著差異，顯示這些模式有不錯的配適能力；至於對個別公司估計出之存活曲線，則或有與經驗存活曲線相差較多的現象，顯示所建立的模式可對個別公司提供可轉債即將結束流通或發生大量交易之預警。 / Convertible bonds are hybrid securities that possess the properties of bonds and the right to convert bonds into shocks. Few articles employed survival analysis to analyze the characteristics of convertible bonds. To investigate the effects of the issuer’s financial information to the duration of circulation and the timing of the massive trading about convertible bonds, Taiwan’s 5-year convertible bonds were collected, and three methods of survival analysis were employed：Cox model、A-G (Anderson-Gill) model and PWP-TT(Prentice-Williams-Petersen) model. We classified convertible bonds as debt-like, equity-like, and hedge-like, and then make the classification as a stratification condition later. In summary, total Assets, total debt ratio, TCRI, and the proportion of holding share in supervisors and directors are significant variables on circulation period of convertible bonds. Apparently, the extent of debt and financial stability of issuers have significant effects on circulation period; the difference between stock price and conversion price, long-term debt ratio, total debt ratio and stock return rate contribute significantly on the timing of massive trading of convertible bonds. While the extent of debt and the return of stock hasten the hazard of the timing of massive trading. Furthermore, there are no significant differences between the survival curves evaluated at the average performance levels and the corresponding empirical survival curves, according to the results of Kolmogorov-Smirnov test. However, the differences between individual survival probabilities and overall empirical survival probabilities might be large, which indicates that the models incorporate companies’ performance overtime may provide a warning message for the termination of circulation or the timing of massive trading for a particular convertible bond.

處理多重事件或再發事件之事件發生時間的資料時，常會以Cox模式為基礎而予以延伸，其中較適合再發事件的模式為：A-G模式、GT-UR 模式、PWP-CP 模式及PWP-GT模式。這些模式又可按照是否以發生次數為分層變數，而分為未分層模式(包含A-G模式、 GT-UR 模式)，及分層模式(包含PWP-CP模式、 PWP-GT 模式)。 本論文將以改良的Cox延伸模式，包括對變異數進行修正或加入事故傾向(或隨機效果)，探討公務人員升等的快慢與哪些變數有關。變異數修正方式利用穩健標準誤以解決事件之再發時間之間的相依問題；事故傾向模式則主要是以隨機效果代表無法觀察到的個體間之異質性，且同一個體的各次發生時間共享相同的異質性，並假定異質性服從某種特定分配。對於各種Cox的延伸模式，我們可比較採用穩健變異數與否對估計及推論結果的差異，以及事故傾向加入前後，估計及推論結果與模式配適上的差異。 由本論文對公務人員升等資料的分析可發現，採用變異數修正方法時，未分層的模式有較小的變異數估計值，所以顯著的變數較多，包括性別、官等、教育程度及年齡；分層模式中顯著變數則只有官等及教育程度。若假定事故傾向服從對數Gamma分配，並加入於上述四種模式中，則顯著的變數與未加入事故傾向時一致，且各模式之下均無法拒絕所有人的事故傾向同為0的假設。這種現象或許是因為我們無法取得教育程度與公務人員考試及格種類之歷史資料，也有可能是因為公務人員升等的體制健全，且法規制定嚴謹，運作也有正常的模式可循所致。

Disposant d’un jeu de données sur des infections nosocomiales, nous utilisons des techniques de vraisemblance empirique pour construire des bandes de confiance pour certaines quantité d’intérêt. Cette étude nous amène à renforcer les outils déjà existants afin qu’ils s’adaptent à notre cadre. Nous présentons dans une première partie les outils mathématiques issus de la littérature que nous utilisons dans ce travail de thèse. Nous les appliquons ensuite à diverses situations et donnons de nouvelles démonstrations lorsque cela est nécessaire. Nous conduisons aussi des simulations et obtenons des résultats concrets concernant notre jeu de données. Enfin, nous détaillons les algorithmes utilisés. / The starting point of this thesis is a data set of nosocomial infectionsin an intensive care unit of a French hostipal. We focused our attention onbuilding confidence bands for some parameters of interest using empiricallikelihood techniques. In order to do so, we had to adapt and develop somealready existing methods so that they fit our setup.We begin by giving a state of the art of the different theories we use.We then apply them to different setups and demonstrate new results whenneeded. Finally, we conduct simulations and describe our algorithms.

De nombreuses études cliniques ou interventions de lutte ont été faites ou sont en cours en Afrique pour la lutte contre le fléau du paludisme. En zone d'endémie, le paludisme est une maladie récurrente. La revue de littérature indique une application limitée des outils statistiques appropriés existants pour l'analyse des données récurrentes de paludisme. Nous avons mis en oeuvre des méthodes statistiques appropriées pour l'analyse des données répétées d'essais thérapeutiques de paludisme. Nous avons également étudié les mesures répétées d'hémoglobine lors du suivi de traitements antipaludiques en vue d'évaluer la tolérance ou sécurité des médicaments en regroupant les données de 13 essais cliniques.Pour l'analyse du nombre d'épisodes de paludisme, la régression binomiale négative a été mise en oeuvre. Pour modéliser la récurrence des épisodes de paludisme, quatre modèles ont été utilisés : i) Les équations d'estimation généralisées (GEE) utilisant la distribution de Poisson; et trois modèles qui sont une extension du modèle Cox: ii) le modèle de processus de comptage d'Andersen-Gill (AG-CP), iii) le modèle de processus de comptage de Prentice-Williams-Peterson (PWP-CP); et iv) le modèle de Fragilité partagée de distribution gamma. Pour l'analyse de sécurité, c'est-à-dire l'évaluation de l'impact de traitements antipaludiques sur le taux d'hémoglobine ou la survenue de l'anémie, les modèles linéaires et latents généralisés mixtes (« GLLAMM : generalized linear and latent mixed models ») ont été mis en oeuvre. Les perspectives sont l'élaboration de guides de bonnes pratiques de préparation et d'analyse ainsi que la création d'un entrepôt des données de paludisme. / Numerous clinical studies or control interventions were done or are ongoing in Africa for malaria control. For an efficient control of this disease, the strategies should be closer to the reality of the field and the data should be analyzed appropriately. In endemic areas, malaria is a recurrent disease. Repeated malaria episodes are common in African. However, the literature review indicates a limited application of appropriate statistical tools for the analysis of recurrent malaria data. We implemented appropriate statistical methods for the analysis of these data We have also studied the repeated measurements of hemoglobin during malaria treatments follow-up in order to assess the safety of the study drugs by pooling data from 13 clinical trials.For the analysis of the number of malaria episodes, the negative binomial regression has been implemented. To model the recurrence of malaria episodes, four models were used: i) the generalized estimating equations (GEE) using the Poisson distribution; and three models that are an extension of the Cox model: ii) Andersen-Gill counting process (AG-CP), iii) Prentice-Williams-Peterson counting process (PWP-CP); and (iv) the shared gamma frailty model. For the safety analysis, i.e. the assessment of the impact of malaria treatment on hemoglobin levels or the onset of anemia, the generalized linear and latent mixed models (GLLAMM) has been implemented. We have shown how to properly apply the existing statistical tools in the analysis of these data. The prospects of this work remain in the development of guides on good practices on the methodology of the preparation and analysis and storage network for malaria data.

再發事件資料常見於醫學、工業、財經、社會等等領域中，對再發資料分析研究時，我們往往無法確知再發事件發生的時間或是發生次數的分配。因此，本論文探討的是分析再發事件的無母數方法，包括Nelson提出的平均累積函數(mean cumulative function)估計量，及Wang、Chiang與Huang介紹的發生率(occurrence rate)之核函數(kernel function)估計量。 就平均累積函數估計量來說，藉由Nelson導出的變異數及自然(naive)變異數，可分別求得平均累積函數的區間估計。本文利用靴環法(bootstrap)計算出平均累積函數在不同時點的變異數，再與Nelson變異數及自然變異數比較，結果顯示Nelson變異數與靴環法算出的變異數較接近。因此，應依據Nelson變異數建構出事件發生累積次數之漸近信賴區間。 本論文亦介紹了兩個或多個母體的平均累積函數的比較方法，包含固定時點之比較與整條曲線之比較。在固定時點之下，比較方法分別為平均累積函數成對差異之漸近信賴區間及靴環信賴區間、變異數分析比較法，與排列檢定法；而整條曲線比較方法包含：類似 統計量、Lawless-Nadeau檢定。這些方法應用在本論文所採之實證資料時，所得到的檢定結論是一致的。 / Recurrent event data arise in many fields, such as medicine, industry, economics, social sciences and so on. When studying recurrent event data, we usually don’t know the exact joint or marginal distributions of the occurrence times or the number of events over time. So, in this article we talk about some nonparametric methods, such as the mean cumulative function (MCF) discussed by Nelson, and kernel estimation of the rate function introduced by Wang, Chiang and Huang. As to the estimator of MCF, we can compute the confidence interval by Nelson’s variance and naive variance. We use bootstrap method to compare the performance of Nelson variance of the estimated MCF and naive variance of the estimated MCF. The results show that Nelson variance is better than naive variance, so we should construct the confidence limits for the MCF by Nelson’s variance except when only grouped data are available. We also introduce methods for comparing MCFs, including pointwise comparison of MCFs and comparison of entire MCFs. Methods for pointwise comparing MCFs include approximate confidence limits for difference between two MCFs, analysis-of-variance comparison, permutation test, and bootstrap’s confidence limits for difference between two MCFs. Methods for comparing entire MCFs include a statistic like Hoetelling’s , and Lawless-Nadeau test. Finally, all approaches are employed to analyze a real data, and the conclusions concordance with each other.