The proportional hazard (PH) cure rate model and the marginal structural Cox model (MSCM) are two broad areas used in analysing survival models with longitudinal data. Cure rate models were introduced to deal with survival models in the presence of a cure fraction and marginal structural models were introduced to adjust for time- ependent confounders through time-dependent weighting in longitudinal studies. However, few studies have tried to combine these two areas in building cure rate models in the presence of time-dependent covariates and time-dependent confounders. This thesis proposes an extension of the maximum likelihood estimation procedure for the PH cure rate model by incorporating (i) case weights, (ii) time-dependent covariates, and (iii) time-dependent weights in the presence of time-dependent covariates and time-dependent confounders into the model. Further, this thesis compares the performance of the PH cure rate model with case weights to the standard unweighted PH cure rate model through simulation studies. Results of these studies suggest that adding case weights in the PH cure rate model improves the estimation of the latency parameter when the sample size is relatively small.

Cure rate models are widely used to model time-to-event data in the presence of long-term survivors. Cure rate models, since introduced by Boag (1949), have gained significance over time due to remarkable advancements in the drug industry resulting in cures for a number of diseases. In this thesis, cure rate models are considered under a competing risk scenario wherein the initial number of competing causes is described by a Conway-Maxwell (COM) Poisson distribution, under the assumption of proportional hazards (PH) lifetime for the susceptibles. This provides a natural extension of the work of Balakrishnan & Pal (2013) who had considered independently and identically distributed (i.i.d.) lifetimes in this setup. By linking covariates to the lifetime through PH assumption, we obtain a flexible cure rate model. First, the baseline hazard is assumed to be of the Weibull form. Parameter estimation is carried out using EM algorithm and the standard errors are estimated using Louis' method. The performance of estimation is assessed through a simulation study. A model discrimination study is performed using Likelihood-based and Information-based criteria since the COM-Poisson model includes geometric, Poisson and Bernoulli as special cases. The details are covered in Chapter 2. As a natural extension of this work, we next approximate the baseline hazard with a piecewise linear functions (PLA) and estimated it non-parametrically for the COM-Poisson cure rate model under PH setup. The corresponding simulation study and model discrimination results are presented in Chapter 3. Lastly, we consider a destructive cure rate model, introduced by Rodrigues et. al (2011), and study it under the PH assumption for the lifetimes of susceptibles. In this, the initial number of competing causes are modeled by a weighted Poisson distribution. We then focus mainly on three special cases, viz., destructive exponentially weighted Poisson, destructive length-biased Poisson and destructive negative binomial cure rate models, and all corresponding results are presented in Chapter 4. / Thesis / Doctor of Philosophy (PhD)

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.

Nesta tese serão estudadas diferentes generalizações de algumas distribuições bem conhecidas na literatura para os tempos de vida, tais como exponencial, Lindley, Rayleigh e exponencial segmentada, entre outras, e compará-las com outras extensões com suporte positivo. A finalidade dessas generalizações é flexibilizar a função de risco de modo que possam assumir formas mais flexíveis. Além disso, pretende-se estudar propriedades importantes dos modelos propostos, tais como os momentos, coeficientes de curtose e assimetria e função quantílica, entre outras. A estimação dos parâmetros é abordada através dos métodos de máxima verossimilhança, via algoritmo EM (quando for possível) ou também, do método dos momentos. O comportamento desses estimadores foi avaliado em estudos de simulação. Foram ajustados a conjuntos de dados reais, usando uma abordagem clássica, e compará-los com outras extensões na literatura. Finalmente, um dos modelos propostos é considerado no contexto de fração de cura. / The main focus of this thesis is the study of generalizations for some positive distributions widely known in the literature of lifetime analysis, such as the exponential, Lindley, Rayleigh and segmented exponential. Comparisons of the proposed extensions and alternative extensions in the literature such as the generalized exponential distribution, are reported. Moreover, of interest is also the study of some properties of the proposed distributions such as moments, kurtosis and asymmetry coefficients, quantile functions and the risk function. Parameter estimation is approached via maximum likelihood (using the EM-algorithm when available) and the method of moments as initial parameter estimators. Results of simulation studies are reported comparing the performance of these estimators with small and moderate sample sizes. Further comparisons are reported for real data applications, where the proposed models show satisfactory performance. Finally, one of the models proposed is considered no context of cure rate.

Os modelos de riscos proporcionais são muito utilizados na análise do tempo de sobrevivência, porém, assumem implicitamente que, observado um conjunto de variáveis explicativas, a população em estudo seja homogênea e que os indivíduos permaneçam sob risco durante todo o período de observação ou até que apresentem o evento de interesse. Tais suposições não são adequadas quando os indivíduos da população em estudo possuem diferentes pré-disposições ao surgimento de uma doença ou quando estão sujeitos à cura após o período de tratamento. Esta dissertação discute o modelo de sobrevivência com fração de cura quando o evento de interesse é caracterizado por fatores latentes competitivos, enquanto a heterogeneidade não observada entre os riscos dos pacientes é modelada através de um fator aleatório denominado termo de fragilidade. / Proportional hazards models are widely used in the analysis of survival time, however, it is implictly assumed that given a set of explanatory variables, the population under study is homogeneous and individuals remain at risk throughout the observation period time or until they have the event of interest. However, such assumptions are not reasonable when individuals from the population under study have dierent pre-dispositions to the emergence of a disease or are subject to the cure after treatment. This work discusses the cure fraction model when the event of interest is characterized by latent competitive factors, with patient risk modeled by a random factor called frailty.

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.

In practice, data are frequently incomplete in one way or another. It can be a significant challenge to make valid inferences about the parameters of interest in this situation. In this thesis, three problems involving such data are addressed. The first two problems involve interval-censored life history data with mismeasured covariates. Data of this type are incomplete in two ways. First, the exact event times are unknown due to censoring. Second, the true covariate is missing for most, if not all, individuals. This work focuses primarily on the impact of covariate measurement error in progressive multi-state models with data arising from panel (i.e., interval-censored) observation. These types of problems arise frequently in clinical settings (e.g. when disease progression is of interest and patient information is collected during irregularly spaced clinic visits). Two and three state models are considered in this thesis. This work is motivated by a research program on psoriatic arthritis (PsA) where the effects of error-prone covariates on rates of disease progression are of interest and patient information is collected at clinic visits (Gladman et al. 1995; Bond et al. 2006). Information regarding the error distributions were available based on results from a separate study conducted to evaluate the reliability of clinical measurements that are used in PsA treatment and follow-up (Gladman et al. 2004). The asymptotic bias of covariate effects obtained ignoring error in covariates is investigated and shown to be substantial in some settings. In a series of simulation studies, the performance of corrected likelihood methods and methods based on a simulation-extrapolation (SIMEX) algorithm (Cook \& Stefanski 1994) were investigated to address covariate measurement error. The methods implemented were shown to result in much smaller empirical biases and empirical coverage probabilities which were closer to the nominal levels. The third problem considered involves an extreme case of interval censoring known as current status data. Current status data arise when individuals are observed only at a single point in time and it is then determined whether they have experienced the event of interest. To complicate matters, in the problem considered here, an unknown proportion of the population will never experience the event of interest. Again, this type of data is incomplete in two ways. One assessment is made on each individual to determine whether or not an event has occurred. Therefore, the exact event times are unknown for those who will eventually experience the event. In addition, whether or not the individuals will ever experience the event is unknown for those who have not experienced the event by the assessment time. This problem was motivated by a series of orthopedic trials looking at the effect of blood thinners in hip and knee replacement surgeries. These blood thinners can cause a negative serological response in some patients. This response was the outcome of interest and the only available information regarding it was the seroconversion time under current status observation. In this thesis, latent class models with parametric, nonparametric and piecewise constant forms of the seroconversion time distribution are described. They account for the fact that only a proportion of the population will experience the event of interest. Estimators based on an EM algorithm were evaluated via simulation and the orthopedic surgery data were analyzed based on this methodology.

In practice, data are frequently incomplete in one way or another. It can be a significant challenge to make valid inferences about the parameters of interest in this situation. In this thesis, three problems involving such data are addressed. The first two problems involve interval-censored life history data with mismeasured covariates. Data of this type are incomplete in two ways. First, the exact event times are unknown due to censoring. Second, the true covariate is missing for most, if not all, individuals. This work focuses primarily on the impact of covariate measurement error in progressive multi-state models with data arising from panel (i.e., interval-censored) observation. These types of problems arise frequently in clinical settings (e.g. when disease progression is of interest and patient information is collected during irregularly spaced clinic visits). Two and three state models are considered in this thesis. This work is motivated by a research program on psoriatic arthritis (PsA) where the effects of error-prone covariates on rates of disease progression are of interest and patient information is collected at clinic visits (Gladman et al. 1995; Bond et al. 2006). Information regarding the error distributions were available based on results from a separate study conducted to evaluate the reliability of clinical measurements that are used in PsA treatment and follow-up (Gladman et al. 2004). The asymptotic bias of covariate effects obtained ignoring error in covariates is investigated and shown to be substantial in some settings. In a series of simulation studies, the performance of corrected likelihood methods and methods based on a simulation-extrapolation (SIMEX) algorithm (Cook \& Stefanski 1994) were investigated to address covariate measurement error. The methods implemented were shown to result in much smaller empirical biases and empirical coverage probabilities which were closer to the nominal levels. The third problem considered involves an extreme case of interval censoring known as current status data. Current status data arise when individuals are observed only at a single point in time and it is then determined whether they have experienced the event of interest. To complicate matters, in the problem considered here, an unknown proportion of the population will never experience the event of interest. Again, this type of data is incomplete in two ways. One assessment is made on each individual to determine whether or not an event has occurred. Therefore, the exact event times are unknown for those who will eventually experience the event. In addition, whether or not the individuals will ever experience the event is unknown for those who have not experienced the event by the assessment time. This problem was motivated by a series of orthopedic trials looking at the effect of blood thinners in hip and knee replacement surgeries. These blood thinners can cause a negative serological response in some patients. This response was the outcome of interest and the only available information regarding it was the seroconversion time under current status observation. In this thesis, latent class models with parametric, nonparametric and piecewise constant forms of the seroconversion time distribution are described. They account for the fact that only a proportion of the population will experience the event of interest. Estimators based on an EM algorithm were evaluated via simulation and the orthopedic surgery data were analyzed based on this methodology.

Os modelos de riscos proporcionais são muito utilizados na análise do tempo de sobrevivência, porém, assumem implicitamente que, observado um conjunto de variáveis explicativas, a população em estudo seja homogênea e que os indivíduos permaneçam sob risco durante todo o período de observação ou até que apresentem o evento de interesse. Tais suposições não são adequadas quando os indivíduos da população em estudo possuem diferentes pré-disposições ao surgimento de uma doença ou quando estão sujeitos à cura após o período de tratamento. Esta dissertação discute o modelo de sobrevivência com fração de cura quando o evento de interesse é caracterizado por fatores latentes competitivos, enquanto a heterogeneidade não observada entre os riscos dos pacientes é modelada através de um fator aleatório denominado termo de fragilidade. / Proportional hazards models are widely used in the analysis of survival time, however, it is implictly assumed that given a set of explanatory variables, the population under study is homogeneous and individuals remain at risk throughout the observation period time or until they have the event of interest. However, such assumptions are not reasonable when individuals from the population under study have dierent pre-dispositions to the emergence of a disease or are subject to the cure after treatment. This work discusses the cure fraction model when the event of interest is characterized by latent competitive factors, with patient risk modeled by a random factor called frailty.

Nesta tese serão estudadas diferentes generalizações de algumas distribuições bem conhecidas na literatura para os tempos de vida, tais como exponencial, Lindley, Rayleigh e exponencial segmentada, entre outras, e compará-las com outras extensões com suporte positivo. A finalidade dessas generalizações é flexibilizar a função de risco de modo que possam assumir formas mais flexíveis. Além disso, pretende-se estudar propriedades importantes dos modelos propostos, tais como os momentos, coeficientes de curtose e assimetria e função quantílica, entre outras. A estimação dos parâmetros é abordada através dos métodos de máxima verossimilhança, via algoritmo EM (quando for possível) ou também, do método dos momentos. O comportamento desses estimadores foi avaliado em estudos de simulação. Foram ajustados a conjuntos de dados reais, usando uma abordagem clássica, e compará-los com outras extensões na literatura. Finalmente, um dos modelos propostos é considerado no contexto de fração de cura. / The main focus of this thesis is the study of generalizations for some positive distributions widely known in the literature of lifetime analysis, such as the exponential, Lindley, Rayleigh and segmented exponential. Comparisons of the proposed extensions and alternative extensions in the literature such as the generalized exponential distribution, are reported. Moreover, of interest is also the study of some properties of the proposed distributions such as moments, kurtosis and asymmetry coefficients, quantile functions and the risk function. Parameter estimation is approached via maximum likelihood (using the EM-algorithm when available) and the method of moments as initial parameter estimators. Results of simulation studies are reported comparing the performance of these estimators with small and moderate sample sizes. Further comparisons are reported for real data applications, where the proposed models show satisfactory performance. Finally, one of the models proposed is considered no context of cure rate.