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)

Nesta tese são propostas duas distribuições discretas: COM-Poisson correlacionada (CPC) e COM-Poisson generalizada parcialmente correlacionada (CPGPC). Também foram propostos modelos de regressão para a distribuição Poisson generalizada parcialmente correlacionada (PGPC) (proposto por Luceño (1995)). Calculamos a função massa de probabilidade (fmp) para todas as distribuições com duas parametrizações. As distribuições foram construídas usando a mesma expansão feita por (Luceño, 1995) na construção da distribuição Poisson generalizada parcialmente correlacionada. A distribuição CPC(l;f;r) é a mesma expansão da distribuição COM-Poisson zero inflacionada ZICMP(m;f;r). Para a distribuição CPGPC(l;f;r;L;K) foi determinada a função característica, função geradora de probabilidade, momentos e a estimação pelo método de máxima verossimilhança para as duas parametrizações. Fizemos a fmp, quantil e gerador de números aleatórios das distribuições citadas no programa R. / In this thesis two discrete distributions are proposed: Correlated COM-Poisson (CPC) and Generalized partially correlated COM-Poisson (CPGPC). We have also proposed regression models for the Generalized partially correlated Poisson distribution (PGPC) (proposed by Luceño (1995)). We calculated the probability mass function for all distributions with two parametrizations. The distributions were constructed using the same expansion made by Luceño (1995) in the construction of the correlated generalized Poisson distribution. The CPC(l;f;r) Correlated COM-Poisson distribution is the same expansion of the zero-inflated COM-Poisson distribution ZICMP(m;f;r). For the CPGPC(l;f;r;L;K) Generalized partially correlated COM-Poisson distribution, the characteristic function, probability-generating function, moments, and the maximum likelihood estimation for the two parametrizations were determined. We performed the probability mass function, quantile and random number generator of the distributions quoted in program R.

Made available in DSpace on 2016-06-02T20:06:06Z (GMT). No. of bitstreams: 1 4336.pdf: 1594022 bytes, checksum: ff2370b4d516b9cdf6dd6da3be557c42 (MD5) Previous issue date: 2012-03-16 / Financiadora de Estudos e Projetos / Experiments involving chemical induction of carcinogens in animals are common in the biological area. Interest in these experiments is, in general, evaluating the chemopreventive effect of a substance in the destruction of damaged cells. In this type of study, two variables of interest are the number of induced tumors and their development times. We explored the use of statistical model proposed by Kokoska (1987) for the analysis of experimental data of chemoprevention of cancer in animals. We flexibility the Kokoska s model, subsequently used by Freedman (1993), whereas for the variable number of tumors induced Conway-Maxwell Poisson (COM-Poisson) distribution. This distribution has demonstrated efficiency due to its great flexibility, when compared to other discrete distributions to accommodate problems related to sub-dispersion and super-dispersion often found in count data. The purpose of this paper is to adapt the theory of long-term destructive model (Rodrigues et al., 2011) for experiments chemoprevention of cancer in animals, in order to evaluate the effectiveness of cancer treatments. Unlike the proposed Rodrigues et al. (2011), we formulate a model for the variable number of detected malignant tumors per animal, assuming that the probability of detection is no longer constant, but dependent on the time step. This is an extremely important approach to cancer chemoprevention experiments, because it makes the analysis more realistic and accurate. We conducted a simulation study, in order to evaluate the efficiency of the proposed model and to verify the asymptotic properties of maximum likelihood estimators. We also analyze a real data set presented in the article by Freedman (1993), to demonstrate the efficiency of the COM-Poisson model compared to results obtained by him with the Poisson and Negative Binomial distributions. / Experimentos que envolvem a indução química de substâncias cancerígenas em animais são comuns na área biológica. O interesse destes experimentos é, em geral, avaliar o efeito de uma substância quimiopreventiva na destruição das células danificadas. Neste tipo de estudo, duas variáveis de interesse são o número de tumores induzidos e seus tempos de desenvolvimento. Exploramos o uso do modelo estatístico proposto por Kokoska (1987) para a análise de dados de experimentos de quimioprevenção de câncer em animais. Flexibilizamos o modelo de Kokoska (1987), posteriormente utilizado por Freedman (1993), considerando para a variável número de tumores induzidos a distribuição Conway-Maxwell Poisson (COM-Poisson). Esta distribuição tem demonstrado eficiência devido à sua grande flexibilidade, quando comparada a outras distribuições discretas, para acomodar problemas relacionados à subdispersão e sobredispersão encontrados frequentemente em dados de contagem. A proposta deste trabalho consiste em adaptar a teoria de modelo destrutivo de longa duração (Rodrigues et al., 2011) para experimentos de quimioprevenção do câncer em animais, com o propósito de avaliar a eficiência de tratamentos contra o câncer. Diferente da proposta de Rodrigues et al. (2011), formulamos um modelo para a variável número de tumores malignos detectados por animal, supondo que sua probabilidade de detecção não é mais constante, e sim dependente do instante de tempo. Esta é uma abordagem extremamente importante para experimentos quimiopreventivos de câncer, pois torna a análise mais realista e precisa. Realizamos um estudo de simulação com o propósito de avaliar a eficiência do modelo proposto e verificar as propriedades assintóticas dos estimadores de máxima verossimilhança. Analisamos também um conjunto de dados reais apresentado no artigo de Freedman (1993), visando demonstrar a eficiência do modelo COM-Poisson em relação aos resultados por ele obtidos com as distribuições Poisson e Binomial Negativa.

<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)

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)