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Over the last decades, the challenges in survival models have been changing considerably and full probabilistic modeling is crucial in many medical applications. Motivated from a new biological interpretation of cancer metastasis, we introduce a general method for obtaining more flexible cure rate models. The proposal model extended the promotion time cure rate model. Furthermore, it includes several well-known models as special cases and defines many new special models. We derive several properties of the hazard function for the proposed model and establish mathematical relationships with the promotion time cure rate model. We consider a frequentist approach to perform inferences, and the maximum likelihood method is employed to estimate the model parameters. Simulation studies are conducted to evaluate its performance with a discussion of the obtained results. A real dataset from population-based study of incident cases of melanoma diagnosed in the state of São Paulo, Brazil, is discussed in detail.

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Mixture cure rate models have been developed to analyze failure time data where a proportion never fails. For such data, standard survival models are usually not appropriate because they do not account for the possibility of non-failure. In this context, mixture cure rate models assume that the studied population is a mixture of susceptible subjects who may experience the event of interest and non-susceptible subjects that will never experience it. More specifically, mixture cure rate models are a class of survival time models in which the probability of an eventual failure is less than one and both the probability of eventual failure and the timing of failure depend (separately) on certain individual characteristics. In this paper, we propose a Bayesian approach to estimate parametric mixture cure rate models with covariates. The probability of eventual failure is estimated using a binary regression model, and the timing of failure is determined using a Weibull distribution. Inference for these models is attained using Markov Chain Monte Carlo methods under the proposed Bayesian framework. Finally, we illustrate the method using data on the return-to-prison time for a sample of prison releases of men convicted of sexual crimes against women in England and Wales and we use mixture cure rate models to investigate the risk factors for long-term and short-term survival of recidivism.

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Models for situations where some individuals are long-term survivors, immune or non-susceptible to the event of interest, are extensively studied in biomedical research. Fitting a regression can be problematic in situations involving small sample sizes with high censoring rate, since the maximum likelihood estimates of some coefficients may be infinity. This phenomenon is called monotone likelihood, and it occurs in the presence of many categorical covariates, especially when one covariate level is not associated with any failure (in survival analysis) or when a categorical covariate perfectly predicts a binary response (in the logistic regression). A well known solution is an adaptation of the Firth method, originally created to reduce the estimation bias. The method provides a finite estimate by penalizing the likelihood function. Bias correction in the mixture cure model is a topic rarely discussed in the literature and it configures a central contribution of this work. In order to handle this point in such context, we propose to derive the adjusted score function based on the Firth method. An extensive Monte Carlo simulation study indicates good inference performance for the penalized maximum likelihood estimates. The analysis is illustrated through a real application involving patients with melanoma assisted at the Hospital das Clínicas/UFMG in Brazil. This is a relatively novel data set affected by the monotone likelihood issue and containing cured individuals.

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In this article, we introduce a long-term survival model in which the number of competing causes of the event of interest follows the zero-modified geometric (ZMG) distribution. Such distribution accommodates equidispersion, underdispersion, and overdispersion and captures deflation or inflation of zeros in the number of lesions or initiated cells after the treatment. The ZMG distribution is also an appropriate alternative for modeling clustered samples when the number of competing causes of the event of interest consists of two subpopulations, one containing only zeros (cure proportion), while in the other (noncure proportion) the number of competing causes of the event of interest follows a geometric distribution. The advantage of this assumption is that we can measure the cure proportion in the initiated cells. Furthermore, the proposed model can yield greater or lower cure proportion than that of the geometric distribution when modeling the number of competing causes. In this article, we present some statistical properties of the proposed model and use the maximum likelihood method to estimate the model parameters. We also conduct a Monte Carlo simulation study to evaluate the performance of the estimators. We present and discuss two applications using real-world medical data to assess the practical usefulness of the proposed model.

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Frailty models are generally used to model heterogeneity between the individuals. The distribution of the frailty variable is often assumed to be continuous. However, there are situations where a discretely-distributed frailty may be appropriate. In this paper, we propose extending the proportional hazards frailty models to allow a discrete distribution for the frailty variable. Having zero frailty can be interpreted as being immune or cured (long-term survivors). Thus, we develop a new survival model induced by discrete frailty with zero-inflated power series distribution, which can account for overdispersion. A numerical study is carried out under the scenario that the baseline distribution follows an exponential distribution, however this assumption can be easily relaxed and some other distributions can be considered. Moreover, this proposal allows for a more realistic description of the non-risk individuals, since individuals cured due to intrinsic factors (immune) are modeled by a deterministic fraction of zero-risk while those cured due to an intervention are modeled by a random fraction. Inference is developed by the maximum likelihood method for the estimation of the model parameters. A simulation study is performed in order to evaluate the performance of the proposed inferential method. Finally, the proposed model is applied to a data set on malignant cutaneous melanoma to illustrate the methodology.

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Cure fraction models are useful to model lifetime data with long-term survivors. We propose a flexible four-parameter cure rate survival model called the log-sinh Cauchy promotion time model for predicting breast carcinoma survival in women who underwent mastectomy. The model can estimate simultaneously the effects of the explanatory variables on the timing acceleration/deceleration of a given event, the surviving fraction, the heterogeneity, and the possible existence of bimodality in the data. In order to examine the performance of the proposed model, simulations are presented to verify the robust aspects of this flexible class against outlying and influential observations. Furthermore, we determine some diagnostic measures and the one-step approximations of the estimates in the case-deletion model. The new model was implemented in the generalized additive model for location, scale and shape package of the R software, which is presented throughout the paper by way of a brief tutorial on its use. The potential of the new regression model to accurately predict breast carcinoma mortality is illustrated using a real data set.

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A new flexible cure rate survival model is developed where the initial number of competing causes of the event of interest (say lesions or altered cells) follow a compound negative binomial (NB) distribution. This model provides a realistic interpretation of the biological mechanism of the event of interest as it models a destructive process of the initial competing risk factors and records only the damaged portion of the original number of risk factors. Besides, it also accounts for the underlying mechanisms that leads to cure through various latent activation schemes. Our method of estimation exploits maximum likelihood (ML) tools. The methodology is illustrated on a real data set on malignant melanoma, and the finite sample behavior of parameter estimates are explored through simulation studies.