Nonproportional hazards model with a frailty term for modeling subgroups with evidence of long-term survivors: Application to a lung cancer dataset.

With advancements in medical treatments for cancer, an increase in the life expectancy of patients undergoing new treatments is expected. Consequently, the field of statistics has evolved to present increasingly flexible models to explain such results better. In this paper, we present a lung cancer dataset with some covariates that exhibit nonproportional hazards (NPHs). Besides, the presence of long-term survivors is observed in subgroups. The proposed modeling is based on the generalized time-dependent logistic model with each subgroup's effect time and a random term effect (frailty). In practice, essential covariates are not observed for several reasons. In this context, frailty models are useful in modeling to quantify the amount of unobservable heterogeneity. The frailty distribution adopted was the weighted Lindley distribution, which has several interesting properties, such as the Laplace transform function on closed form, flexibility in the probability density function, among others. The proposed model allows for NPHs and long-term survivors in subgroups. Parameter estimation was performed using the maximum likelihood method, and Monte Carlo simulation studies were conducted to evaluate the estimators' performance. We exemplify this model's use by applying data of patients diagnosed with lung cancer in the state of São Paulo, Brazil.

Weighted Lindley frailty model: estimation and application to lung cancer data.

In this paper, we propose a novel frailty model for modeling unobserved heterogeneity present in survival data. Our model is derived by using a weighted Lindley distribution as the frailty distribution. The respective frailty distribution has a simple Laplace transform function which is useful to obtain marginal survival and hazard functions. We assume hazard functions of the Weibull and Gompertz distributions as the baseline hazard functions. A classical inference procedure based on the maximum likelihood method is presented. Extensive simulation studies are further performed to verify the behavior of maximum likelihood estimators under different proportions of right-censoring and to assess the performance of the likelihood ratio test to detect unobserved heterogeneity in different sample sizes. Finally, to demonstrate the applicability of the proposed model, we use it to analyze a medical dataset from a population-based study of incident cases of lung cancer diagnosed in the state of São Paulo, Brazil.

A new cure rate model with flexible competing causes with applications to melanoma and transplantation data.

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.

Improved Bayes estimators and prediction for the Wilson-Hilferty distribution.

In this paper, we revisit the Wilson-Hilferty distribution and presented its mathematical properties such as the r-th moments and reliability properties. The parameters estimators are discussed using objective reference Bayesian analysis for both complete and censored data where the resulting marginal posterior intervals have accurate frequentist coverage. A simulation study is presented to compare the performance of the proposed estimators with the frequentist approach where it is observed a clear advantage for the Bayesian method. Finally, the proposed methodology is illustrated on three real datasets.

Defective regression models for cure rate modeling with interval-censored data.

Regression models in survival analysis are most commonly applied for right-censored survival data. In some situations, the time to the event is not exactly observed, although it is known that the event occurred between two observed times. In practice, the moment of observation is frequently taken as the event occurrence time, and the interval-censored mechanism is ignored. We present a cure rate defective model for interval-censored event-time data. The defective distribution is characterized by a density function whose integration assumes a value less than one when the parameter domain differs from the usual domain. We use the Gompertz and inverse Gaussian defective distributions to model data containing cured elements and estimate parameters using the maximum likelihood estimation procedure. We evaluate the performance of the proposed models using Monte Carlo simulation studies. Practical relevance of the models is illustrated by applying datasets on ovarian cancer recurrence and oral lesions in children after liver transplantation, both of which were derived from studies performed at A.C. Camargo Cancer Center in São Paulo, Brazil.

Incorporation of frailties into a cure rate regression model and its diagnostics and application to melanoma data.

Cure rate models have been widely studied to analyze time-to-event data with a cured fraction of patients. Our proposal consists of incorporating frailty into a cure rate model, as an alternative to the existing models to describe this type of data, based on the Birnbaum-Saunders distribution. Such a distribution has theoretical arguments to model medical data and has shown empirically to be a good option for their analysis. An advantage of the proposed model is the possibility to jointly consider the heterogeneity among patients by their frailties and the presence of a cured fraction of them. In addition, the number of competing causes is described by the negative binomial distribution, which absorbs several particular cases. We consider likelihood-based methods to estimate the model parameters and to derive influence diagnostics for this model. We assess local influence on the parameter estimates under different perturbation schemes. Deriving diagnostic tools is needed in all statistical modeling, which is another novel aspect of our proposal. Numerical evaluation of the considered model is performed by Monte Carlo simulations and by an illustration with melanoma data, both of which show its good performance and its potential applications. Particularly, the illustration confirms the importance of statistical diagnostics in the modeling.

Birnbaum-Saunders frailty regression models: Diagnostics and application to medical data.

In survival models, some covariates affecting the lifetime could not be observed or measured. These covariates may correspond to environmental or genetic factors and be considered as a random effect related to a frailty of the individuals explaining their survival times. We propose a methodology based on a Birnbaum-Saunders frailty regression model, which can be applied to censored or uncensored data. Maximum-likelihood methods are used to estimate the model parameters and to derive local influence techniques. Diagnostic tools are important in regression to detect anomalies, as departures from error assumptions and presence of outliers and influential cases. Normal curvatures for local influence under different perturbations are computed and two types of residuals are introduced. Two examples with uncensored and censored real-world data illustrate the proposed methodology. Comparison with classical frailty models is carried out in these examples, which shows the superiority of the proposed model.

Two new defective distributions based on the Marshall-Olkin extension.

The presence of immune elements (generating a fraction of cure) in survival data is common. These cases are usually modeled by the standard mixture model. Here, we use an alternative approach based on defective distributions. Defective distributions are characterized by having density functions that integrate to values less than 1, when the domain of their parameters is different from the usual one. We use the Marshall-Olkin class of distributions to generalize two existing defective distributions, therefore generating two new defective distributions. We illustrate the distributions using three real data sets.

Survival models induced by zero-modified power series discrete frailty: Application with a melanoma data set.

Survival models with a frailty term are presented as an extension of Cox's proportional hazard model, in which a random effect is introduced in the hazard function in a multiplicative form with the aim of modeling the unobserved heterogeneity in the population. Candidates for the frailty distribution are assumed to be continuous and non-negative. However, this assumption may not be true in some situations. In this paper, we consider a discretely distributed frailty model that allows units with zero frailty, that is, it can be interpreted as having long-term survivors. We propose a new discrete frailty-induced survival model with a zero-modified power series family, which can be zero-inflated or zero-deflated depending on the parameter value. Parameter estimation was obtained using the maximum likelihood method, and the performance of the proposed models was performed by Monte Carlo simulation studies. Finally, the applicability of the proposed models was illustrated with a real melanoma cancer data set.

Objective bayesian analysis for multiple repairable systems.

This article focus on the analysis of the reliability of multiple identical systems that can have multiple failures over time. A repairable system is defined as a system that can be restored to operating state in the event of a failure. This work under minimal repair, it is assumed that the failure has a power law intensity and the Bayesian approach is used to estimate the unknown parameters. The Bayesian estimators are obtained using two objective priors know as Jeffreys and reference priors. We proved that obtained reference prior is also a matching prior for both parameters, i.e., the credibility intervals have accurate frequentist coverage, while the Jeffreys prior returns unbiased estimates for the parameters. To illustrate the applicability of our Bayesian estimators, a new data set related to the failures of Brazilian sugar cane harvesters is considered.

Improved objective Bayesian estimator for a PLP model hierarchically represented subject to competing risks under minimal repair regime.

In this paper, we propose a hierarchical statistical model for a single repairable system subject to several failure modes (competing risks). The paper describes how complex engineered systems may be modelled hierarchically by use of Bayesian methods. It is also assumed that repairs are minimal and each failure mode has a power-law intensity. Our proposed model generalizes another one already presented in the literature and continues the study initiated by us in another published paper. Some properties of the new model are discussed. We conduct statistical inference under an objective Bayesian framework. A simulation study is carried out to investigate the efficiency of the proposed methods. Finally, our methodology is illustrated by two practical situations currently addressed in a project under development arising from a partnership between Petrobras and six research institutes.

Long-term frailty modeling using a non-proportional hazards model: Application with a melanoma dataset.

The semiparametric Cox regression model is often fitted in the modeling of survival data. One of its main advantages is the ease of interpretation, as long as the hazards rates for two individuals do not vary over time. In practice the proportionality assumption of the hazards may not be true in some situations. In addition, in several survival data is common a proportion of units not susceptible to the event of interest, even if, accompanied by a sufficiently large time, which is so-called immune, "cured," or not susceptible to the event of interest. In this context, several cure rate models are available to deal with in the long term. Here, we consider the generalized time-dependent logistic (GTDL) model with a power variance function (PVF) frailty term introduced in the hazard function to control for unobservable heterogeneity in patient populations. It allows for non-proportional hazards, as well as survival data with long-term survivors. Parameter estimation was performed using the maximum likelihood method, and Monte Carlo simulation was conducted to evaluate the performance of the models. Its practice relevance is illustrated in a real medical dataset from a population-based study of incident cases of melanoma diagnosed in the state of São Paulo, Brazil.

New defective models based on the Kumaraswamy family of distributions with application to cancer data sets.

An alternative to the standard mixture model is proposed for modeling data containing cured elements or a cure fraction. This approach is based on the use of defective distributions to estimate the cure fraction as a function of the estimated parameters. In the literature there are just two of these distributions: the Gompertz and the inverse Gaussian. Here, we propose two new defective distributions: the Kumaraswamy Gompertz and Kumaraswamy inverse Gaussian distributions, extensions of the Gompertz and inverse Gaussian distributions under the Kumaraswamy family of distributions. We show in fact that if a distribution is defective, then its extension under the Kumaraswamy family is defective too. We consider maximum likelihood estimation of the extensions and check its finite sample performance. We use three real cancer data sets to show that the new defective distributions offer better fits than baseline distributions.