Likelihood Inference for Flexible Cure Rate Models with Gamma Lifetimes

A New Approach to Modeling the Cure Rate in the Presence of Interval Censored Data

We consider interval censored data with a cured subgroup that arises from longitudinal followup studies with a heterogeneous population where a certain proportion of subjects is not susceptible to the event of interest. We propose a two component mixture cure model, where the first component describing the probability of cure is modeled by a support vector machine-based approach and the second component describing the survival distribution of the uncured group is modeled by a proportional hazard structure. Our proposed model provides flexibility in capturing complex effects of covariates on the probability of cure unlike the traditional models that rely on modeling the cure probability using a generalized linear model with a known link function. For the estimation of model parameters, we develop an expectation maximization-based estimation algorithm. We conduct simulation studies and show that our proposed model performs better in capturing complex effects of covariates on the cure probability when compared to the traditional logit link-based two component mixture cure model. This results in more accurate (smaller bias) and precise (smaller mean square error) estimates of the cure probabilities, which in-turn improves the predictive accuracy of the latent cured status. We further show that our model's ability to capture complex covariate effects also improves the estimation results corresponding to the survival distribution of the uncured. Finally, we apply the proposed model and estimation procedure to an interval censored data on smoking cessation.

A generalized Gompertz promotion time cure model and its fitness to cancer data

The cure models based on standard distributions like exponential, Weibull, lognormal, Gompertz, gamma, are often used to analyze survival data from cancer clinical trials with long-term survivors. Sometimes, the data is simple, and the standard cure models fit them very well, however, most often the data are complex and the standard cure models don't fit them reasonably well. In this article, we offer a novel generalized Gompertz promotion time cure model and illustrate its fitness to gastric cancer data by three different methods. The generalized Gompertz distribution is as simple as the generalized Weibull distribution and is not computationally as intensive as the generalized F distribution. One detailed real data application is provided for illustration and comparison purposes.

On the estimation of interval censored destructive negative binomial cure model

In this article, a competitive risk survival model is considered in which the initial number of risks, assumed to follow a negative binomial distribution, is subject to a destructive mechanism. Assuming the population of interest to have a cure component, the form of the data as interval-censored, and considering both the number of initial risks and risks remaining active after destruction to be missing data, we develop two distinct estimation algorithms for this model. Making use of the conditional distributions of the missing data, we develop an expectation maximization (EM) algorithm, in which the conditional expected complete log-likelihood function is decomposed into simpler functions which are then maximized independently. A variation of the EM algorithm, called the stochastic EM (SEM) algorithm, is also developed with the goal of avoiding the calculation of complicated expectations and improving performance at parameter recovery. A Monte Carlo simulation study is carried out to evaluate the performance of both estimation methods through calculated bias, root mean square error, and coverage probability of the asymptotic confidence interval. We demonstrate the proposed SEM algorithm as the preferred estimation method through simulation and further illustrate the advantage of the SEM algorithm, as well as the use of a destructive model, with data from a children's mortality study.

A support vector machine-based cure rate model for interval censored data

The mixture cure rate model is the most commonly used cure rate model in the literature. In the context of mixture cure rate model, the standard approach to model the effect of covariates on the cured or uncured probability is to use a logistic function. This readily implies that the boundary classifying the cured and uncured subjects is linear. In this article, we propose a new mixture cure rate model based on interval censored data that uses the support vector machine to model the effect of covariates on the uncured or the cured probability (i.e. on the incidence part of the model). Our proposed model inherits the features of the support vector machine and provides flexibility to capture classification boundaries that are nonlinear and more complex. The latency part is modeled by a proportional hazards structure with an unspecified baseline hazard function. We develop an estimation procedure based on the expectation maximization algorithm to estimate the cured/uncured probability and the latency model parameters. Our simulation study results show that the proposed model performs better in capturing complex classification boundaries when compared to both logistic regression-based and spline regression-based mixture cure rate models. We also show that our model's ability to capture complex classification boundaries improve the estimation results corresponding to the latency part of the model. For illustrative purpose, we present our analysis by applying the proposed methodology to the NASA's Hypobaric Decompression Sickness Database.

On the integration of decision trees with mixture cure model

The mixture cure model is widely used to analyze survival data in the presence of a cured subgroup. Standard logistic regression-based approaches to model the incidence may lead to poor predictive accuracy of cure, specifically when the covariate effect is non-linear. Supervised machine learning techniques can be used as a better classifier than the logistic regression due to their ability to capture non-linear patterns in the data. However, the problem of interpret-ability hangs in the balance due to the trade-off between interpret-ability and predictive accuracy. We propose a new mixture cure model where the incidence part is modeled using a decision tree-based classifier and the proportional hazards structure for the latency part is preserved. The proposed model is very easy to interpret, closely mimics the human decision-making process, and provides flexibility to gauge both linear and non-linear covariate effects. For the estimation of model parameters, we develop an expectation maximization algorithm. A detailed simulation study shows that the proposed model outperforms the logistic regression-based and spline regression-based mixture cure models, both in terms of model fitting and evaluating predictive accuracy. An illustrative example with data from a leukemia study is presented to further support our conclusion.

Likelihood inference for COM-Poisson cure rate model with interval-censored data and Weibull lifetimes

In this paper, we consider a competing cause scenario and assume the number of competing causes to follow a Conway–Maxwell Poisson distribution which can capture both over and under dispersion that is usually encountered in discrete data. Assuming the population of interest having a component cure and the form of the data to be interval censored, as opposed to the usually considered right-censored data, the main contribution is in developing the steps of the expectation maximization algorithm for the determination of the maximum likelihood estimates of the model parameters of the flexible Conway–Maxwell Poisson cure rate model with Weibull lifetimes. An extensive Monte Carlo simulation study is carried out to demonstrate the performance of the proposed estimation method. Model discrimination within the Conway–Maxwell Poisson distribution is addressed using the likelihood ratio test and information-based criteria to select a suitable competing cause distribution that provides the best fit to the data. A simulation study is also carried out to demonstrate the loss in efficiency when selecting an improper competing cause distribution which justifies the use of a flexible family of distributions for the number of competing causes. Finally, the proposed methodology and the flexibility of the Conway–Maxwell Poisson distribution are illustrated with two known data sets from the literature: smoking cessation data and breast cosmesis data.

On the comparison of risk of death according to different stages of breast cancer via the long-term exponentiated Weibull hazard model

Long-term survivor models have been extensively used for modelling time-to-event data with a significant proportion of patients who do not experience poor outcome. In this paper, we propose a new long-term survivor hazard model, which accommodates comprehensive families of cure rate models as particular cases, including modified Weibull, exponentiated Weibull, Weibull, exponential and Rayleigh distribution, among others. The maximum likelihood estimation procedure is presented. A simulation study evaluates bias and mean square error of the considered estimation procedure as well as the coverage probabilities of the parameters asymptotic and bootstrap confidence intervals. A real Brazilian dataset on breast cancer illustrates the methodology. From the practical point of view, under our modelling, we provide a parameter that works as a metric to quantify and compare the risk between different stages of the disease. We emphasize that, we developed an online platform for oncologists to calculate the probability of survival of patients diagnosed with breast cancer according to the stage of the disease in real time.