A Model Based on Mixture of Weibull Distributions for Depending Competing Risks Data in the Presence of Long-Term Survivors, and Its Application to Malignant Melanoma Cancer Data.

The flexibility of finite mixture models makes them suitable candidates for analyzing survival data with complex, multimodal distributions. Such data is often available if the event of interest occurs due to multiple failure modes. Here, we explore the modeling of competing risks time-to-event data with covariates in the presence of long-term survivors in the population using finite mixture models. The mixture cure rate model is used to describe the uncertainty in the population, where the susceptible part of the population is modeled using a finite mixture of Weibull distributions with different shape and scale parameters. Moreover, if information on covariates is available, the cure rate may be modeled using a binary regression model on the covariates. Here, we use the logistic function to relate covariates to the cure rate. The distribution corresponding to the susceptible part may also depend on covariates. To explore such dependency, we model the scale parameter of the Weibull distribution using covariates. Then, we discuss the classical parametric inference for the constructed model based on random and non-informative right-censored competing risks time-to-event data. An efficient method based on the expectation-maximization algorithm is proposed to estimate model parameters, thereby avoiding the complexity of directly maximizing the likelihood function. Additionally, a method for constructing confidence intervals for all model parameters is addressed. A simulation study is performed in the presence of two competing causes to investigate the finite sample properties of the proposed estimation methodologies. Finally, the methods are illustrated by analyzing a real data set on malignant melanoma cancer. Predicting the conditional survival function of an alive patient is of natural interest to an experimenter or medical researcher. A method for estimating such a conditional survival probability is also discussed.