Modeling survival data based on a reparameterized weighted Lindley distribution

Abstract

In this work, we propose different statistical modeling for survival data based on a reparameterized weighted Lindley distribution. Initially, we present this distribution and study its mathematical properties, maximum likelihood estimation, and numerical simulations. Then, we propose a novel frailty model by using the reparameterized weighted Lindley distribution for modeling unobserved heterogeneity in univariate survival data. The frailty is introduced multiplicatively on the baseline hazard function. We obtain unconditional survival and hazard functions through the Laplace transform function of the frailty distribution. We assume hazard functions of the Weibull and Gompertz distributions as the baseline hazard functions and use the maximum likelihood method for estimating the resulting model parameters. 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. Also, we propose a frailty long-term model where the frailties are described by reparameterized weighted Lindley distribution. An advantage of the proposed model is to jointly model the heterogeneity among patients by their frailties and the presence of a cured fraction of them. We assume that the unknown number of competing causes that can influence the survival time follows a negative binomial distribution and that the time for the $k$-th competing cause to produce the event of interest follows the reparameterized weighted Lindley frailty model with Weibull baseline distribution. Some special cases of the model are presented. The cure fraction is modeled by using the logit link function. Again, we use the maximum likelihood method under random right-censoring to estimate the proposed model parameters. Further, we present a Monte Carlo simulation study to verify the maximum likelihood estimators' behavior assuming different sample sizes and censoring proportions. Finally, we extend the non-proportional generalized time-dependent logistic regression model by incorporating reparameterized weighted Lindley frailties. This proposed modeling has several important characteristics, such as non-proportional hazards, identifies the presence of long-term survivors without the addition of new parameters, captures the unobserved heterogeneity, allows the intersection of survival curves, and allows decreasing or unimodal hazard function. Again, parameter estimation is performed using the maximum likelihood method. Monte Carlo simulation studies are conducted to evaluate the asymptotic properties of the estimators as well as some properties of the model. The potentiality of all the proposed models is analyzed by employing real datasets and model comparisons are performed.