Bivariate lifetime modelling using copula functions in presence of mixture and non-mixture cure fraction models, censored data and covariates

Oliveira

Achcar

et al. 2022

AJS

In medical studies, it is common the presence of a fraction of patients who do not experience the event of interest. These patients are people who are not at risk of the event or are patients who were cured during the research. The proportion of immune or cured patients is known in the literature as cure rate. In general, the traditional existing lifetime statistical models are not appropriate to model data sets with cure rate, including bivariate lifetimes. In this paper, it is proposed a bivariate model based on a defective Gompertz distribution and also using a Clayton copula function to capture the possible dependence structure between the lifetimes. An extensive simulation study was carried out in order to evaluate the biases and the mean squared errors for the maximum likelihood estimators of the parameters associated to the proposed distribution. Some applications using medical data are presented to show the usefulness of the proposed model. Maximum likelihood and Bayesian methods were used to estimate the parameters of the model.

Section: Introductionmentioning

confidence: 99%

The Bivariate Defective Gompertz Distribution Based on Clayton Copula with Applications to Medical Data

Oliveira

Achcar

et al. 2022

AJS

“…Copula functions are used to build multivariate probability distributions with different dependence structures assuming specified marginal probability distributions for the random variables [56]. In the special case of only two lifetimes, bivariate models derived from copula functions have been used by many authors,see for example, [29], [35], [2], [43], [39], [52], [65] and [20].…”

Section: Introductionmentioning

confidence: 99%

A Bayesian Inference Approach for Bivariate Weibull Distributions Derived from Roy and Morgenstern Methods

Oliveira

Stat., optim. inf. comput.

Santos

et al. 2021

Bivariate lifetime distributions are of great importance in studies related to interdependent components, especially in engineering applications. In this paper, we introduce two bivariate lifetime assuming three- parameter Weibull marginal distributions. Some characteristics of the proposed distributions as the joint survival function, hazard rate function, cross factorial moment and stress-strength parameter are also derived. The inferences for the parameters or even functions of the parameters of the models are obtained under a Bayesian approach. An extensive numerical application using simulated data is carried out to evaluate the accuracy of the obtained estimators to illustrate the usefulness of the proposed methodology. To illustrate the usefulness of the proposed model, we also include an example with real data from which it is possible to see that the proposed model leads to good fits to the data.

“…Copula functions were introduced by Sklar [9] where, in the bivariate case, they are related to bivariate distribution functions whose marginal distributions are univariate uniform distributions in the interval [0, 1]. Bivariate models for the lifetime data analysis based on copula functions have been introduced by a number of authors, such as Kundu and Gupta [10], Achcar et al [11], Peres et al [12], Nair et al [13] and Romeo et al [14].…”

Section: Introductionmentioning

confidence: 99%

Bivariate lifetime models in presence of cure fraction: a comparative study with many different copula functions

Achcar

Martinez

2020

Heliyon

Self Cite

In time-to-event studies it is common the presence of a fraction of individuals not expecting to experience the event of interest; these individuals who are immune to the event or cured for the disease during the study are known as long-term survivors. In addition, in many studies it is observed two lifetimes associated to the same individual, and in some cases there exists a dependence structure between them. In these situations, the usual existing lifetime distributions are not appropriate to model data sets with long-term survivors and dependent bivariate lifetimes. In this study, it is proposed a bivariate model based on a Weibull standard distribution with a dependence structure based on fifteen different copula functions. We assumed the Weibull distribution due to its wide use in survival data analysis and its greater flexibility and simplicity, but the presented methods can be adapted to other continuous survival distributions. Three examples, considering real data sets are introduced to illustrate the proposed methodology. A Bayesian approach is assumed to get the inferences for the parameters of the model where the posterior summaries of interest are obtained using Markov Chain Monte Carlo simulation methods and the Openbugs software. For the data analysis considering different real data sets it was assumed fifteen different copula models from which is was possible to find models with satisfactory fit for the bivariate lifetimes in presence of long-term survivors.