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Survival Analysis
Published in Jianrong Wu, Statistical Methods for Survival Trial Design, 2018
where . For positive values of , the hazard increases and . If , then . Thus, the Gompertz distribution with describes a population with a non-zero portion of cure (long-term survivors). The survival distribution S(t) is discontinuous at ; that discontinuity is removable by defining when . The Gompertz hazard function increases and decreases monotonically, according to whether or , respectively (Figure 2.6).
Zero-adjusted defective regression models for modeling lifetime data
Published in Journal of Applied Statistics, 2019
Vinicius F. Calsavara, Agatha S. Rodrigues, Ricardo Rocha, Francisco Louzada, Vera Tomazella, Ana C. R. L. A. Souza, Rafaela A. Costa, Rossana P. V. Francisco
The Gompertz distribution is commonly used to model survival data in many areas [17], especially when an approximately exponential hazard is suspected. Its probability density function is a>0, scale parameter b>0, a. In this scenario, when the parameter a is negative the Gompertz distribution naturally become an improper distribution, which it is useful to model survival data in presence of a surviving fraction. The cure fraction in the population is calculated as the limit of the survival function, when a<0, and it is given by
An improper form of Weibull distribution for competing risks analysis with Bayesian approach
Published in Journal of Applied Statistics, 2019
A. R. Baghestani, F. S. Hosseini-Baharanchi
Another extension is to provide the characteristics of impropriety for parametric distributions in order to model CIF. In 2009, Cheng modeled CIF throughout a three-parameter logistic function especially with a sigmoid-shaped hazard function [2]. Recently, Shayan et al. showed that four-parameter log-logistic model is an improper distribution in a subset of parameter space that could capture unimodal and bathtub hazards. Their results revealed that CIF estimation was more accurate than nonparametric estimation in some settings [15]. Moreover, in 2016, an extension to Gompertz model named 3-parameter Gompertz distribution has been developed by Haile et al. for competing risks modeling through considering a constraint on the parameter space to obtain impropriety characteristics. However, this model can generate monotone as well as unimodal hazard function, it could not model bathtub hazard rate. In addition, this model showed a weak convergence in a simulation study [6]. The original two-parameter Weibull distribution of the form of
Defective models induced by gamma frailty term for survival data with cured fraction
Published in Journal of Applied Statistics, 2019
Juliana Scudilio, Vinicius F. Calsavara, Ricardo Rocha, Francisco Louzada, Vera Tomazella, Agatha S. Rodrigues
The Gompertz distribution is often used to model survival data in various areas of knowledge [17]. The probability density function for the Gompertz distribution is given by