Submit Manuscript | http://medcraveonline.com Introduction In many health studies, powerful tools for the statistical analysis are the survival analysis techniques that could be useful, for example, to identify risk factors or treatments that infuences the survival or cure probability of a certain disease. In general, survival analysis consists in a set of techniques and statistical models commonly used when the random variable of interest is the time until the occurrence of a specifc event, such as the time until the occurrence of a disease or the time until the patient’s death. A concept that differs survival analysis from others statistical analysis is the presence of censored data that occur when we have partial individual information about the time of occurrence of the variable of interest, however we do not know the exact time of occurrence of the event, that is, the real time of occurrence may exceed the observed time. The censored data can occur for a variety of reasons as the loss of monitoring of the patient over time and the non–occurrence of the event of interest until the end of the experiment. According to Colosimo and Giolo 1 there are two reasons that justify the use of censored data in statistical analysis: (I) although these observations are not complete, they provide information about the patient’s lifetime and (II) the omission of the observations censored can lead to the calculation of biased estimates. In survival analysis, usual parametric and non–parametric tools are widely used for analyzing data from time to event data. These tools are useful when some observations are censored and the event of interest was not seen in all patients during the follow–up period. The most used procedures include the mortality table, the Kaplan–Meier estimator for the survival function, the Cox proportional hazards model and parametric survival models. Parametric models are more fexible than Cox proportional hazards model, especially when there is no proportionality of risks between groups and are based mainly on two important functions, the survival function and the hazard function. These techniques are described in several textbooks as Kalbeisch and Prentice, 2 Klein and Moeschberger 3 and Kleinbaum and Klein. 4 Let a non–negative random variable T related to the failure time, then the survival function is defned as the probability that an observation will not fail until a certain time t, that is, the probability that an observation will survive time t. In probabilistic terms, () ( ) St PT t = ≥ On other hand, the hazard function () t λ represents the instantaneous failure rate at time t conditional on survival time t and is very useful to describe the distribution of patient’s lifetime. In probabilistic terms, 0 ( ) () lim . t Pt T t tT t t t λ ∆→ ≤ < +∆ | ≥ = ∆ Nonetheless, a common situation in many lifetime studies, particularly in cancer research, occurs when it is expected that a fraction of individuals will not experience the event of interest, this fraction of individuals often are immune or are cured. The presence of immune or cured individuals in a data set is usually suggested by a Kaplan–Meier plot of the survival function, which shows a long and stable plateau, with several censored date at the extreme right of the plot. 5–8 However, an effcient and commonly applied technique is to consider a mixture of two populations, one susceptible to the event of interest adopting a base probability distribution to model the survival time of susceptible patients, and one of the most common distributions as, for example, the Weibull distribution 9–11 for the non– susceptible population. According to Hjorth, 12 one or two parameters distributions have some important limitations such as the inability to model data that presents a bathtub risk function, for example. However, the most fexible distributions and with the largest number of parameters, may have inaccurate estimates, when there is a small sample size. In this way, this paper present a mixture cure rate model based on the Mirra distribution 13 to estimate survival and hazard curves. The choice of Mirra distribution is justifed due it number of parameters (two–parameters) and the bathtub shapes for the hazard function. Two Biom Biostat Int J. 2020;9(4):132‒137. 132 ©2020 Peres et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Estimation of survival and hazard curves of mixture Mirra cure rate model: Application to gastric and breast cancer data Volume 9 Issue 4 - 2020 Marcos Vinicius de Oliveira Peres, 1 Franchesco Sanches dos Santos, 1 Ricado Puziol de Oliveira 2 1 Department of Mathematics, State University of Paraná, Brazil 2 Department of the Environment, State University of Maringá, Brazil Correspondence: Marcos Vinicius de Oliveira Peres, Department of Mathematics, State University of Paraná, Brazil, Email Received: June 08, 2020 | Published: July 21 2020 Abstract In many applications related to time to event data, especially in the medical feld, it is common the presence of a fraction of individuals not expecting to experience the event of interest, these individuals immune to the event or cured for the disease during the study are known as long–term survivors. To estimate survival and hazard curves, in this situation, it is common the use of Weibull cure rate model due to its great fexibility and simplicity. In this paper, we present the estimation of survival and hazard curves using a extension of Mirra model using the classical cure rate approach and applying it to gastric and breast cancer data. The inferences of interest were obtained using a Bayesian approach and the results achieved from this study showed that the Mirra model has a good ft and could be an useful alternative for estimation and shape prediction of survival and hazard curves for long– term survivors, especially for cancer data. The results could be extended using regression approach in order to identify risk factor that affects the survival probability. Keywords: bayesian approach, breast cancer, cure rate models, gastric cancer, survival analysis Biometrics & Biostatistics International Journal Research Article Open Access