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Biometrics & Biostatistics International Journal

Research Article Volume 9 Issue 4

Estimation of survival and hazard curves of mixture Mirra cure rate model: Application to gastric and breast cancer data

Marcos Vinicius de Oliveira Peres,1 Franchesco Sanches dos Santos,1 Ricado Puziol de Oliveira2

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

Received: June 08, 2020 | Published: July 21, 2020

Citation: Peres MVDO, Santos FSD, Oliveira RPD. Estimation of survival and hazard curves of mixture Mirra cure rate model: Application to gastric and breast cancer data. Biom Biostat Int J. 2020;9(4):132-137. DOI: 10.15406/bbij.2020.09.00310

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Abstract

In many applications related to time to event data, especially in the medical field, 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 flexibility 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 fit 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

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 influences 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 specific 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 Giolo1 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 flexible 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 Moeschberger3 and Kleinbaum and Klein.4

Let a non–negative random variable T related to the failure time, then the survival function is defined 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,

S(t)=P(Tt) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaacIcacaWG0bGaaiykaiabg2da9iaadcfacaGGOaGaamiv aiabgwMiZkaadshacaGGPaaaaa@4072@

On other hand, the hazard function λ(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4UdWMaaiikaiaadshacaGGPaaaaa@3A82@ represents the instantaneous failure rate at time   t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaaaa@3775@ conditional on survival time t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaaaa@3775@ and is very useful to describe the distribution of patient’s lifetime. In probabilistic terms,

λ(t)= lim Δt0 P(tT<t+Δt|Tt) Δt . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4UdWMaaiikaiaadshacaGGPaGaeyypa0ZdamaaxababaWdbiaa bYgacaqGPbGaaeyBaaWcpaqaa8qacqqHuoarcaWG0bGaeyOKH4QaaG imaaWdaeqaaOWdbmaalaaapaqaa8qacaWGqbGaaiikaiaadshacqGH KjYOcaWGubGaeyipaWJaamiDaiabgUcaRiaabs5acaWG0bGaaiiFai aadsfacqGHLjYScaWG0bGaaiykaaWdaeaapeGaaeiLdiaadshaaaGa aiOlaaaa@564B@

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 efficient 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 distribution9–11 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 flexible 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 distribution13 to estimate survival and hazard curves. The choice of Mirra distribution is justified due it number of parameters (two–parameters) and the bathtub shapes for the hazard function. Two dataset sets are considered to illustrate the proposed methodology. The first, related to gastric cancer, presents the bathtub shape for the empirical hazard function; and the second, related to breast cancer, presents a decreasing shape for the empirical hazard function.

The paper is organized as follows: in Section 2, it is presented the mixture Mirra cure rate model as its likelihood function. A brief description of the Bayesian approach and some discrimination criteria are also presented in Section 2. Section 3 presents the statistical analysis based on mixture Mirra cure rate model for two cancer data: gastric and breast cancer. The estimation of the survival and hazard curves are also presented in Section 3. Finally, Section 4 close the paper with some concluding remarks.

Material and methods

The mirra distribution

A first approach of the Mirra distribution (TPM) was introduced in the literature by Subhradev Sen.13 These authors studied most of it properties and illustrate how the Mirra distribution was synthesized as a special finite mixture of exponential and gamma distributions. In this way, let T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamivaaaa@3755@ be a continuous random variable following Mirra distribution with parameters α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySdegaaa@381B@ and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUdehaaa@3832@ . The probability density function (pdf) and survival function (sf) of the random variable T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamivaaaa@3755@ is given, respectively, by,

f( t )= θ 3 θ 2 +α ( 1+ 1 2  α  t 2 ) e θt , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzamaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaGaeyypa0Za aSaaa8aabaWdbiabeI7aX9aadaahaaWcbeqaa8qacaaIZaaaaaGcpa qaa8qacqaH4oqCpaWaaWbaaSqabeaapeGaaGOmaaaakiabgUcaRiab eg7aHbaadaqadaWdaeaapeGaaGymaiabgUcaRmaalaaapaqaa8qaca aIXaaapaqaa8qacaaIYaaaaiaacckacqaHXoqycaGGGcGaamiDa8aa daahaaWcbeqaa8qacaaIYaaaaaGccaGLOaGaayzkaaGaamyza8aada ahaaWcbeqaa8qacqGHsislcqaH4oqCcaWG0baaaOGaaiilaaaa@53F6@ (3)

 and,

S( t )= θ 2 θ 2 +α ( 1+ 1 θ 2  α+ 1 θ  α t+ 1 2  α  t 2 ) e θt . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uamaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaGaeyypa0Za aSaaa8aabaWdbiabeI7aX9aadaahaaWcbeqaa8qacaaIYaaaaaGcpa qaa8qacqaH4oqCpaWaaWbaaSqabeaapeGaaGOmaaaakiabgUcaRiab eg7aHbaadaqadaWdaeaapeGaaGymaiabgUcaRmaalaaapaqaa8qaca aIXaaapaqaa8qacqaH4oqCpaWaaWbaaSqabeaapeGaaGOmaaaaaaGc caGGGcGaeqySdeMaey4kaSYaaSaaa8aabaWdbiaaigdaa8aabaWdbi abeI7aXbaacaGGGcGaeqySdeMaaiiOaiaadshacqGHRaWkdaWcaaWd aeaapeGaaGymaaWdaeaapeGaaGOmaaaacaGGGcGaeqySdeMaaiiOai aadshapaWaaWbaaSqabeaapeGaaGOmaaaaaOGaayjkaiaawMcaaiaa dwgapaWaaWbaaSqabeaapeGaeyOeI0IaeqiUdeNaamiDaaaakiaac6 caaaa@63DB@ (4)

 where t>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabg6da+iaaicdaaaa@3937@ and α,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySdeMaaiilaiabeI7aXjabg6da+iaaicdaaaa@3C43@ . Thus, the hazard function (hf) is represented by,

λ( t )= θ( 1+ 1 2  α  t 2 ) ( 1+ 1 θ 2  α+ 1 θ  α t+ 1 2  α  t 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4UdW2aaeWaa8aabaWdbiaadshaaiaawIcacaGLPaaacqGH9aqp daWcaaWdaeaapeGaeqiUde3aaeWaa8aabaWdbiaaigdacqGHRaWkda WcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmaaaacaGGGcGaeqySdeMa aiiOaiaadshapaWaaWbaaSqabeaapeGaaGOmaaaaaOGaayjkaiaawM caaaWdaeaapeWaaeWaa8aabaWdbiaaigdacqGHRaWkdaWcaaWdaeaa peGaaGymaaWdaeaapeGaeqiUde3damaaCaaaleqabaWdbiaaikdaaa aaaOGaaiiOaiabeg7aHjabgUcaRmaalaaapaqaa8qacaaIXaaapaqa a8qacqaH4oqCaaGaaiiOaiabeg7aHjaacckacaWG0bGaey4kaSYaaS aaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaGaaiiOaiabeg7aHjaa cckacaWG0bWdamaaCaaaleqabaWdbiaaikdaaaaakiaawIcacaGLPa aaaaGaaiOlaaaa@647C@ (5)

As noted by Subhradev Sen13 the hf of this distribution has the shape of a bathtub, decreasing to t< 2 α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabgYda8maakaaapaqaa8qadaWcaaWdaeaapeGaaGOmaaWd aeaapeGaeqySdegaaaWcbeaaaaa@3B5C@ and increasing to t> 2 α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabg6da+maakaaapaqaa8qadaWcaaWdaeaapeGaaGOmaaWd aeaapeGaeqySdegaaaWcbeaaaaa@3B60@ . Moreover, these authors also showed that the hf is limited with the following limits,

2 θ 3 α+2 θ 2 +θ 2α <λ(t)<θ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaaikdacqaH4oqCpaWaaWbaaSqabeaapeGaaG4m aaaaaOWdaeaapeGaeqySdeMaey4kaSIaaGOmaiabeI7aX9aadaahaa Wcbeqaa8qacaaIYaaaaOGaey4kaSIaeqiUde3aaOaaa8aabaWdbiaa ikdacqaHXoqyaSqabaaaaOGaeyipaWJaeq4UdWMaaiikaiaadshaca GGPaGaeyipaWJaeqiUdeNaaiOlaaaa@4E01@ (6)

In Figure 1, it is illustrated the shape of pdf, sf and hf. From those plots, it is possible to see the bathtub shape for the hf (green line). Also, for example, assuming α=5,θ=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySdeMaeyypa0JaaGynaiaacYcacqaH4oqCcqGH9aqpcaaIYaaa aa@3E08@ (green line), we have that the limits of the hf are given by 0.828<λ(t)<2.000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaGimaiaac6cacaaI4aGaaGOmaiaaiIdacqGH8aapcqaH7oaBcaGG OaGaamiDaiaacMcacqGH8aapcaaIYaGaaiOlaiaaicdacaaIWaGaaG imaaaa@43D2@ .

Figure 1 Behavior of the pdf (left panel), sf (middle panel) and hf (right-panel) for TPM distribution assuming arbitrary values for the parameters α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHXoqyaaa@383A@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH4oqCaaa@3851@ .

A great advantage for the use of TPM distribution is that the special cases maintain the bathtub shape for the hf. These cases are described by, 

  1. When we consider α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySdeMaeyypa0JaaGymaaaa@39DC@ , we get the Mirra–silence distribution (MSI), with probability density function and survival function respectively given by

f( t )= θ 3 1+ θ 2 ( 1+ 1 2   t 2 ) e θt     and    S( t )= θ 2 1+ θ 2 ( 1+ 1 θ 2 + 1 θ  t+ 1 2   t 2 ) e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzamaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaGaeyypa0Za aSaaa8aabaWdbiabeI7aX9aadaahaaWcbeqaa8qacaaIZaaaaaGcpa qaa8qacaaIXaGaey4kaSIaeqiUde3damaaCaaaleqabaWdbiaaikda aaaaaOWaaeWaa8aabaWdbiaaigdacqGHRaWkdaWcaaWdaeaapeGaaG ymaaWdaeaapeGaaGOmaaaacaGGGcGaamiDa8aadaahaaWcbeqaa8qa caaIYaaaaaGccaGLOaGaayzkaaGaamyza8aadaahaaWcbeqaa8qacq GHsislcqaH4oqCcaWG0baaaOGaaiiOaiaacckacaGGGcGaaiiOaiaa bggacaqGUbGaaeizaiaacckacaGGGcGaaiiOaiaacckacaWGtbWaae Waa8aabaWdbiaadshaaiaawIcacaGLPaaacqGH9aqpdaWcaaWdaeaa peGaeqiUde3damaaCaaaleqabaWdbiaaikdaaaaak8aabaWdbiaaig dacqGHRaWkcqaH4oqCpaWaaWbaaSqabeaapeGaaGOmaaaaaaGcdaqa daWdaeaapeGaaGymaiabgUcaRmaalaaapaqaa8qacaaIXaaapaqaa8 qacqaH4oqCpaWaaWbaaSqabeaapeGaaGOmaaaaaaGccqGHRaWkdaWc aaWdaeaapeGaaGymaaWdaeaapeGaeqiUdehaaiaacckacaWG0bGaey 4kaSYaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaGaaiiOaiaa dshapaWaaWbaaSqabeaapeGaaGOmaaaaaOGaayjkaiaawMcaaiaadw gapaWaaWbaaSqabeaapeGaeyOeI0IaeqiUdeNaamiDaaaaaaa@7EF1@ (7)

  1. In case of θ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUdeNaeyypa0JaaGymaaaa@39F3@ , is obtained the Mirra–surrender distribution (MSII), with probability density function and survival function respectively given by

f( t )= 1 1+α ( 1+ 1 2  α  t 2 ) e t     and    S( t )= 1 1+α ( 1+ 1 θ 2  α+ 1 θ  α t+ 1 2  α  t 2 ) e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzamaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaGaeyypa0Za aSaaa8aabaWdbiaaigdaa8aabaWdbiaaigdacqGHRaWkcqaHXoqyaa WaaeWaa8aabaWdbiaaigdacqGHRaWkdaWcaaWdaeaapeGaaGymaaWd aeaapeGaaGOmaaaacaGGGcGaeqySdeMaaiiOaiaadshapaWaaWbaaS qabeaapeGaaGOmaaaaaOGaayjkaiaawMcaaiaadwgapaWaaWbaaSqa beaapeGaeyOeI0IaamiDaaaakiaacckacaGGGcGaaiiOaiaacckaca qGHbGaaeOBaiaabsgacaGGGcGaaiiOaiaacckacaGGGcGaam4uamaa bmaapaqaa8qacaWG0baacaGLOaGaayzkaaGaeyypa0ZaaSaaa8aaba Wdbiaaigdaa8aabaWdbiaaigdacqGHRaWkcqaHXoqyaaWaaeWaa8aa baWdbiaaigdacqGHRaWkdaWcaaWdaeaapeGaaGymaaWdaeaapeGaeq iUde3damaaCaaaleqabaWdbiaaikdaaaaaaOGaaiiOaiabeg7aHjab gUcaRmaalaaapaqaa8qacaaIXaaapaqaa8qacqaH4oqCaaGaaiiOai abeg7aHjaacckacaWG0bGaey4kaSYaaSaaa8aabaWdbiaaigdaa8aa baWdbiaaikdaaaGaaiiOaiabeg7aHjaacckacaWG0bWdamaaCaaale qabaWdbiaaikdaaaaakiaawIcacaGLPaaacaWGLbWdamaaCaaaleqa baWdbiabgkHiTiabeI7aXjaadshaaaaaaa@81DA@ (8)

  1. If we consider α=θ=β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySdeMaeyypa0JaeqiUdeNaeyypa0JaeqOSdigaaa@3D7E@ , we have in this case the the xgamma distribution (XG), proposed by Sen et al.,14 with probability density function and survival function respectively given by

f( t )= β 2 1+β ( 1+ 1 2  β  t 2 ) e βt     and    S( t )= β 1+β ( 1+t+ 1 β + 1 2  β  t 2 ) e βt . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzamaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaGaeyypa0Za aSaaa8aabaWdbiabek7aI9aadaahaaWcbeqaa8qacaaIYaaaaaGcpa qaa8qacaaIXaGaey4kaSIaeqOSdigaamaabmaapaqaa8qacaaIXaGa ey4kaSYaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaGaaiiOai abek7aIjaacckacaWG0bWdamaaCaaaleqabaWdbiaaikdaaaaakiaa wIcacaGLPaaacaWGLbWdamaaCaaaleqabaWdbiabgkHiTiabek7aIj aadshaaaGccaGGGcGaaiiOaiaacckacaGGGcGaaeyyaiaab6gacaqG KbGaaiiOaiaacckacaGGGcGaaiiOaiaadofadaqadaWdaeaapeGaam iDaaGaayjkaiaawMcaaiabg2da9maalaaapaqaa8qacqaHYoGya8aa baWdbiaaigdacqGHRaWkcqaHYoGyaaWaaeWaa8aabaWdbiaaigdacq GHRaWkcaWG0bGaey4kaSYaaSaaa8aabaWdbiaaigdaa8aabaWdbiab ek7aIbaacqGHRaWkdaWcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmaa aacaGGGcGaeqOSdiMaaiiOaiaadshapaWaaWbaaSqabeaapeGaaGOm aaaaaOGaayjkaiaawMcaaiaadwgapaWaaWbaaSqabeaapeGaeyOeI0 IaeqOSdiMaamiDaaaakiaac6caaaa@7C78@ (9)

Mixture mirra cure rate model

Let us denote by T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamivaaaa@3755@ the event of interest. Following Maller and Zhou,15 the standard cure rate model (or mixture cure rate model) assuming that the probability of the time–to–event to be greater than a specified time t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaaaa@3775@ is given by the survival function,

S(t)=ρ+(1ρ) S 0 (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaacIcacaWG0bGaaiykaiabg2da9iabeg8aYjabgUcaRiaa cIcacaaIXaGaeyOeI0IaeqyWdiNaaiykaiaadofapaWaaSbaaSqaa8 qacaaIWaaapaqabaGcpeGaaiikaiaadshacaGGPaaaaa@4667@ (10)

where ρ(0,1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdiNaeyicI4SaaiikaiaaicdacaGGSaGaaGymaiaacMcaaaa@3D3E@ is the mixing parameter which represents the proportion of “long–term survivors”, “non–susceptible” or “cured patients”, and S 0 (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4ua8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGOaGaamiD aiaacMcaaaa@3AD4@ denotes a proper survival function for the non–cured or susceptible group in the population. Observe that if t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDaiabgkziUkabg6HiLcaa@3AD3@ , then S(t)ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaacIcacaWG0bGaaiykaiabgkziUkabeg8aYbaa@3D53@ , that is, the survival function has an asymptote at the cure rate ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdihaaa@383C@ . The probability density and the hazard functions corresponding to (10) are given, respectively, by,

f(t)=(1ρ) f 0 (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzaiaacIcacaWG0bGaaiykaiabg2da9iaacIcacaaIXaGaeyOe I0IaeqyWdiNaaiykaiaadAgapaWaaSbaaSqaa8qacaaIWaaapaqaba GcpeGaaiikaiaadshacaGGPaaaaa@43EB@ (11)

 and,

λ(t)= (1ρ) f 0 (t) ρ+(1ρ) S 0 (t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4UdWMaaiikaiaadshacaGGPaGaeyypa0ZaaSaaa8aabaWdbiaa cIcacaaIXaGaeyOeI0IaeqyWdiNaaiykaiaadAgapaWaaSbaaSqaa8 qacaaIWaaapaqabaGcpeGaaiikaiaadshacaGGPaaapaqaa8qacqaH bpGCcqGHRaWkcaGGOaGaaGymaiabgkHiTiabeg8aYjaacMcacaWGtb WdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiaacIcacaWG0bGaaiyk aaaaaaa@50BD@ (12)

Now, assuming TPM distribution by the Equation (4) as the baseline sf for the susceptible individuals in the Equation (10), we get the mixture TPM cure rate model with sf and pdf given, respectively, by

S(t)=ρ+ (1ρ) θ 2 θ 2 +α ( 1+ 1 θ 2  α+ 1 θ  α t+ 1 2  α  t 2 ) e θt , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaacIcacaWG0bGaaiykaiabg2da9iabeg8aYjabgUcaRmaa laaapaqaa8qacaGGOaGaaGymaiabgkHiTiabeg8aYjaacMcacqaH4o qCpaWaaWbaaSqabeaapeGaaGOmaaaaaOWdaeaapeGaeqiUde3damaa CaaaleqabaWdbiaaikdaaaGccqGHRaWkcqaHXoqyaaWaaeWaa8aaba WdbiaaigdacqGHRaWkdaWcaaWdaeaapeGaaGymaaWdaeaapeGaeqiU de3damaaCaaaleqabaWdbiaaikdaaaaaaOGaaiiOaiabeg7aHjabgU caRmaalaaapaqaa8qacaaIXaaapaqaa8qacqaH4oqCaaGaaiiOaiab eg7aHjaacckacaWG0bGaey4kaSYaaSaaa8aabaWdbiaaigdaa8aaba WdbiaaikdaaaGaaiiOaiabeg7aHjaacckacaWG0bWdamaaCaaaleqa baWdbiaaikdaaaaakiaawIcacaGLPaaacaWGLbWdamaaCaaaleqaba WdbiabgkHiTiabeI7aXjaadshaaaGccaGGSaaaaa@6AED@ (13)

 and,

f(t)= (1ρ) θ 3 θ 2 +α ( 1+ 1 2  α  t 2 ) e θt . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzaiaacIcacaWG0bGaaiykaiabg2da9maalaaapaqaa8qacaGG OaGaaGymaiabgkHiTiabeg8aYjaacMcacqaH4oqCpaWaaWbaaSqabe aapeGaaG4maaaaaOWdaeaapeGaeqiUde3damaaCaaaleqabaWdbiaa ikdaaaGccqGHRaWkcqaHXoqyaaWaaeWaa8aabaWdbiaaigdacqGHRa WkdaWcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmaaaacaGGGcGaeqyS deMaaiiOaiaadshapaWaaWbaaSqabeaapeGaaGOmaaaaaOGaayjkai aawMcaaiaadwgapaWaaWbaaSqabeaapeGaeyOeI0IaeqiUdeNaamiD aaaakiaac6caaaa@586A@ (14)

The likelihood function

Let T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamivaaaa@3755@ be a positive random variable denoting the survival time of a patient and U MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyvaaaa@3756@ another positive random variable denoting the censoring or dropout time of the patient. Also define an indicator variable (binary variable) of censoring for the i   th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiaabckapaWaaWbaaSqabeaapeGaamiDaiaadIgaaaaaaa@3ABF@ patient defined by d i =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiza8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGH9aqpcaaI Xaaaaa@3A88@ , for T= t i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamivaiabg2da9iaadshapaWaaSbaaSqaa8qacaWGPbaapaqabaaa aa@3A9C@ and U t i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyvaiabgwMiZkaadshapaWaaSbaaSqaa8qacaWGPbaapaqabaaa aa@3B5D@ and d i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiza8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGH9aqpcaaI Waaaaa@3A87@ for T> t i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamivaiabg6da+iaadshapaWaaSbaaSqaa8qacaWGPbaapaqabaaa aa@3A9E@ and U= t i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyvaiabg2da9iaadshapaWaaSbaaSqaa8qacaWGPbaapaqabaaa aa@3A9D@ , where t i =min( T i , U i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiDa8aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacqGH9aqpcaqG TbGaaeyAaiaab6gacaGGOaGaamiva8aadaWgaaWcbaWdbiaadMgaa8 aabeaak8qacaGGSaGaamyva8aadaWgaaWcbaWdbiaadMgaa8aabeaa k8qacaGGPaaaaa@432A@ . For the i   th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiaabckapaWaaWbaaSqabeaapeGaamiDaiaadIgaaaaaaa@3ABF@ patient, i=1,2,,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqGHMacVcaGG SaGaamOBaaaa@3E78@ , the contribution for the log–likelihood function is given by,

= i=1 n d i log[ f( t i ) ]+ i=1 n (1 d i )log[ S( t i ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeS4eHWMaeyypa0ZaaabmaeaacaWGKbWaaSbaaSqaaiaadMgaaeqa aaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aOGaci iBaiaac+gacaGGNbWaamWaa8aabaWdbiaadAgacaGGOaGaamiDa8aa daWgaaWcbaWdbiaadMgaa8aabeaak8qacaGGPaaacaGLBbGaayzxaa Gaey4kaSYaaabmaeaacaGGOaGaaGymaiabgkHiTiaadsgapaWaaSba aSqaa8qacaWGPbaapaqabaGcpeGaaiykaiaabYgacaqGVbGaae4zam aadmaapaqaa8qacaWGtbGaaiikaiaadshapaWaaSbaaSqaa8qacaWG PbaapaqabaGcpeGaaiykaaGaay5waiaaw2faaaWcbaGaamyAaiabg2 da9iaaigdaaeaacaWGUbaaniabggHiLdaaaa@5F13@ (15)

where f( t i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzaiaacIcacaWG0bWdamaaBaaaleaapeGaamyAaaWdaeqaaOWd biaacMcaaaa@3B1B@ and S( t i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4uaiaacIcacaWG0bWdamaaBaaaleaapeGaamyAaaWdaeqaaOWd biaacMcaaaa@3B08@ denotes, respectively, the pdf and sf associated to the i   th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiaabckapaWaaWbaaSqabeaapeGaamiDaiaadIgaaaaaaa@3ABF@ patient. Now, assuming the mixture TPM cure rate model with a parameter vector ψ=(α,θ,ρ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiYdKNaeyypa0Jaaiikaiabeg7aHjaacYcacqaH4oqCcaGGSaGa eqyWdiNaaiykaaaa@411E@ , we have, for the i   th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiaabckapaWaaWbaaSqabeaapeGaamiDaiaadIgaaaaaaa@3ABF@ patient, i=1,2,,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqGHMacVcaGG SaGaamOBaaaa@3E78@ , the contribution for the log–likelihood function is given by,

(ψ)=ln( 1ρ ) i=1 n d i +ln( θ 3 ) i=1 n d i ln( θ 2 +α ) i=1 n d i + i=1 n d i ln( 1+ 1 2 α i 2 ) +θ i=1 n d i t i + i=1 n ( 1 d i ) [ ρ+ ( 1ρ ) θ 2 θ 2 +α ( 1+ 1 θ 2 α+ 1 θ α t i + 1 2 α t i 2 ) e θ t i ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqWItecBcaGGOaGaeqiYdKNaaiykaiabg2da9iaabYgacaWG UbWaaeWaaeaacaaIXaGaeyOeI0IaeqyWdihacaGLOaGaayzkaaWaaa bmaeaacaWGKbWaaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacqGH9aqp caaIXaaabaGaamOBaaqdcqGHris5aOGaey4kaSIaaeiBaiaad6gada qadaqaaiabeI7aXnaaCaaaleqabaGaaG4maaaaaOGaayjkaiaawMca amaaqadabaGaamizamaaBaaaleaacaWGPbaabeaaaeaacaWGPbGaey ypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoakiabgkHiTiaadYgacaWG UbWaaeWaaeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcq aHXoqyaiaawIcacaGLPaaadaaeWaqaaiaadsgadaWgaaWcbaGaamyA aaqabaaabaGaamyAaiabg2da9iaaigdaaeaacaWGUbaaniabggHiLd GccqGHRaWkdaaeWaqaaiaadsgadaWgaaWcbaGaamyAaaqabaaabaGa amyAaiabg2da9iaaigdaaeaacaWGUbaaniabggHiLdGccaqGSbGaae OBamaabmaabaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaaikda aaGaeqySde2aa0baaSqaaiaadMgaaeaacaaIYaaaaaGccaGLOaGaay zkaaaabaGaey4kaSIaeqiUde3aaabmaeaacaWGKbWaaSbaaSqaaiaa dMgaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHri s5aOGaamiDamaaBaaaleaacaWGPbaabeaakiabgUcaRmaaqadabaWa aeWaaeaacaaIXaGaeyOeI0IaamizamaaBaaaleaacaWGPbaabeaaaO GaayjkaiaawMcaaaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGUbaa niabggHiLdGcdaWadaqaaiabeg8aYjabgUcaRmaalaaabaWaaeWaae aacaaIXaGaeyOeI0IaeqyWdihacaGLOaGaayzkaaGaeqiUde3aaWba aSqabeaacaaIYaaaaaGcbaGaeqiUde3aaWbaaSqabeaacaaIYaaaaO Gaey4kaSIaeqySdegaamaabmaabaGaaGymaiabgUcaRmaalaaabaGa aGymaaqaaiabeI7aXnaaCaaaleqabaGaaGOmaaaaaaGccqaHXoqycq GHRaWkdaWcaaqaaiaaigdaaeaacqaH4oqCaaGaeqySdeMaamiDamaa BaaaleaacaWGPbaabeaakiabgUcaRmaalaaabaGaaGymaaqaaiaaik daaaGaeqySdeMaamiDamaaDaaaleaacaWGPbaabaGaaGOmaaaaaOGa ayjkaiaawMcaaiaadwgadaahaaWcbeqaaiabgkHiTiabeI7aXjaads hadaWgaaadbaGaamyAaaqabaaaaaGccaGLBbGaayzxaaaaaaa@C259@ (16)

Bayesian analysis

In general, the statistical inference is the process of data analysis to deduce properties of a population from a sampled data of that population. According to Ibrahim et al.,16 the Bayesian paradigm is based on specifying a probability model for the observed data D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiraaaa@3745@ , given a vector of unknown parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUdehaaa@3832@ and provides a rational method for updating the new information using the Bayes’ rule and prior distributions for the uncertainty about θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUdehaaa@3832@ . That is, the Bayesian paradigm is the process of fitting a probability model to a set of data and summarizing the result by a probability distribution, called posterior distribution, on the parameters of the model and on unobserved quantities such as predictions for new observations. In this way, assuming the proposed mixture TPM cure rate model, we simulate samples from the joint posterior distribution using the MCMC (Markov Chain Monte Carlo) algorithm implemented in the OpenBUGS software, a free version of WinBUGS software,17 by the package “BRugs"18 from R software.19

For the Bayesian analysis, as prior distributions, we adopted approximately non–informative gamma prior distributions, G(a,b) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4raiaacIcacaWGHbGaaiilaiaadkgacaGGPaaaaa@3B1E@ , for the parameters α,θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySdeMaaiilaiabeI7aXbaa@3A81@ , where a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyyaaaa@3762@ and b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOyaaaa@3763@ are known hyperparameters, and G(a,b) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4raiaacIcacaWGHbGaaiilaiaadkgacaGGPaaaaa@3B1E@ denotes a gamma distribution with mean a/b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyyaiaac+cacaWGIbaaaa@38FC@ and variance a/ b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyyaiaac+cacaWGIbWdamaaCaaaleqabaWdbiaaikdaaaaaaa@3A04@ . Moreover, for the parameter ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdihaaa@383C@ we assume a prior beta distribution ρBeta(1,1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdiNaeyipI4NaamOqaiaadwgacaWG0bGaamyyaiaacIcacaaI XaGaaiilaiaaigdacaGGPaaaaa@40B3@ since ρ(0,1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdiNaeyicI4SaaiikaiaaicdacaGGSaGaaGymaiaacMcaaaa@3D3E@ . The posterior summaries of interest are computed adopting a burn–in sample of size 50,000 to eliminate the effect of the initial values and a final Gibbs sample of size 4,000 taking every 50th sample from 250,000 simulated Gibbs samples. Also, the  highest probability density (95% HPD)20 interval was considered for the Bayesian estimates. The convergence procedures based on traceplots were verified using the “coda"21 package from R software.

To discriminate between models in the statistical analysis, two criteria are considered here: the deviance information criterion (DIC) and the extended Bayesian information criteria (EBIC). The DIC is a criterion specially useful for selection models under the Bayesian approach where samples of the posterior distribution for the parameters of the model are obtained using MCMC methods. It is similar to AIC criteria with two changes: replace the maximum likelihood estimate θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeI7aXzaata aaaa@382C@ with posterior mean θ Bayes =EAθ|yA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqbeI7aXzaata WaaSbaaSqaaabaaaaaaaaapeGaamOqaiaadggacaWG5bGaamyzaiaa dohaa8aabeaak8qacqGH9aqptuuDJXwAK1uy0HMmaeHbfv3ySLgzG0 uy0HgiuD3BaGqbaiab=ri8fjab=bi8bjabeI7aXjaacYhacaqG5bGa e8hGWheaaa@5233@ and replace k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Aaaaa@376C@ with a data–based bias correction. The new measure of predictive accuracy, according to Spiegelhalter et al.,22 is,

elpd DIC =logp(y| θ Bayes ) p DIC , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaicaaeaacaWGLbGaamiBaiaadchacaWGKbaacaGLImcapaWaaSba aSqaa8qacaqGebGaaeysaiaaboeaa8aabeaak8qacqGH9aqpcaqGSb Gaae4BaiaabEgacaWGWbGaaiikaiaabMhacaGG8bWdaiqbeI7aXzaa taWaaSbaaSqaa8qacaWGcbGaamyyaiaadMhacaWGLbGaam4CaaWdae qaaOWdbiaacMcacqGHsislcaWGWbWdamaaBaaaleaapeGaaeiraiaa bMeacaqGdbaapaqabaGcpeGaaiilaaaa@5250@ (17)

where PDIC is the effective number of parameters, defined as,

p DIC =2( logp(y| θ Bayes E post (logp(y|θ) ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiCa8aadaWgaaWcbaWdbiaabseacaqGjbGaae4qaaWdaeqaaOWd biabg2da9iaaikdadaqadaWdaeaapeGaaeiBaiaab+gacaqGNbGaam iCaiaacIcacaqG5bGaaiiFa8aacuaH4oqCgaWeamaaBaaaleaapeGa amOqaiaadggacaWG5bGaamyzaiaadohaa8aabeaak8qacqGHsisltu uDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab=ri8f9aa daWgaaWcbaWdbiaadchacaWGVbGaam4Caiaadshaa8aabeaak8qaca GGOaGaaeiBaiaab+gacaqGNbGaamiCaiaacIcacaqG5bGaaiiFaiab eI7aXjaacMcaaiaawIcacaGLPaaacaGGSaaaaa@6624@ (18)

 where the expectation in the second term is an average of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUdehaaa@3832@ over its posterior distribution. The posterior mean of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiUdehaaa@3832@ will produce the maximum log predictive density when it happens to be the same as the mode, and negative p DIC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamiCa8aadaWgaaWcbaWdbiaabseacaqGjbGaae4qaaWdaeqaaaaa @3A24@ can be produced if posterior mean is far from the mode (Spiegelhalter et al.,22). Finally, the actual quantity called DIC is defined in terms of the deviance rather than the log predictive density. Thus,

DIC=2logp(y| θ Bayes )+2 p DIC MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiraiaabMeacaqGdbGaeyypa0JaeyOeI0IaaGOmaiaabYgacaqG VbGaae4zaiaadchacaGGOaGaaeyEaiaacYhapaGafqiUdeNbambada WgaaWcbaWdbiaadkeacaWGHbGaamyEaiaadwgacaWGZbaapaqabaGc peGaaiykaiabgUcaRiaaikdacaWGWbWdamaaBaaaleaapeGaaeirai aabMeacaqGdbaapaqabaaaaa@4EB0@ (19)

 Smaller values of DIC indicate better models with a difference at least 5 by each model in DIC values.23 Note that these values could be negative. On other hand, the extended Bayesian information criteria (EBIC), proposed by Chen and Chen24, is given

EBIC= D(θ) ¯ +kln(n), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyraiaabkeacaqGjbGaae4qaiabg2da9maanaaabaGaamiraiaa cIcacqaH4oqCcaGGPaaaaiabgUcaRiaadUgacaqGSbGaaeOBaiaacI cacaWGUbGaaiykaiaacYcaaaa@4538@ (20)

where D ¯ =E[D(θ)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaanaaabaGaam iraaaaqaaaaaaaaaWdbiabg2da9iaadweacaGGBbGaamiraiaacIca cqaH4oqCcaGGPaGaaiyxaaaa@3EBE@ , is the posterior mean of the deviance, k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Aaaaa@376C@ is the number of model parameters and n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=grVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOBaaaa@376F@ is the sample size. The EBIC has the advantage of penalizing the model by the number of parameters.

Applications in cancer data

In this section, we present the usefulness of the proposed mixture TPM cure rate to estimate survival and hazard curves in cancer research under a Bayesian approach. For the hyperparameters of the gamma prior distributions, we adopted a=b=1. The results showed that the mixture TPM cure rate model is great do describe the survival probabilities as well the bathtub shape from the empirical hazard function of the data. The model also shows good lengths (small values for the difference between lower and upper bound) for the 95% HPD interval which is a indication of good fit.

Gastric cancer data

The gastric cancer is one of the leading causes of cancer–related death. Several studies are carried out to advance our understanding of the biologic behavior of gastric cancer and improve surgical management and outcome.25 To illustrate the usefulness of the proposed methodology, we considered a dataset related to the times until death in months since surgery of 201 patients of different clinical stages, obtained by Jacome et al.26 who carried out a retrospective study in patients with gastric adenocarcinoma who underwent curative resection with D2 lymphadenectomy in the Barretos Cancer Hospital (Brazil) between January 2002 and December 2007. For more details of the dataset, the reader should consult Martinez et al.27 The posterior summaries of interest for parameters of the mixture TPM cure rate model are presented in Table 1 and the fit of mixture TPM cure rate model was compared to the fit of the mixture MSI, MSII and XG cure rate models (the special cases of the TPM model). By the both discrimination criteria, it could be concluded that the mixture TPM cure rate model is the best model fitted for the dataset.

Model

Parameter

Posterior Median

95% HPD

DIC

EBIC

TPM

  α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHXoqyaaa@383A@

 0.0700

 (0.0086, 0.1573)

 894.2

 907.4

  θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH4oqCaaa@3851@

 0.1731

 (0.1265, 0.2154)

 

 

  ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCaaa@385A@

 0.4675

 (0.3794, 0.5588)

 

 

MS I

θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH4oqCaaa@3851@

 0.2314

 (0.1990, 0.2603)

 924.5

 932.6

  ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCaaa@385A@

 0.4954

 (0.4203, 0.5697)

 

 

MS II

  α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHXoqyaaa@383A@

 5.8830

 (3.1216, 9.1146)

 1851

 1859,6

  ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCaaa@385A@

 0.5272

 (0.4583, 0.5967)

 

 

XG

  β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHYoGyaaa@383C@  

 0.1983

 (0.1670, 0.2296)

 898.8

 907,5

  ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCaaa@385A@

 0.4805

 (0.4013, 0.5555)

 

 

Table 1 Posterior summaries for the parameters of the models including the cure fraction for the gastric cancer data

In Figure 2, it is presented the estimated survival (left panel) and hazard (right panel) curves for each model considered in the analysis. From the Kaplan–Meier curve for the empirical survival function, it could be seen a plateau close to the value 0.5 which was great estimated by all models. However, assuming the MSII model, the survival curve is poorly estimated. On other hand, for the hazard curve, only the TPM model has a good fit to capture the bathtub shape from the empirical hazard function (obtained using the package “bshazard").28 The other models present some values out of the confidence bounds from the empirical hazard function. Finally, in 3, it is present the probability plots for each model where it could be seen that the TPM model has a good fit for the gastric cancer data.

Figure 2 Plots of the survival functions estimated by Kaplan-Meier method and from the mixture cure fraction model based on TPM distribution and special cases (left panel) and respective hazard functions (right panel), considering gastric cancer data

Figure 3 Plots of the Kaplan-Meier estimates for the survival function versus the respective predict values obtained from the mixture models based on TPM distribution and special cases, considering gastric cancer data.

Figure 4 Plots of the survival functions estimated by Kaplan-Meier method and from the models based on TPM distribution and special cases (left panel) and respective hazard functions (right panel), considering breast cancer data.

Figure 5 Plots of the Kaplan-Meier estimates for the survival function versus the respective predict values obtained from the mixture models based on TPM distribution and special cases, considering breast cancer data.

In conclusion, the proposed mixture TPM cure rate model is adequate to model the lifetime of the patients with gastric cancer and the shape of the hazard function which could be useful in medical studies that the main interest is how to describe or predict hazard curves. In addition, comparing with the results obtained by Martinez et al.,27 no significant differences were found in the estimated cure fraction, however, it is worth mentioning that the model proposed in this manuscript has simpler equation, less number of parameters and flexibility of the hazard function which could be more useful that the model adopted in Martinez et al.,27 especially for the bathtub shape of hazard function of the data.

Breast Cancer Data

According Bray et al.29 the breast cancer is the most common cancer in women worldwide, other than non–melanoma skin cancer. However, with the advancement of treatments the proportion of cure increased considerably.30 To illustrate the usefulness of the proposed methodology, we considered now a dataset related related to a cohort study where 97 patients underwent surgical treatment for breast cancer followed up for a period ranging from the year 2000 to 2011. More details of this study can be found in Shigemizu et al.,31 For more details of the dataset, the reader should consult Shigemizu et al.31 As lifetime, it was considered the overall survival time (OS), that is, the lifetime after the diagnosis or started treatment. The posterior summaries of interest for parameters of the mixture TPM cure rate model are presented in Table 2. By the both discrimination criteria, it could be concluded that there is no significant difference for DIC values among the considered models and, by the EBIC criteria, the mixture TPM cure model may not be the most suitable for this data. This fact occurs due to the penalty that the EBIC put on the number of parameters of the model.

Model

Parameter

Posterior Median

95% HPD

DIC

EBIC

TPM

  α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHXoqyaaa@383A@ >

 0.9817

 (0.0003, 3.3707)

172.3

184.0

  θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH4oqCaaa@3851@

 0.6964

 (0.3061, 1.0197)

 

 

 

 0.7765

 (0.6783, 0.8640)

 

 

MS I

  θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH4oqCaaa@3851@

 0.7180

 (0.5257, 0.9524)

172,0

179.1

  ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCaaa@385A@

 0.7794

 (0.6842, 0.8622)

 

 

MS II

  α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHXoqyaaa@383A@

 2.4050

 (0.6351, 5.3031)

172.5

 179.9

  ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCaaa@385A@

 0.7910

 (0.7087, 0.8698)

 

 

XG

β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHYoGyaaa@383C@  

 0.6762

 (0.4270, 0.9452)

172.4

179.6

  ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCaaa@385A@

 0.7759

 (0.6773, 0.8605)

 

 

Table 2 Posterior summaries for the parameters of the models including the cure fraction for the breast cancer data

In Figure 2, it is presented the estimated survival (left panel) and hazard (right panel) curves for each model considered in the analysis. From the Kaplan–Meier curve for the empirical survival function, it could be seen a plateau close to the value 0.8 which was great estimated by all models. For the hazard curve, all the models have a good fit to capture the decreasing shape from the empirical hazard function. Finally, in 3, it is present the probability plots for each model where it could be seen that the probability plots are basically identical for each model and implying in a reasonable fit for each model assuming the breast cancer data.

Conclusion

In this study, it was introduced a new univariate cure rate model using the mixture approach and the TPM distribution introduced by Subhradev Sen13 in order to estimate the survival and hazard curves in medical application related to cancer data. The main advantage of the proposed model is the number of parameters and the incorporation of the bathtub shape from hazard function that is common in cancer data.

In the applications considered here, we can conclude that the proposed cure rate model could be really useful. For example, in the application with gastric cancer, the proposed mixture TPM cure model showed a good fit and was the only one that captured the bathtub shape from the empirical hazard function within the confidence bounds. On other hand, despite the good fit from all considered models in the breast cancer application, the proposed model also capture the shape of the empirical hazard function within confidence bounds. Now, assuming the survival curves, in both applications, the proposed model captured the cure rate as well the entire survival curve, being great to predict survival probabilities.

In conclusion, the results emerging from this study reinforce the fact that the search of appropriate lifetime distribution could be extremely difficult, especially, depending on the shape of the empirical hazard function of the data. However, the proposed methodology could be very useful in the medical data analysis where the interest is the estimation of the fraction of patients in the studied population who never experience the event of interest. The results could be also extended to other cross–over trials in clinical research; reliability analysis in engineering; risk analysis in economics; among many others areas.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Acknowledgments

None.

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