Applicability of modified weibull extension distribution in modeling censored medical datasets: a bayesian perspective

There are some contributions analyzing the censored medical datasets using modifications of the conventional lifetime distribution; however most of the said contributions did not considered the modification of the Weibull distribution (WD). The WD is an important lifetime model. Due to its prime importance in modeling life data, many researchers have proposed different modifications of WD. One of the most recent modifications of WD is Modified Weibull Extension distribution (MWED). However, the ability of MWED to model the censored medical data has not yet been explored in the literature. We have explored the suitability of the model in modeling censored medical datasets. The analysis has been carried out using Bayesian methods under different loss functions and informative priors. The approximate Bayes estimates have been computed using Lindley’s approximation. Based on detailed simulation study and real life analysis, it has been concluded that Bayesian methods performed better as compared to maximum likelihood estimates. In case of small samples, the performance of Bayes estimates under ELF and informative prior was the best. However, in case of large samples, the choice of prior and loss function did not affect the efficiency of the results to a large extend. The MWED performed efficiently in modeling real censored datasets relating to survival times of the leukemia and bile duct cancer patients. The MWED was explored to be a very promising candidate model for modeling censored medical datasets.

The literature contains many valuable contributions for analysis of lifetime data using different modifications of the Weibull distribution. Silva et al. 1 introduced beta modified Weibull distribution and showed that it is suitable in modeling data with monotone failure rates. Almalki and Yuan 2 introduced a new modified Weibull distribution and estimated its model parameters based on order statistics using moment estimates, MLE and Bayes estimates. Sarhan and Apalo 3 proposed exponentiated modified Weibull extension distribution and discussed its applications in different fields. Peng and Yan 4 introduced extended Weibull distribution, estimated the model parameters and explored applicability of the model. Ahmad and Iqbal 5 developed the generalized flexible Weibull extension distribution and compared its modeling capabilities with some conventional life models. El-Morshedy et al. 6 proposed three parametric exponentiated inverse flexible Weibull extension distribution. The proposed model was shown to be better than other modifications of Weibull distributions in modeling life datasets. Tahir et al. 7 introduced transmuted Weibull-Pareto and evaluated its important properties. Lindley-Weibull distribution was introduced by Cordeiro et al. 8 , as a better alternate to Weibull distribution.

Modified Weibull extension distribution (MWED). This section includes the introduction of MWED.
The MWED is very useful lifetime model, especially when the hazard rate has bathtub shape (Yang et al. 19 ). In addition, the modeling of failure times and reliability using MWED is quite convenient due to its closed form expressions for cumulative distribution function (Xie et al. 18 ). The additional feature of MWED is that the confidence interval for the shape parameter and joint confidence interval can be derived explicitly. Due to these features, the MWED is very suitable candidate to model the censored lifetimes. The analysis of applicability of MWED to model censored datasets relating to medical field can be very interesting. After defining the basic formulation about MWED in this Section, the estimates based MWED has been used to model the right censored medical datasets. The density function and some important characteristics of MWED have been reported in the following equations.
The probability density function (PDF) of the MWED is where θ, σ , µ ≥ 0 are the parameters of the model and x ≥ 0. The cumulative distribution function (CDF) of the MWED is The reliability function for the MWED is (1) f x|θ, σ , µ = σ µ (x/θ) σ −1 Exp (x/θ) σ + µθ 1 − Exp(x/θ) σ where θ is a scale parameter and σ, µ are shape parameters and 'u' is uniformly distributed over range (0, 1). This model has Weibull distribution as a special and asymptotic case, so it can be considered as a Weibull extension distribution. When σ ≥ 1 the hazard rate function is an increasing function and when σ ≤ 1 the hazard rate function is a bathtub-shaped function.
Bayesian estimation of the MWED using right censored datasets. The important part of the Bayesian estimation is to obtain the likelihood function for the sampling distribution. The likelihood function under type-II censored samples can be defined as. Suppose that 'n' items are put on a test and the test was terminated when the 'r' failures were observed. Hence the 'n − r' items were type-II right censored. Then the likelihood function for the said type-II right censored dataset is The Likelihood function under censored samples Prior and posterior distributions. The additional advantage of the Bayesian methods is that they can incorporate the prior information to update the current state of knowledge about the model parameters. This study will include the assumption of non-informative and informative priors for the derivation of Bayes estimates under different loss functions. The joint informative prior assuming gamma prior for each parameter of MWED is.
where a 1 , a 2 , a 3 , b 1 , b 2 , b 3 are hyper-parameters. The values of the hyper-parameters have been chosen by using prior mean approach. In prior mean, the values of the prior-parameters are selected in the way that prior means becomes approximately equal to the true parametric values. In case of real datasets, the true parametric values are not available, so the values of the hyperparameters has been chosen to be so that the prior mean become approximately equal to the MLEs for the model parameters. The MLE estimators have been obtained by maximizing (7) with respect to model parameters. The R Code for obtaining MLEs has been given in Supplementary information.
The posterior distribution under Gamma prior As the closed form expressions for the Bayes estimates of model parameters under SELF, PLF, QLF and ELF are not possible, the Bayesian approximate method, namely, Lindley's approximate has been used to obtain the numerical solutions for model parameters under the said loss functions. The results under Bayes estimates have also been compared with most commonly used classical method, namely, MLE.
Lindley's approximation (LA). Having sufficiently large samples, Lindley 24 proposed that function of the form where � = (θ, σ , µ) , h(�) is some function involving , I(�|x) is the logarithmic of likelihood function and G(�) in the logarithmic of g(�) given in (8), can be given in the following form where is MLE of the parametric set ,

Results
This section deals with the analytical and numerical estimation for the parameters of MWED using MLE and Bayesian method. The Bayes estimates have been obtained using different loss functions and informative priors. The LA has been used to obtain the numerical results for the Bayes estimates. The performance of different estimates has been compared using different simulated datasets. The suitability of the MWED has been explored in modeling the censored medical datasets. In particular, two censored medical datasets have been used for analysis.
Simulation study using right censored datasets. The MLEs, Bayes estimates (BEs) and amounts of mean square errors (MSEs) for MWED under different loss functions SELF, PLF, QLF and ELF using different parametric spaces and sample sizes have been reported in this section. The results using censored simulated datasets have been reported in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. The simulated datasets have been generated using different sample sizes and different true parametric values. In particular, the samples of size The numerical values for the prior parameters have been chosen using prior mean methodology. The prior mean approach chooses the values of the hyper-parameters in such a way that prior mean approximate to true parametric value. The comparison between MLE and Bayesian estimation methods has been carried out using the amount of MSEs associated with respective estimates. In Tables, the amount of MSEs has been presented in the bold fonts. The steps to generate the right censored simulated samples and to compute estimates have been given in the following.
Step 1: Generate a sample of size 'n' from MWED using inverse transformation technique.
Step 2: Sort the generated sample in ascending order of magnitudes of the values.
Step 3: Decide the censoring rate, that is, what number/proportion of values will be censored.
Step 4: Let we have starting 'r' number of items are completely observed, then remaining 'n -r' number of items are assumed censored. www.nature.com/scientificreports/ Step 5: Take rth observed item as the value of x r .
Step 6: Apply the LA given in Section "Lindley's Approximation (LA)" to obtain the numerical estimates.
Step 7: Repeat Step-1 to Step-6 10,000 times and report the average of the estimates and their MSEs. The graphs for amounts of MSEs associated with estimates using simulated datasets of size n = 20 and 100 for different parametric values, have been placed in Figs. 1, 2, 3

Discussions
The MWED is very important distribution to model failure times and reliability of the data. It is often preferred over other modifications of the Weibull distribution owing to the fact that the model possesses closed form CDF and hazard rate. The use of MWED is especially advantageous when the hazard rate of the data is of bathtub shape. The models with closed form CDF and hazard rate are also preferred to model the censored datasets. www.nature.com/scientificreports/ It is worth mentioning here that the survival times of the patients often possess bathtub shaped hazard rate (Kayid 26 ). So, the MWED having bathtub shaped hazard rate is very relevant in modeling the survival times and reliability of the patients. However, according to the best of our knowledge, no earlier study has reported this aspect of MWED. In addition, the Bayesian analysis of the censored datasets using different modifications of the Weibull distribution has been quite frequent in literature. A careful review of the literature suggests that the Bayesian analysis of censored datasets using MWED has not been discussed in detail in literature. Especially, the suitability of the MWED in modeling censored medical datasets using Bayesian methods has not been discussed in literature. The gap has been bridged, in this paper, by considering Bayesian analysis of the censored    www.nature.com/scientificreports/ medical datasets using MWED. The detailed simulation study suggests that the estimates based on MWED possess the consistency property. The estimates using Bayesian methods were found to be better than those under MLE method. In case of Bayesian methods, the estimates under ELF were quite better as compared to their counterparts. These finding are in agreement with the earlier studies conducted for generalized exponential distribution (Mitra abd Kundu 27 ) and for Weibull model (Kundu 28 ). The suitability of the MWED in modeling censored medical datasets was evaluated by modeling two right censored datasets regarding survival times of the cancer patients. It was encouraging to observe that MWED was able to represent the behavior of both the datasets.

Conclusion
Although the literature contains the analysis of censored medical datasets using the modified versions of the lifetime distributions, most of the proposed models were not modifications of the Weibull distribution. Especially the Bayesian estimation of censored medical datasets using the modified version of Weibull distribution is rarely found in literature. The Weibull distribution is very important lifetime model and many authors have proposed different modifications of this model. The recent modification of Weibull distribution, namely MWED, has been shown to perform better than Weibull and mixture of Weibull distribution in modeling lifetime datasets. We have proposed Bayesian analysis of censored medical datasets using MWED. The results have been compared with most frequently used MLE method. The informative priors and different loss functions have been used for the analysis. The reliability characteristics of the said datasets have also been evaluated. The detailed simulation study has been conducted to prove the consistency and efficiency of the proposed estimates as compared to MLE. The applicability and suitability of the MWED is modeling censored medical datasets has been explored using two real datasets. The results confirmed the consistency property of the estimates. In addition, the performance of the Bayes estimates was better as compared to MLE. This feature of Bayes estimates was more evident in the small samples. In particular, the Bayes estimates under ELF and informative prior were the best. The proposed estimators were quite insensitive with respect the different choices of true parametric values. Further, the performance of the proposed Bayes estimates, in modeling the censored real medical datasets, was better as compared to their counterparts. In particular, the survivors of the patients are more accurately modeled using the Bayes estimates under ELF. Finally, the MWED was explored to be a very potential candidate for modeling censored medical datasets. The proposed model was able to represent the behavior of both censored real medical datasets. The study is useful for the researchers dealing with censored medical datasets, especially when more flexibility in modeling is needed.