Abstract

The exploration of suitable models for modeling censored medical datasets is of great importance. There are numerous studies dealing with modeling the censored medical datasets. However, majority of the earlier contributions have utilized the conventional models for modeling the said datasets. Unfortunately, the conventional models are not capable of capturing the behavior of the heterogeneous datasets involving the mixture of two or more subpopulations. In addition, the earlier contributions have considered conventional censoring schemes by replacing all the censored items with the largest failed item. This paper is aimed at proposing the analysis of right-censored mixture medical datasets. The mixture of the generalized exponential distribution has been proposed to model the right-censored heterogeneous medical datasets. In converse to conventional censoring schemes, we have proposed censoring schemes which replace the censored items with conditional expectation (CE) of the random variable. In addition, the Bayesian methods have been proposed to estimate the model parameters. The performance and sensitivity of the proposed estimators have been evaluated using a detailed simulation study. The detailed simulation study suggests that censoring schemes based on CE provide improved estimation as compared to conventional censoring schemes. The suitability of the model in modeling heterogeneous datasets has been verified by modeling two real right-censored medical datasets. The comparison of the proposed model with existing mixture model under Bayesian methods advocated the improved performance of the proposed model.

1. Introduction

The exploration of suitable models for modeling censored medical datasets is of great importance. There are numerous studies dealing with modeling the censored medical datasets. However, majority of the earlier contributions have utilized the conventional models for modeling the said datasets. Unfortunately, the conventional models are not capable of capturing the behavior of the heterogeneous datasets involving the mixture of two or more subpopulations. Some researchers have considered mixture models for analysis of medical datasets. Hanson [1] proposed mixture of Gamma distributions to model the survival times of the lung cancer patients. Noor et al. [2] considered the mixture of exponential models to analyze the data regarding incidents of mortality due the different types of the cancer. Bussy et al. [3] introduced a supervised learning mixed model for modeling censored mixture data. The estimation of the model parameters was carried out using Expectation Maximization (EM) algorithm. The applicability of the proposed model was illustrated using three real datasets relating to genetic cancer. Mahmud et al. [4] proposed a mixture of log-skew-normal distributions for analysis of data from a Diary of Asthma and Viral Infectious Study conducted during 2004. Geissen et al. [5] presented multiexperiment mixture model that enables the researchers to simultaneously model censored and uncensored data. Cheung et al. [6] proposed a family of mixture models for undiagnosed prevalent disease with interval-censored incidents. Xiang et al. [7] proposed a mixture cure model for analysis of the survival data with a cure fraction. The estimation under the proposed model was carried out using EM algorithm. However, earlier contributions regarding analysis of censored mixture medical data have utilized the classical method of estimation, such as maximum likelihood estimation and EM algorithm. Recently, Noor et al. [2] proposed Bayesian methods for analysis of heterogeneous medical datasets. However, the model proposed by Noor et al. [2] was less flexible as it has no shape parameters. Resultantly, the corresponding analysis was quite straight forward. In addition, all of the above contributions have considered conventional censoring environment which replaces the censored items either by test termination time (for type-I censoring) or by the largest observed value (for type-II censoring). In other words, the censored items have been simply replaced by a common predetermined value. Statistical properties of estimates based on such censoring environment can be unclear [8]. Steiner and Mackay [9] addressed this issue by proposing the use of conditional expectation (CE) of random variable as a replacement for the censored items. However, the said proposal was for the single models. A careful review of literature suggests that the use of CE has never been considered for the censored mixture models.

The generalized exponential distribution (GED) is very important lifetime distribution. It has more features as compared to exponential and Rayleigh distribution as it has the shape parameter. The utilization of GED is convenient as compared to lognormal and gamma distribution as it has closed form expressions for cumulative distribution function and the hazard rate function. Few studies have also concluded that the performance of GED is better as compared to Weibull distribution in modeling censored data [10]. The mixture of GED (MGED) has been introduced more recently. The analysis of MGED under progressive censoring was considered by Wang et al. [11]. Teng and Zhang [12] showed that the Gaussian mixture and Laplacian mixture can be obtained as special cases of MGED. The estimation of model parameters was considered using EM algorithm. The superiority of MGED over Weibull distribution was explored by Ateya [13], and the industrial applications of MGED were reported by Ali et al. [14]. Mohamed et al. [15] introduced the methodology to obtain the Bayesian predictions using MGED. Kazmi and Aslam [16] considered the Bayesian analysis for right censored using MGED assuming shape parameters to be known. The above discussion suggests that the MGED is very relevant distribution in modeling the censored datasets. However, the suitability of MGED in modeling right-censored mixture datasets from medical fields using Bayesian methods is still to be explored.

We have considered the Bayesian estimation of two-component MGED (2CMGED) when samples are right censored. The applicability of the proposed model in modeling right-censored heterogeneous datasets from medical sciences has been explored using real datasets. The main feature of the paper is the introduction of censoring environment in which the censored items are replaced by the CE of the random variable. We have compared the proposed censoring environment with the existing censoring environment in which the censored items are replaced by the largest observed value. The Bayesian estimation has been carried out assuming noninformative (NIP) and informative priors (IP). Four loss functions, namely, squared error loss function (SELF), precautionary loss function (PLF), entropy loss function (ELF) and, LINEX loss function (LLF) have been used for the analysis. Since the Bayesian estimates (BEs) were unavailable in closed form, the Bayesian approximate methods, namely, Lindley’s approximation (LA) and importance sampling (IS) were used for the numerical computations. The performance of the proposed model was compared with two-component mixture of exponential distribution (2CMED). The said comparison advocated the superiority of the 2CMGED over 2CMED. In addition, the results based on proposed CE censoring environment provided improved estimation as compared to conventional censoring environment.

2. Materials and Methods

The probability density function (PDF) of the generalized exponential distribution is where is random variable and and are the parameters of the distribution.

The CDF of the generalized exponential distribution is

The two-component mixture of generalized exponential distributions (2CMGED) with mixing weights (,) is where ; ; .

The cumulative distribution function for the 2CMGED is

2.1. Bayesian Estimation under Right-Censored Samples

In this section, the right-censored samples have been used to estimate the parameters of 2CMGED. Unfortunately, the proposed BEs do not exist in the explicit form; hence the approximate Bayesian methods have been used for the estimation.

2.1.1. The Likelihood Function under Right-Censored Samples

Consider a sample of size ‘’ from 2CMGED from which and number of observation are assumed to come from component-I and component-II of the mixture. Suppose and number of failed items observed from component-I and component-II, respectively. The remaining and items have been assumed to be censored from each component. Then the likelihood function for such right-censored mixture data can be written as where , is an order statistic.

Putting the results in Equation (5), we have

The conventional censoring schemes replace the censored items by the largest observed value. This assumption is not suitable because the censoring items are surely greater than the largest observed value. The appropriate solution to this issue is to replace the censored items with CE of the random variable. The conditional distribution for model given in Equation (1) is

The CE for Equation (1) is

Since the analytical solutions for CE are not possible. The values for CE have been obtained numerically using numerical integrations.

So, the resulting dataset is of the form

2.1.2. Priors and Posterior Distributions

We have proposed two sets of priors for the parameters of the 2CMGED. One set contains the NIP, while the other set is the combination of IPs. The description of each set of priors is presented in the followings.

The combined NIP for the parametric set is where the model parameters have uniform priors over the rage .

Based on NIP given in Equation (10), the posterior distribution for is

Again, let , _, and , where

are the hyperparameters.

Then, the combined prior distribution for is

The posterior distribution under Equation (12) is

As the posterior distributions under both priors do not provide closed form estimators, we have proposed approximate estimation in the coming sections.

2.1.3. Lindley’s Approximation (LA)

This section considers the LA for approximate solution of the model parameters. If sample size is sufficiently large, then according to Lindley, any ratio of the integrals of the form where is any function of , is the log-likelihood function and is the logarithmic of joint prior for the parametric set , can be evaluated as where is the maximum likelihood estimator (MLE) of the parametric set . where is the element of the inverse of the matrix , all evaluated at the MLEs of the parameters.

The log-likelihood function from Equation (6) is

The MLEs of the parameters are obtained by iterative solution of the following:

The MLEs for the parametric set is denoted by . As mentioned in the previous sections, the second order and third order derivatives from the Equation ((13) have not been presented here. Based on the second order derivatives, the elements of the matrix are obtained and denoted by , where .

Using LA, the BEs for , under SELF and NIP, are given in the following:

Similarly considering LA, the BEs for , under SELF and NIP are

The BEs for the model parameters under ELF using NIP are

Finally, the BEs for the model parameters under LLF using NIP are