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Bayesian Inference of a Dependent Competing Risk Data

Analysis of competing risks data plays an important role in the lifetime data analysis. Recently Feizjavadian and Hashemi (Computational Statistics and Data Analysis, vol. 82, 19-34, 2015) provided a classical inference of a competing risks data set using four-parameter Marshall-Olkin bivariate Weibull distribution when the failure of an unit at a particular time point can happen due to more than one cause. The aim of this paper is to provide the Bayesian analysis of the same model based on a very flexible Gamma-Dirichlet prior on the scale parameters. It is observed that the Bayesian inference has certain advantages over the classical inference in this case. We provide the Bayes estimates of the unknown parameters and the associated highest posterior density credible intervals based on Gibbs sampling technique. We further consider the Bayesian inference of the model parameters assuming partially ordered Gamma-Dirichlet prior on the scale parameters when one cause is more severe than the other cause. We have extended the results for different censoring schemes also.


Introduction
In lifetime data analysis an experimenter often wants to analyze data which have multiple failure modes. In the statistical literature it is known as the competing risks problem. There are mainly two different approaches to handle competing risks data. One is known as the latent failure time model of Cox (1959) and the other is known as the cause specific hazard rate model of Prentice et al. (1978). In case of exponential and Weibull lifetime distributions it has been shown by Kundu (2004) that both the models lead to the same likelihood function, although their interpretations are different. An extensive list of literature exits in this area, see for example Kalbfleish and Prentice (1980), Lawless (1982) or Crowder (2001) and the references cited therein. Most of the existing studies are based on the assumptions that the causes of failures are independent, although it may not be true in practice. It may be mentioned that there are some identifiability issues in this respect, see for example Tsiatis (1975).
The bivariate or multivariate lifetime distributions play an important role in analyzing dependent competing risk model. Bayesian inference of a dependent competing risk model assuming absolute continuous bivariate exponential distribution is studied by Wang and Ghosh (2003). When there is a positive probability of simultaneous occurrence of two causes of failure then Marshall-Olkin bivariate exponential (MOBE) distribution, introduced by Marshall and Olkin (1967), can be used to analyze the data. If the data indicate that the marginals have unimodal probability density functions (PDFs) then MOBE distribution will not fit the data. Due to this limitations, Marshall-Olkin bivariate Weibull (MOBW) distribution was introduced by Lu (1989). Later this distribution has been studied by several authors including Jose, Ristic and Joseph (2011), Dey and Kundu (2009) and Kundu and Gupta (2013). The analysis of dependent competing risk model using MOBW distribution is considered by Feizjavdian and Hashemi (2015). Bayesian inference of a series system with dependent causes of failure using MOBW distribution is provided by Xu and Zhou (2017).
Different methods of estimating parameters of dependent competing risks using MOBW model has been studied by Shen and Xu (2018). Order restriction among model parameters in a reliability model has been considered by several authors. Order restricted inference of step-stress model has been considered by Balakrishnan, Beutner and Kateri (2009) and Samanta and Kundu (2018) to incorporate the fact that the increased stress level will reduce the expected lifetime of the experimental units. Recently Mondal and Kundu (2020) considered order restricted inference for two exponential populations. They have mentioned that this order restricted inference can be used in an accelerated life test if one sample is put under higher stress keeping the other one in normal stress. In competing risk model when it is known apriori that one cause of failure is higher risk than the other then we may incorporate this information by considering an order restriction on the model parameters.
The motivation of this paper came from a recent paper by Feizjavdian and Hashemi (2015). They have analyzed a data set obtained from the Diabetic Retinopathy Study (DRS) conducted by the National Eye Institute to estimate the effect of laser treatment in delaying the onset of blindness in patients with diabetic retinopathy. At the beginning of the experiment, for each patient, one eye was selected for laser treatment and the other eye was not given the laser treatment. For each patient the minimum time to blindness (T ) and the indicator specifying whether the treated eye (δ = 1) or the untreated eye (δ = 2) has first failed has been recorded. If both the eyes have failed simultaneously then δ = 0 has been recorded. The data set is presented in Table 1. The main objective of this experiment is to study whether the laser treatment has any effect on delaying the onset of blindness in patients with diabetic retinopathy. Clearly, the time to blindness of the two eyes cannot be independent and there are some ties in the data set. Due to this reason Feizjavdian and Hashemi (2015) considered a dependent competing risks model and they have proposed to use the Marshall-Olkin bivariate Weibull distribution for this purpose. They provided the maximum likelihood estimators (MLEs) of the unknown parameters and obtained the associated asymptotic confidence intervals. The maximum likelihood estimators cannot be obtained in explicit forms, hence they have obtained the approximate maximum likelihood estimators which can be obtained in explicit forms. It is observed that the proposed model works quite well for fitting purposes. They have observed that even for highly censored data, the MLEs perform quite well.
The main aim of this paper is to provide the Bayesian analysis of the same data set under a very flexible Gamma-Dirichlet (GD) prior on the scale parameters and for a very general log-concave prior on the shape parameter. It is observed that the Bayesian inference has some natural advantages in this case. The Gamma-Dirichlet prior was originally introduced by Pena and Gupta (1990) for Marshall-Olkin bivariate exponential distribution (MOBE).
The GD prior is a very flexible prior, and its joint PDF can take variety of shapes depending on the hyper parameters. It can be both positively and negatively correlated. In case of MOBW distribution the GD distribution is a conjugate prior of the scale parameters for a fixed shape parameter. Hence the posterior distribution of the scale parameters for the fixed shape parameter can be obtained in a very convenient form. We have used a very general log-concave prior on the shape parameter, and they are assumed to be independent. The Bayes estimators cannot be obtained in closed form. We have used Gibbs sampling technique to compute the Bayes estimates and the associated highest posterior density (HPD) credible intervals.
We further consider the Bayesian inference of the model parameters with partially order restriction on scale parameters. This order restriction comes naturally when it is apriori known that one cause of failure is more severe than the other. In this case we consider partially order restricted Gamma-Dirichlet prior for scale parameters and we use importance sampling technique for Bayes estimates and the credible intervals. We re-analyze the same data set assuming the order restriction on scale parameters. One major advantage of the Bayesian inference is that different forms of data for example; Type-I, Type-II, hybrid censored data can be handled quite conveniently also, unlike the classical inference. Finally the Bayesian testing of hypothesis has been considered to test the hypothesis that there is no significant difference between two causes of failure. We propose to use Bayes factor to test the hypothesis and interestingly in this case it can be obtained in explicit form. We have reanalyzed the data set and it is observed that the laser treatment does not have any effect in delaying the onset of blindness.
The rest of the paper is organized as follows. In Section 2 we describe the Marshall-Olkin bivariate Weibull distribution and provide the likelihood function based on competing risk data. Prior assumptions and posterior analysis is provided in Section 2.1. The order restricted Bayesian inference is given in Section 3. The inference under different censoring scheme is provided in Section 4. In Section 5, we discuss the Bayesian testing of hypothesis problem. The analysis of the data set has been provided in Section 6 and an extensive simulation results have been discussed in Section 7. Finally we have concluded the article in Section 8. the failure times are recorded. We assume that the units are failed due to several causes of failure. Here we restrict ourselves to two causes of failure, although the results can be easily generalized for more than two causes also. Let X 1 be a random variable associated with the lifetime of an unit under the first cause and X 2 be a random variable associated with the second cause. An unit is failed if the minimum of the two occurs. Therefore T = min{X 1 , X 2 } is the random variable associated with the lifetime of an experimental unit which is exposed to both the risk factors. Along with the failure time, the cause of failure is also recorded. In reality the two causes are related and hence we assume MOBW distribution for two causes of failure. In this model it is assumed that the failure can occur due to both the causes simultaneously. The MOBW distribution is defined as follows: Suppose for i = 0, 1, 2, U i follows independent Weibull distribution with shape parameter α and scale parameter λ i . We will denote it by U i ∼ W E(α, λ i ). The PDF and the survival function of U i for u i > 0 are, respectively, Now let X 1 = min{U 0 , U 1 } and X 2 = min{U 0 , U 2 }, then the bivariate random variable (X 1 , X 2 ) is said to be follow the MOBW distribution with shape parameter α and scale parameters λ 0 , λ 1 and λ 2 and it is denoted by (X 1 , X 2 ) ∼ MOBW (α, λ 0 , λ 1 , λ 2 ). The survival function of MOBW random variable (X 1 , X 2 ) is The joint PDF of (X 1 , X 2 ) is where Let us define Therefore, T = min{X 1 , X 2 } and ∆ be the random variables corresponding to the failure time and the cause of failure of an experimental unit respectively. Thus the available data set on a competing risk model is of the form: {(t 1:n , δ 1 ), . . . , (t n:n , δ n )}, where (t i:n , δ i ) denotes the i-th ordered observed value of (T, ∆). The likelihood function based on the data set can be obtained using the below equation where δ i0 , δ i1 , δ i2 are the indicators for failure of i-th observation due to both the causes simultaneously, first cause and second cause respectively. From the survival function in (1) of MOBW distribution we have Therefore using (3) and (4) the likelihood of the data is where n 0 = n i=1 δ i0 , n 1 = n i=1 δ i1 and n 2 = n i=1 δ i2 (n i > 0 for i = 0, 1, 2 and n = 2 i=0 n i ) are the number of failures due to both the causes, the first cause and the second cause, respectively.

Prior Assumption and Posterior Analysis
In this section we will provide the Bayesian inference of the model parameters under squared error loss function. Since we have considered a dependent competing risk model, in the Bayesian analysis we assume a dependent prior distribution of (λ 0 , λ 1 , λ 2 ). Using the concept of Pena and Gupta (1990) we have assumed the multivariate Gamma-Dirichlet prior for (λ 0 , λ 1 , λ 2 ). Therefore the joint prior distribution of (λ 0 , λ 1 , λ 2 ) with hyper parameters a > 0, b > 0, a 0 > 0, a 1 > 0 and a 2 > 0 is given by where a = a 0 + a 1 + a 2 and λ = λ 0 + λ 1 + λ 2 . This distribution will be denoted by GD(a, b, a 0 , a 1 , a 2 ). In general this is a dependent prior but if a = a then λ i 's are independent gamma priors with parameter b and a i (i = 0, 1, 2). The prior distribution of α is Gamma with hyper parameters c 1 > 0 and c 2 > 0 (denoted by GA(c 1 , c 2 )) and is independent with the joint prior distribution of (λ 0 , λ 1 , λ 2 ). Thus the joint prior of (α, λ 0 , λ 1 , λ 2 ) is given by Therefore the joint posterior distribution of (α, λ 0 , λ 1 , λ 2 ) is given by where, In this case the explicit form of the Bayes estimates cannot be obtained and hence we propose to use Gibbs sampling technique to obtain the Bayes estimates and associated credible intervals. The form of the π 1 (α) is not any standard distributional form but in Theorem (1) we will show that π 1 (α) is a log-concave density function. On the other hand, for a given α the joint posterior distribution of (λ 0 , λ 1 , λ 2 ), i.e., π 2 (λ 0 , λ 1 , λ 2 |α) is GD(a + n, b + n i=1 t α i:n , a 0 + n 0 , a 1 + n 1 , a 2 + n 2 ).
Proof. See in the Appendix.
The method proposed by Devroye (1984) for generation of random sample from a logconcave density function can be used to generate sample from π 1 (α). Generation of sample from Gamma-Dirichlet distribution is quite straight forward which is given explicitly in Kundu and Pradhan (2011). Thus we propose to execute the following algorithm to obtain the Bayes estimates and the associated credible intervals of the unknown parameters.
Step 3. Repeat Step 1 and Step 2, M times to obtain (α 1 , Step 4. Bayes estimate of α, λ 0 , λ 1 and λ 2 with respect to squared error loss function are respectively given by Step 5. The corresponding posterior variance can be obtained respectively as Step 6. To obtain credible interval of α, we order α 1 , . . . , α M as α (1) < . . . < α (M ) . Then Step 7 Similar to Step 6 and Step 7 we can obtain the symmetric and HPD credible intervals for other parameters.

Order Restricted Inference
In this section we provide the order restricted Bayesian inference of the model parameters.
Between two causes, let cause -1 be more severe than cause -2. Therefore, there is a ordering between the parameters related to two causes. In this model assumption, the ordering is We want to incorporate this information in our inference. In order restricted inference, we consider the following joint prior distribution of (λ 0 , λ 1 , λ 2 ) assuming λ 1 < λ 2 . Let Note that the above prior distribution is the joint PDF of partially ordered random vari- . We denote the prior in (9) as P OGD(a, b, a 0 , a 1 , a 2 ). Here also we assume that the prior distribution of α is Gamma with hyper parameters c 1 > 0 and c 2 > 0 and is independent with the joint prior distribution of (λ 0 , λ 1 , λ 2 ). The explicit form of the Bayes estimates under squared error loss function cannot be obtained. Hence we propose to use importance sampling technique to obtain the Bayes estimates and the associated credible intervals. The joint posterior distribution can be written as where As before π 1 (α) is a log-concave density function and hence we can generate α from π 1 (α) easily. Also note that π 2 (λ 0 , λ 1 , λ 2 |α) is POGD(a + n, b + n i=1 t α i:n , a 0 + 2n 0 , a 1 + n 1 + n 2 , a 2 + n 1 + n 2 ) and generation from this distribution is quite straight forward. For given α, first generate (λ * 0 , λ * 1 , λ * 2 ) from GD(a + n, b + n i=1 t α i:n , a 0 + 2n 0 , a 1 + n 1 + n 2 , a 2 + n 1 + n 2 ) and then take (λ 0 , λ 1 , . Now we propose the following algorithm for Bayes estimates and the associated credible intervals.

Inference under Different Censoring Schemes
There are several censoring schemes available in the literature. One major advantage of the Bayesian inference is that we can easily extend the inference to different censoring schemes.
In this section we discuss the inference of dependent competing risk model under different censoring schemes. Before proceeding, we define the following notations. τ * = termination time of the experiment; n * = total number of failure before τ * .

Type-I Censoring
In Type-I censoring scheme we stop the experiment at a prefix time, say τ * and the number of observations failed before τ * is n * . In this case observed data is of the form {(t 1:n , δ 1 ), . . . , (t n * :n , δ n * )}.

Type-II Censoring
In this censoring scheme the life testing experiment is terminated when the r-th (prefixed number) failure occurs, i.e, the total number of failure is fixed but the termination time of the experiment is random. Available data under this censoring scheme is of the forms {(t 1:n , δ 1 ), . . . , (t r:n , δ r )}. Inference of Type-II censored data is very similar to that of Type-I censored data. In this case we have to take n * = r, τ * = t r:n and D(α, τ * ) = r i=1 t α i:n + (n − r)t r:n . Also note that r = n 0 + n 1 + n 2 . All other expressions and the following analysis are same as the Type-I censoring scheme.

Type-I Hybrid Censoring
The termination time in Type-I hybrid censoring scheme (HCS) is τ * = min{t r:n , τ }, where r is a pre-fixed number and τ is pre-fixed time. If n 1 is the number of failures before τ then the available data under this censoring scheme is one of the following forms (a) {(t 1:n , δ 1 ), . . . , (t n 1 :n , δ n 1 )} if τ ≤ t r:n , (b) {(t 1:n , δ 1 ), . . . , (t r:n , δ r )} if t r:n < τ.
Based on Type-I Hybrid censored data, the posterior analysis is same as that of Type-I censoring scheme with, for case (a) n * = n 1 , τ * = τ, D(α, τ * ) = n 1 i=1 t α i:n + (n − n 1 )τ , and for case (b) n * = r, τ * = t r:n and D(α, τ * ) = r i=1 t α i:n + (n − r)t r:n . All other expressions and the following analysis are same as the Type-I censoring scheme.
Based on Type-II Hybrid censored data, the posterior analysis is same as that of Type-I censoring scheme with, for case (a) n * = r, τ * = t r:n and D(α, τ * ) = r i=1 t α i:n + (n − r)t r:n , and for case (b) n * = n 1 , τ * = τ, D(α, τ * ) = n 1 i=1 t α i:n + (n − n 1 )τ . All other expressions and the following analysis are same as the Type-I censoring scheme.

Type-I Progressive Censoring
Let τ 1 , . . . , τ k be k pre-fixed time points and R 1 , . . . , R k−1 be pre-fixed nonnegative integers less than n. Also let n i (i = 1, . . . , k) be the number of failures between time τ i−1 to τ i (τ 0 = 0). At the time τ i (i = 1, . . . , k − 1), R i randomly chosen units from the survived units are removed from the experiment. Finally R k = n − k i=1 n i − k−1 i=1 R i units are removed at time τ k . The available data in this censoring scheme is of the form {(t 1:n , δ 1 ), . . . , (t n * :n , δ n * )}.
Based on Type-I progressive censored data, the posterior analysis is same as that of Type-I censoring scheme with, n * = k i=1 n i , τ * = τ k and D(α, τ * ) = n * i=1 t α i:n + k i=1 R i τ α i . All other expressions and the following analysis are same as the Type-I censoring scheme.  1:n , δ 1 ), . . . , (t m:n , δ m )}. Based on Type-II progressive censored data, the posterior analysis is same as that of Type-I censoring scheme with, n * = m, τ * = t m:n and D(α, τ * ) = m i=1 (R i + 1)t α i:n . All other expressions and the following analysis are same as the Type-I censoring scheme.

Testing of Hypothesis
In this section we provide a method of testing the hypothesis that both the causes have equal effect. Mathematically, we want to test the null hypothesis H 0 : λ 1 = λ 2 against the alternative H 1 : λ 1 = λ 2 . Therefore under H 0 , i.e. under the assumption of equality of two causes of failure we may assume that the data t = (t 1:n , . . . , t n:n ) is coming from a Weibull distribution with parameters α * and λ * . We propose to use Bayes factor for testing the hypothesis. Under H 1 , the likelihood function and the joint prior distribution are given in equation (5) and equation (7) respectively. Under H 0 , the likelihood function is given by Assume that the prior distributions of α * and λ * are GA(d 1 , d 2 ) and GA(d 3 , d 4 ) respectively.
Also assume that the prior distributions of α * and λ * are independent. Hence the joint density function of α * and λ * is Therefore, under H 0 , the marginal distribution of t is given by Similarly the marginal distribution of t under H 1 is given by Therefore the Bayes factor (BF) for testing H 0 against H 1 is .
Hence for given data, we reject H 0 if BF is low. We illustrate this testing of hypothesis in data analysis section.

Data Analysis
Diabetic Retinopathy is one of the major causes of vision loss and blindness of diabetes patients. National Eye Institute conducted DRS to estimate the effect of laser treatment in reducing the risk of blindness. The study was conducted on 71 patients. For each patient, one eye was selected at random and the laser treatment was given on that eye. For each patient the time to blindness and the indicator mentioning whether treated or untreated or both eyes became blind has been recorded. The main purpose of this study is to verify whether the laser treatment has any effect in delaying the onset of blindness in patients with diabetic retinopathy. The treatment or lack of treatment can be regarded as two causes of blindness, hence this data set can be treated as a competing risks data. Clearly, the two competing causes in this case cannot be taken as independent. Moreover, there is a positive probability of simultaneous occurrence of both the causes. Hence, MOBW distribution is a plausible model to analyze this data set.
We  Table 2. Next we analyze the data assuming λ 1 < λ 2 , i.e., the expected time to blindness of the treated eye is higher than the eye without the laser treatment. The Bayes estimates of α, λ 0 , λ 1 and λ 2 assuming order restriction are 1.5388, 0.0707, 0.1789 and 0.2281, respectively. The symmetric and HPD credible intervals assuming order restriction are provided in Table 3.
Next we have checked the goodness of fit of the data using Kolmogorov-Smirnov (KS) test statistics. The KS distance between empirical and fitted CDF using Bayes estimates without order restriction is 0.0579, which indicates that the empirical and fitted CDF are very close. The p-value of the test for testing the equality of two CDFs is 0.9598, i.e., based on the data we cannot reject the hypothesis that the empirical and fitted CDF are equal. We have performed the test for order restricted case also. The KS distance and p-value in this case are 0.0572 and 0.9637 respectively. Note that the KS distance in case of order restricted   inference is smaller than the without order restricted inference. The graphical representation of empirical and fitted CDF is provided in Figure 1.
Next we have tested the hypothesis that there is no significance difference between two causes of failure, i.e., we have tested the null hypothesis H 0 : λ 1 = λ 2 against the alternative  H 1 : λ 1 = λ 2 using the method proposed in Section 5. The Bayes factor for the given data using almost non-informative prior is 2.326479×10 32 . Since the BF is very high, we conclude that two causes are not significantly different. Therefore, the conclusion from the study is that the laser treatment does not have any effect in delaying the onset of blindness to the patients with diabetic retinopathy. Now, we consider the data without the causes of failure and fitted the data assuming Weibull distribution with shape parameter α * and scale parameter λ * . Assuming almost non-informative gamma prior for both α * and λ * , the Bayes estimates of α * and λ * are respectively 1.5358 and 0.4795. The KS distance and p-value of the fit are respectively 0.0582 and 0.9581, which indicates the good fit of the data. The empirical and the fitted CDF of the data assuming Weibull distribution is presented in Figure 2. This can be used to estimate E(T ), i.e. the expected time to the onset of blindness, or to estimate E(T |T > a), for some a > 0, etc.

Simulation
In this section we provide an extensive simulation study based on complete sample to verify how the proposed estimators behave for different sample sizes and for different set of parameters. Simulation results are provided for both, without order restriction and with order restriction on scale parameters. We consider three sets of parameter values of (α, λ 0 , λ 1 , λ 2 ): Set I (2.0, 0.5, 1.0, 1.2), Set II (2.0, 1.0, 1.0, 1.2) and Set III (2.0, 1.5, 1.0, 1.2). We have taken n = 30, 40, 50. We have considered almost non-informative proper priors, as suggested by Congdon (2003); the hyper parameters are a = b = c 1 = c 2 = 0.001 and a 0 = a 1 = a 2 = 1.
We provide the average estimates (AEs) along with the mean square errors (MSEs) of the model parameters. The average lengths (AL) and coverage percentages (CP) of 95% symmetric and highest posterior density (HPD) credible intervals are also provided. All the simulation results are provided from Table 4 to Table 9  Now if we compare the inference based on partially order restriction on scale parameters with unrestricted inference then it has been observed that order restricted inference provides lower MSEs for λ 1 and λ 2 than unrestricted inference. Also the ALs of different CRIs of λ 1 and λ 2 under order restricted inference is is lower than that of unrestricted inference.

Conclusion
In this article we have provided the Bayesian inference of a dependent competing risk model.
We assume Marshall-Olkin bivariate Weibull distribution to explain the dependency structure between two causes of failure. Bayesian inference has been provided under two scenario, in one case we assume order restriction between two causes of failures and in other case we do not assume any order restriction. We have also shown that the inference procedure for both the cases can easily be extended to different censoring schemes. We propose to use Bayes factor to test the hypothesis that there is no significant difference between two causes of failure. An extensive simulation results and a data analysis shows that the proposed method works quite well. If it is known apriori that one cause of failure is higher risk than the other then it is better to use the order restricted inference.