Subsurvival Functions Research Articles

Nonparametric Bayesian reliability analysis of masked data with dependent competing risks

The characteristic of dependence widely exists among different failure modes of systems, which brings extra difficulty for the reliability analysis. In this paper, a nonparametric Bayesian analysis method is proposed for dependent masked data under accelerated lifetime test with censoring. Using the copula function, the dependence structure is constructed among the competing failure modes which can be viewed as the components in a series system. By establishing the transformational relationship between the subsurvival functions and the survival functions, the estimators of components’ reliability can be derived from the nonparametric Bayesian estimators of the subsurvival functions when a Dirichlet multivariate process prior is considered. A simulation study is given to illustrate the effectiveness of the proposed methods and the influence of the degree of dependence on the performance of the estimation procedure. It is shown that the dependence between failure modes should not be ignored because ignoring the dependence may lead to serious errors. A numerical example based on a life test of rolling ball bearings is presented as an application of the proposed methodology.

Nonparametric Bayesian Analysis for Masked Data From Hybrid Systems in Accelerated Lifetime Tests

Under constant stress accelerated test, we analyze the masked data from hybrid systems including series–parallel system and parallel–series system. The Bayesian posterior distribution and estimates of components’ subsurvival functions are obtained by assuming a prior of the multivariate Dirichlet process. By establishing a relationship between subsurvival and survival functions of components, the estimates of reliabilities for components and system are derived from the estimates of subsurvival functions. It is not confined to the common restriction that the sets of discontinuity points of the survival functions have to be disjointed. For a complex system, we represent it to the proposed series–parallel system or parallel–series system. Thus, the nonparametric Bayesian approach is also applicable for the complex system with s-independent components. A simulated example is presented to demonstrate the efficiency of the method. Finally, the method is applied to a real data of coal wine monitoring power.

Reliability nonparametric Bayesian estimation for the masked data of parallel systems in step-stress accelerated life tests

The accelerated life tests with two groups of step-stress levels are considered for the parallel systems, in which the masked data are observed. We assume the power function as accelerated function for the life transformation between different stress levels, which covers the shortage of linear accelerated function. Then the estimators of coefficient vectors in accelerated function are obtained. The relationship among survival and subsurvival functions of components is discussed under a regular condition. It does not require that the discontinuity points of survival functions have to be disjointed. With the help of that transformational relationship, nonparametric Bayesian estimators of reliability functions corresponding to any components’ set are derived. Due to the complexity of masked data, a group of computational algorithms are developed to obtain the estimates. Finally, a simulated example is presented to illustrate the proposed nonparametric Bayesian method.

Empirical likelihood ratio test for symmetry against type I bias with applications to competing risks

A random variable X with cumulative distribution function F is said to have a symmetric distribution about θ if and only if X−θ and−X+θ are identically distributed. Different types of partial skewness and one-sided bias are obtained by looking at different types of orderings between the distributions of X−θ and−X+θ. For example, X, or equivalently F, is said to have type I bias about θ if X−θ is stochastically larger than−X+θ. In this paper, we assume that F is continuous, θ is known and develops an empirical likelihood ratio type test for testing for symmetry about θ against this type of alternative. This test is shown to be asymptotically distribution free and the results of a simulation study show that it outperforms in terms of power, a test developed for the same problem in Alfieri and El Barmi [(2005), ‘Nonparametric Estimation of a Distribution Function with Type I Bias with Applications to Competing Risks’, Journal of Nonparametric Statistics, 17, 319–333]. It turns out that the results developed here can be extended in a natural way to compare the sub-survival functions corresponding to two risks in a competing risks setting. We show how this can be done and illustrate our theoretical results with a real life example.

On Bayesian estimation of a survival curve: comparative study and examples

A nonparametric Bayesian approach is applied to estimate a survival curve by means of a functional of the subsurvival functions associated with censored and non-censored events. In order to actually compute the Bayesian estimator, a numerical algorithm based on the Runge-Kutta fourth-order method is introduced. It provides good accuracy and is simple to program. Using a simulated data set, the performance of the Bayesian estimator is compared to the Product-Limit. A descriptive analysis of the results from the simulations is presented. The conclusions are given in terms of the proportion of the censored data and sample size. Also, the predictive value of the estimations is investigated using a cross-validation measure, namely the mean square predictive value. The numerical methodology is illustrated considering the original Kaplan-Meier data. Finally, the Bayesian analysis is applied to a real case of cervix uterine cancer, where the elicitation of the prior distribution considers the high proportion of censoring in the sample.

Comparing relative importance of the same component in a series system in two environments

The relative performance of a component of a series system in two different environments is considered. The conditional probability of the failure of the system due to the failure of the specified component given that the system failed before time t is regarded as a measure of relative importance of the component to the system. A U-statistic test for checking the equality of the relative importance of the component to the system in two different environments against the alternative that the relative importance is smaller in one of the environments, is proposed. Some simulation results for estimating the power of the test are reported. The proposed test is applied to one real data set and it is seen that a different aspect of the data is brought out by this comparison than that by the comparisons of the absolute importance functions such as the subsurvival functions, considered in earlier studies.

Competing risk and the Cox proportional hazard model

We propose a heuristic for evaluating model adequacy for the Cox proportional hazard model by comparing the population cumulative hazard with the baseline cumulative hazard. We illustrate how recent results from the theory of competing risk can contribute to analysis of data with the Cox proportional hazard model. A classical theorem on independent competing risks allows us to assess model adequacy under the hypothesis of random right censoring, and a recent result on mixtures of exponentials predicts the patterns of the conditional subsurvival functions of random right censored data if the proportional hazard model holds.

On Testing Dependence between Time to Failure and Cause of Failure via Conditional Probabilities

Abstract. Dependence structures between the failure time and the cause of failure are expressed in terms of the monotonicity properties of the conditional probabilities involving the cause of failure and the failure time. These properties of the conditional probabilities are used for testing four types of departures from the independence of the failure time and the cause of failure and tests based on U‐statistics are proposed. In the process, a concept of concordance and discordance between a continuous and a binary variable is introduced to propose a statistical test. The proposed tests are applied to two illustrative applications.

Nonparametric estimation from current status data with competing risks

SUMMARY A great deal of recent attention has focused on the estimation of survival distributions based on current status data, an extreme form of interval censored data. This particular data structure arises in a wide variety of applications where cross-sectional observation either naturally occurs or is preferred to more traditional forms of follow-up. Here we consider current status data in the context of competing risks. We briefly consider simple parametric models as a backdrop to nonparametric procedures. We make some brief comparisons and remarks regarding the nonparametric maximum likelihood estimator. The ideas are illustrated on the data of Krailo & Pike (1983) which considers estimation of the age distribution at both natural and operative menopause. We also consider the case where there is exact observation of failure times due to one of the competing risks when failure occurs prior to the monitoring time. Basic techniques of survival analysis focus on estimation of the survival distribution based on various forms of censored data. An extreme form of censoring arises where the only information on studied individuals is their survival status at a single monitoring time. This particular data structure is known as current status data. Nonparametric estimation of the survival function and semiparametric techniques for related regression models, based on current status data, have been much studied of late. Here, we consider estimation of sub-survival functions based on current status observations in the presence of competing risks. For simplicity, we describe only two competing risks, but the ideas are easily extended when there are more than two possible causes of failure. Let )j denote the cause-specific hazard function for cause j = 1, 2 (Kalbfleisch & Prentice, 2002, p. 252). If T and J are the random variables that measure time to failure and cause of failure, respectively, the two sub-distribution functions of interest are

Bayesian nonparametric estimation in a series system or a competing-risks model

This article presents a Bayesian nonparametric approach to the estimation of a system and its components' survival functions arising from observing the failure of a series system or a competing risk model. A Dirichlet multivariate process is used as a prior for the vector of the components' random subsurvival function to derive Bayes estimator of the survival function when the cause of failure belongs to a certain risk subset. This is done as follows. First, Peterson's formula is evaluated using the Bayes estimators of the subsurvival functions corresponding to the risk subset, to obtain a plugged-in nonparametric estimator of the survival function associated with the risk subset. Then, using the product-integration approach, it is proved that this nonparametric estimator is in fact the Bayes estimator of the survival function corresponding to the risk subset under quadratic loss function and the Dirichlet multivariate process. The weak convergence and the strong consistency of the estimator is established. The special case when the system has only two components corresponds to well studied randomly censored model.

Comparing sub-survival functions in a competing risks model.

In the competing risks literature, one usually compares whether two risks are equal or whether one is "more serious." In this paper, we propose tests for the equality of two competing risks against an ordered alternative specified by their sub-survival functions. These tests are naturally developed as extensions of those based on hazard rates and cumulative incidence functions. We note that the interpretation of the new test results is more direct compared to the situation when the hypotheses are framed in terms of their cumulative incidence functions. The proposed tests are of the Kolmogrov-Smirnov type, based on maximum differences between sub-survival functions. Our simulation studies indicate that they are excellent competitors of the existing tests, that are based mainly on differences between cumulative incidence functions. A numerical example will demonstrate the advantages of the proposed tests.

Comparative evaluation of maintenance performance using subsurvival functions

Subsurvival functions are applied to operational data for the control rod drive systems of Nordic nuclear reactors to evaluate maintenance performance. Competing failure modes are preventive and corrective maintenance. Maintenance indicators are defined and evaluated for 8 plants.

A characterization of marginal distributions of (possibly dependent) lifetime variables which right censor each other

It is well known that the joint distribution of a pair of lifetime variables $X_1$ and $X_2$ which right censor each other cannot be specified in terms of the subsurvival functions $$P(X_2 > X_1 > x), \quad P(X_1 > X_2 > x)$ \quad \text{and} \quad $P(X_1 = X_2 > x)$$ without additional assumptions such as independence of $X_1$ and $X_2$. For many practical applications independence is an unacceptable assumption, for example, when $X_1$ is the lifetime of a component subjected to maintenance and $X_2$ is the inspection time. Peterson presented lower and upper bounds for the marginal distributions of $X_1$ and $X_2$, for given subsurvival functions. These bounds are sharp under nonatomicity conditions. Surprisingly, not every pair of distribution functions between these bounds provides a feasible pair of marginals. Crowder recognized that these bounds are not functionally sharp and restricted the class of functions containing all feasible marginals. In this paper we give a complete characterization of the possible marginal distributions of these variables with given sub-survival functions, without any assumptions on the underlying joint distribution of $X_1, X_2$. Furthermore, a statistical test for an hypothesized marginal distribution of $(X_1$ based on the empirical subsurvival functions is developed. The characterization is generalized from two to any number of variables.

Hierarchical Bayes estimation for the exponential-multinomial model in reliability and competing risks

The exponential-multinomial distribution arises from: (1) observing the system failure of a series system with p components having independent exponential lifetimes, or (2) a competing-risks model with p sources of failure, as well as (3) the Marshall-Olkin multivariate exponential distribution under a series sampling scheme. Hierarchical Bayes (HB) estimators of the component sub-survival function and the system reliability are obtained using the Gibbs sampler. A large-sample approximation of the posterior pdf is used to derive the HB estimators of the parameters of the model with respect to the quadratic loss function. The exact risk of the HE estimator is obtained and is compared with those corresponding to some other estimators such as Bayes, maximum likelihood, and minimum variance unbiased estimators.

The total time on test statistic and age-dependent censoring

It is known that observations from any censored life (or competing risk) process can be described by a random censoring (or independent risk) model. On the other hand it is impossible to verify that the censoring is really random. A class of age-dependent censoring processes are defined and the class of corresponding subsurvival functions is described. Exponential life variables censored by independent variables can also be described by age-dependent censoring models if the censoring variable is DFR. The total time on test statistic consistently estimates the expected life of an exponential life variable under random right censoring. If the censoring is age-dependent, the Total Time on Test statistic will severly overestimate the expected life of the variable of interest. Care should be taken to motivate the application of the total time on test statistic in such situations.

Estimation of the subsurvival function for time-to-tumor in survival/sacrifice experiments

We propose a new closed-form nonparametric estimator for the subsurvival function for time-to-tumor. This estimator combines sacrifice data and data from natural deaths of animals without assumptions concerning the lethality of the tumor and without specification of cause of death.

Estimation and Asymptotic Properties of the Distribution of Tune-to-Tumour in Carcinogenesis Experiments

The problem of the estimation of the cumulative hazard function for time-to-tumour is considered for tumours that are not clinically observable and irreversible. The problem is posed in a compartmental model framework like that of Dewanji and Kalbfleisch (1986). The essential data from a single animal that are considered are: its group, age at death, mode of death (natural or sacrifice), and a list of the tumours present at death. No assumption concerning the lethality of the tumour of interest is required. First an isotonic estimator is proposed for the subsurvival function for time-to-tumour, based on the estimator derived by Gómez and Van Ryzin (1987) for the subdistribution function for time-to-tumour which maintains the asymptotic behaviour, that is, it is consistent and asymptotically normal. Then, the cumulative hazard estimator is defined which is itself monotonically increasing, consistent, and asymptotically normal. The limiting variance is computed and a consistent estimator is given. The observable data are summarized in a 4-variate counting process. The intensity of this process and the corresponding martingales were derived by Gómez and Van Ryzin (1987). The martingale theory can then be applied to obtain the desired results. In particular, the limiting variance is easily achieved. Finally, the proposed methods are illustrated with Berlin, Brodsky, and Clifford data on glomerulosclerosis disease in female RF mice.

Expressing the Kaplan-Meier Estimator as a Function of Empirical Subsurvival Functions

Abstract The Kaplan-Meier estimator for the survival function in the censored data problem can be expressed for finite samples as an explicit function of two empirical subsurvival functions. This function is the natural one that expresses the survival function in terms of the sub-survival functions. As an illustration of the usefulness of expressing the Kaplan-Meier estimator in this way, the strong consistency of the Kaplan-Meier estimator is easily proved.

Expressing the Kaplan-Meier Estimator as a Function of Empirical Subsurvival Functions

Expressing the Kaplan-Meier Estimator as a Function of Empirical Subsurvival Functions