Confidence bounds for the joint survival function in the dependent competing risks setup

In the competing risks set up with two dependent competing risks, the joint distribution of (X₁,X₂), the latent lifetimes of the system under the two risks, is not identifiable on the basis of the distribution of the actual observation (T, δ) where T = min(X₁, X₂) and δ = I(T=X₁), Using Peterson's (1976) bounds, we have obtained conservative pointwise as well as simultaneous confidence bounds for the unidentifiable joint survival function. In an example we evaluate the confidence bounds and Indicate where the estimated joint survival function in the independent case, lies within them.     

Detecting change in a hazard regression model with right-censoring

The hazard function plays an important role in survival analysis and reliability, since it quantifies the instantaneous failure rate of an individual at a given time point t, given that this individual has not failed before t. In some applications, abrupt changes in the hazard function are observed, and it is of interest to detect the location of such a change. In this paper, we consider testing of existence of a change in the parameters of an exponential regression model, based on a sample of right-censored survival times and the corresponding covariates. Likelihood ratio type tests are proposed and non-asymptotic bounds for the type II error probability are obtained. When the tests lead to acceptance of a change, estimators for the location of the change are proposed. Non-asymptotic upper bounds of the underestimation and overestimation probabilities are obtained. A short simulation study illustrates these results.     

Nonparametric bounds for the survivor function under general dependent truncation

Truncation occurs in cohort studies with complex sampling schemes. When truncation is ignored or incorrectly assumed to be independent of the event time in the observable region, bias can result. We derive completely nonparametric bounds for the survivor function under truncation and censoring; these extend prior nonparametric bounds derived in the absence of truncation. We also define a hazard ratio function that links the unobservable region in which event time is less than truncation time, to the observable region in which event time is greater than truncation time, under dependent truncation. When this function can be bounded, and the probability of truncation is known approximately, it yields narrower bounds than the purely nonparametric bounds. Importantly, our approach targets the true marginal survivor function over its entire support, and is not restricted to the observable region, unlike alternative estimators. We evaluate the methods in simulations and in clinical applications.     

A Mixture Model of Proportional Reversed Hazard Rate

This article investigates properties of mixture model of proportional reversed hazard rate. Firstly, the mixing random variable and the overall population variable are proved to be positively likelihood dependent. Secondly, lower bounds for the distribution function as well as the conditional distribution are established in the case that the mixing variable belongs to certain nonparametric classes. Finally, some stochastic orders on the mixing (baseline) variables are proved to be translated to the corresponding overall population variables.     

A Kernel Smooth Approach for Joint Modeling of Accelerated Failure Time and Longitudinal Data

Joint likelihood approaches have been widely used to handle survival data with time-dependent covariates. In construction of the joint likelihood function for the accelerated failure time (AFT) model, the unspecified baseline hazard function is assumed to be a piecewise constant function in the literature. However, there are usually no close form formulas for the regression parameters, which require numerical methods in the EM iterations. The nonsmooth step function assumption leads to very spiky likelihood function which is very hard to find the globe maximum. Besides, due to nonsmoothness of the likelihood function, direct search methods are conducted for the maximization which are very inefficient and time consuming. To overcome the two disadvantages, we propose a kernel smooth pseudo-likelihood function to replace the nonsmooth step function assumption. The performance of the proposed method is evaluated by simulation studies. A case study of reproductive egg-laying data is provided to demonstrate the usefulness of the new approach.     

Consistency of the Generalized MLE with Interval-Censored and Masked Competing Risks Data

We consider nonparametric estimation based on interval-censored competing risks data with masked failure cause. The generalized maximum likelihood estimator of the joint survival function of the failure time and the failure cause is studied under mixed case interval censorship and random partition masking. Strong consistency in the L ₁(μ)-topology is established for some finite measure μ which is derived from the joint censoring and masking distribution. Under additional regularity assumptions we also establish the strong consistencies in the topologies of weak convergence, point-wise convergence, and uniform convergence.     

Testing dependence between the failure time and failure modes: An application of enlarged filtration

The model of independent competing risks provides no information for the assessment of competing failure modes if the failure mechanisms underlying these modes are coupled. Models for dependent competing risks in the literature can be distinguished on the basis of the functional behaviour of the conditional probability of failure due to a particular failure mode given that the failure time exceeds a fixed time, as a function of time. There is an interesting link between monotonicity of such conditional probability and dependence between failure time and failure mode, via crude hazard rates. In this paper, we propose tests for testing the dependence between failure time and failure mode using the crude hazards and using the conditional probabilities mentioned above. We establish the equivalence between the two approaches and provide an asymptotically efficient weight function under a sequence of local alternatives. The tests are applied to simulated data and to mortality follow-up data.     

Modeling systems with two dependent components under bivariate shock models

Series and parallel systems consisting of two dependent components are studied under bivariate shock models. The random variables N₁ and N₂ that represent respectively the number of shocks until failure of component 1 and component 2 are assumed to be dependent and phase-type. The times between successive shocks are assumed to follow a continuous phase-type distribution, and survival functions and mean time to failure values of series and parallel systems are obtained in matrix forms. An upper bound for the joint survival function of the components is also provided under the particular case when the times between shocks follow exponential distribution.     

Restricted tests for testing independence of time to failure and cause of failure in a competing-risks model

We consider the competing-risks problem without making any assumption concerning the independence of the risks. Maximum-likelihood estimates of the cause-specific hazard rates are obtained under the condition that their ratio is monotone. We also consider the likelihood-ratio test for testing the proportionality of two cause-specific hazard rates against the alternative that the ratio of these two hazard rates is monotonic. This testing problem is equivalent to testing independence against likelihood-ratio dependence of the time to failure and the cause of failure in the competing-risks setup. We allow for random censoring on the right. The asymptotic null distribution of the test statistic is obtained and is found to be of the chi-bar-square type. The problem is extended to the case of more than two risks. A numerical example is given to illustrate the procedure.     

Extended Hazard Regression Model for Reliability and Survival Analysis

We propose an extended hazard regression model which allows the spread parameter to be dependent on covariates. This allows a broad class of models which includes the most common hazard models, such as the proportional hazards model, the accelerated failure time model and a proportional hazards/accelerated failure time hybrid model with constant spread parameter. Simulations based on sub-classes of this model suggest that maximum likelihood performs well even when only small or moderate-size data sets are available and the censoring pattern is heavy. The methodology provides a broad framework for analysis of reliability and survival data. Two numerical examples illustrate the results.     

Non parametric hazard estimation with dependent censoring using penalized likelihood and an assumed copula

This article introduces a novel non parametric penalized likelihood hazard estimation when the censoring time is dependent on the failure time for each subject under observation. More specifically, we model this dependence using a copula, and the method of maximum penalized likelihood (MPL) is adopted to estimate the hazard function. We do not consider covariates in this article. The non negatively constrained MPL hazard estimation is obtained using a multiplicative iterative algorithm. The consistency results and the asymptotic properties of the proposed hazard estimator are derived. The simulation studies show that our MPL estimator under dependent censoring with an assumed copula model provides a better accuracy than the MPL estimator under independent censoring if the sign of dependence is correctly specified in the copula function. The proposed method is applied to a real dataset, with a sensitivity analysis performed over various values of correlation between failure and censoring times.     

A unimodal hazard rate function and its failure distribution

A unimodal hazard rate function is suggested to model a failure rate that has a relatively high rate of failure in the middle of expected life time. This unimodal hazard rate function has two shape parameters. One of the parameters indicates the location (time) of the mode and the other controls the height of the mode. In effect, these two parameters index the class of unimodal hazard rate functions. The reliability function and the failure density function of the unimodal hazard rate function are relatively uncomplicated and mathematically tractable. The properties of the unimodal hazard rate function and the failure density function are investigated. The maximum likelihood method is used for the inference concerning the two parameters and an example based on real data is presented. This unimodal hazard rate function is particularly useful when the time of the peak failure rate is of prime interest. The failure distribution provides a practical way of estimating the peak failure time.     

Estimating the turning point of the log-logistic hazard function in the presence of long-term survivors with an application for uterine cervical cancer data

The hazard function plays an important role in cancer patient survival studies, as it quantifies the instantaneous risk of death of a patient at any given time. Often in cancer clinical trials, unimodal hazard functions are observed, and it is of interest to detect (estimate) the turning point (mode) of hazard function, as this may be an important measure in patient treatment strategies with cancer. Moreover, when patient cure is a possibility, estimating cure rates at different stages of cancer, in addition to their proportions, may provide a better summary of the effects of stages on survival rates. Therefore, the main objective of this paper is to consider the problem of estimating the mode of hazard function of patients at different stages of cervical cancer in the presence of long-term survivors. To this end, a mixture cure rate model is proposed using the log-logistic distribution. The model is conveniently parameterized through the mode of the hazard function, in which cancer stages can affect both the cured fraction and the mode. In addition, we discuss aspects of model inference through the maximum likelihood estimation method. A Monte Carlo simulation study assesses the coverage probability of asymptotic confidence intervals.     

General characterization theorems via the mean absolute deviation

The Chernoff–Borovkov–Utev inequality resulted owing to earlier inequalities established by Chernoff (1981) and Borovkov and Utev (1983), respectively, giving bounds for the variance of functions of normal r.v.’s and leading to characterizations of normality. Subsequently, several analytic properties of variance bounds and other relevant results were established by others. Defining the mean absolute deviation (about a median) as E|X−med(X)| where med(X) is a median of the distribution of the random variable X, Freimer and Mudholkar (1991) gave a bound for the mean absolute deviation of a certain real-valued function of an absolutely continuous random variable (w.r.t. Lebesgue measure) and Korwar (1991) presented an analogue of this in the discrete case; these authors, also, characterized the Laplace and a mixture of two Waring distributions via the respective bounds.We extend these latter results theorems to the case where the distributions are not necessarily purely discrete or absolutely continuous, via the approach of Alharbi and Shanbhag (1996). The results in Freimer and Mudholkar (1991) and Korwar (1991) are now corollaries to our findings. Also, following Alharbi and Shanbhag (1996), we relate these results to the modified version of Cox’s representation for a survival function in terms of the hazard measure, given in Kotz and Shanbhag (1980). (The original version of the representation mentioned had appeared in Cox (1972).)     

The Weibull–Pareto Composite Family with Applications to the Analysis of Unimodal Failure Rate Data

The Weibull distribution is composited with Pareto model to obtain a flexible, reliable long-tailed parametric distribution for modeling unimodal failure rate data. The hazard function of the composite family accommodates decreasing and unimodal failure rates, which are separated by the boundary line of the space of shape parameter, gamma, when it equals to a known constant. The least square and maximum likelihood parameter estimation techniques are discussed. The advantages of using the proposed family are demonstrated and compared by illustrating well-known examples: guinea pigs survival time data, head and neck cancer data, and nasopharynx cancer survival data.     

Detecting multiple change points in piecewise constant hazard functions

The National Cancer Institute (NCI) suggests a sudden reduction in prostate cancer mortality rates, likely due to highly successful treatments and screening methods for early diagnosis. We are interested in understanding the impact of medical breakthroughs, treatments, or interventions, on the survival experience for a population. For this purpose, estimating the underlying hazard function, with possible time change points, would be of substantial interest, as it will provide a general picture of the survival trend and when this trend is disrupted. Increasing attention has been given to testing the assumption of a constant failure rate against a failure rate that changes at a single point in time. We expand the set of alternatives to allow for the consideration of multiple change-points, and propose a model selection algorithm using sequential testing for the piecewise constant hazard model. These methods are data driven and allow us to estimate not only the number of change points in the hazard function but where those changes occur. Such an analysis allows for better understanding of how changing medical practice affects the survival experience for a patient population. We test for change points in prostate cancer mortality rates using the NCI Surveillance, Epidemiology, and End Results dataset.     

Hazard Rate Properties of Systems with Two Components

In this article, we study the hazard rate ordering of lifetimes of two-component systems (series and parallel) by considering some bivariate distributions for the joint distribution of component lifetimes. Such that, we aim to investigate the lifetimes of systems with stochastically dependent and with stochastically independent components, the lifetimes of the components, and a stochastic ordering relation between these lifetimes. In addition to these, the mononotonicity of the hazard rates of the parallel and the series systems for the bivariate Farlie–Gumbel–Morgenstern (FGM) family is studied.     

Expected Power for the False Discovery Rate with Independence

The Benjamini-Hochberg procedure is widely used in multiple comparisons. Previous power results for this procedure have been based on simulations. This article produces theoretical expressions for expected power. To derive them, we make assumptions about the number of hypotheses being tested, which null hypotheses are true, which are false, and the distributions of the test statistics under each null and alternative. We use these assumptions to derive bounds for multiple dimensional rejection regions. With these bounds and a permanent based representation of the joint density function of the largest p-values, we use the law of total probability to derive the distribution of the total number of rejections. We derive the joint distribution of the total number of rejections and the number of rejections when the null hypothesis is true. We give an analytic expression for the expected power for a false discovery rate procedure that assumes the hypotheses are independent.     

A new test for single against competing risks models in duration analysis

This paper shows that the single-risk duration model with two event types is a limiting case of bivariate dependent competing risks model, where the joint distribution of event times are degenerate. Then a new test is proposed for the null hypothesis of single risk against dependent competing risks model under the proportional hazard model assumption.     

Inconsistency of the MLE for the Joint Distribution of Interval-Censored Survival Times and Continuous Marks

Abstract.  This paper considers the non-parametric maximum likelihood estimator (MLE) for the joint distribution function of an interval-censored survival time and a continuous mark variable. We provide a new explicit formula for the MLE in this problem. We use this formula and the mark-specific cumulative hazard function of Huang & Louis (1998) to obtain the almost sure limit of the MLE. This result leads to necessary and sufficient conditions for consistency of the MLE, which imply that the MLE is inconsistent in general. We show that the inconsistency can be repaired by discretizing the marks. Our theoretical results are supported by simulations.