On assessing independence of Competing Risks when failure times are discrete

The traditional approach to modelling for Competing Risks, via a multivariate distribution of latent failure times, is very natural for many applications but suffers from a well-documented problem of identifiability. However, the demonstrations of this problem in the literature apply to essentially continuous latent failure times where any atoms of probability in their distributions are not too intrusive. It is shown in this paper that for discrete failure times the classic results on the identifiability problem concerning the existence of equivalent independent risks are incomplete.     

Characterizations of Competing Risks in Terms of Independent-Risks Proxy Models

In competing risks a failure time T and a cause C , one of p possible, are observed. A traditional representation is via a vector ( T ₁, ..., T_p ) of latent failure times such that T = min( T ₁, ..., T_p ); C is defined by T = T_C in the basic situation of failure from a single cause. There are several results in the literature to the effect that a joint distribution for ( T ₁, ..., T_p ), in which the T_j are independent, can always be constructed to yield any given bivariate distribution for ( C , T ). For this reason the prevailing wisdom is that independence cannot be assessed from competing risks data, not even with arbitrarily large sample sizes (e.g. Prentice et al. , 1978). A result was given by Crowder (1996) which shows that, under certain circumstances, independence can be assessed. The various results will be drawn together and a complete characterization can now be given in terms of independent-risks proxy models.     

Inference for the Dependent Competing Risks Model with Masked Causes of Failure

The competing risks model is useful in settings in which individuals/units may die/fail for different reasons. The cause specific hazard rates are taken to be piecewise constant functions. A complication arises when some of the failures are masked within a group of possible causes. Traditionally, statistical inference is performed under the assumption that the failure causes act independently on each item. In this paper we propose an EM-based approach which allows for dependent competing risks and produces estimators for the sub-distribution functions. We also discuss identifiability of parameters if none of the masked items have their cause of failure clarified in a second stage analysis (e.g. autopsy). The procedures proposed are illustrated with two datasets.     

The Identifiability of Dependent Competing Risks Models Induced by Bivariate Frailty Models

In this paper, we propose to use a special class of bivariate frailty models to study dependent censored data. The proposed models are closely linked to Archimedean copula models. We give sufficient conditions for the identifiability of this type of competing risks models. The proposed conditions are derived based on a property shared by Archimedean copula models and satisfied by several well‐known bivariate frailty models. Compared with the models studied by Heckman and Honoré and Abbring and van den Berg, our models are more restrictive but can be identified with a discrete (even finite) covariate. Under our identifiability conditions, expectation–maximization (EM) algorithm provides us with consistent estimates of the unknown parameters. Simulation studies have shown that our estimation procedure works quite well. We fit a dependent censored leukaemia data set using the Clayton copula model and end our paper with some discussions. © 2014 Board of the Foundation of the Scandinavian Journal of Statistics     

Dealing with Ties in Failure Time Data

In dealing with ties in failure time data the mechanism by which the data are observed should be considered. If the data are discrete, the process is relatively simple and is determined by what is actually observed. With continuous data, ties are not supposed to occur, but they do because the data are grouped into intervals (even if only rounding intervals). In this case there is actually a non–identifiability problem which can only be resolved by modelling the process. Various reasonable modelling assumptions are investigated in this paper. They lead to better ways of dealing with ties between observed failure times and censoring times of different individuals. The current practice is to assume that the censoring times occur after all the failures with which they are tied.     

Modeling the Agreement of Discrete Bivariate Survival Times using Kappa Coefficient

Using local kappa coefficients, we develop a method to assess the agreement between two discrete survival times that are measured on the same subject by different raters or methods. We model the marginal distributions for the two event times and local kappa coefficients in terms of covariates. An estimating equation is used for modeling the marginal distributions and a pseudo-likelihood procedure is used to estimate the parameters in the kappa model. The performance of the estimation procedure is examined through simulations. The proposed method can be extended to multivariate discrete survival distributions.     

A Class of Non-parametric Tests in the Competing Risks Model for Comparing Two Samples

We consider a model when a process involving the production of elements is under inspection. The elements have possible failures due to competing risks. We assume the availability of a data set of failure times, D₁, obtained when the process is under control. Our main goal is to test if the failure rates in D₁ are equal to or less than the failure rates in another data set D₂, against undesirable neighbouring alternatives. A class of tests based on a two-dimensional vector statistic is obtained. Linear test statistics with weight functions giving optimal local asymptotic power are derived. Martingale techniques are used. An example illustrates the derivation of reasonable tests     

Nonparametric Test for Doubly Interval-Censored Failure Time Data

This paper considers comparison of discrete failure time distributions when the survival time of interest measures elapsed time between two related events and observations on the occurrences of both events could be interval-censored. This kind of data is often referred to as doubly interval-censored failure time data. If the occurrence of the first event defining the survival time can be exactly observed, the data are usually referred to as interval-censored data. For the comparison problem based on interval-censored failure time data, Sun (1996) proposed a nonparametric test procedure. In this paper we generalize the procedure given in Sun (1996) to doubly interval-censored data case and the generalized test is evaluated by simulations.     

A Discrete Time Parametric Model for the Analysis of Failure Time Data

This paper introduces a parametric discrete failure time model which allows a variety of smooth hazard function shapes, including shapes which are not readily available with continuous failure time models. The model is easy to fit, and statistical inference is simple. Further, it is readily extended to allow for differences between subjects while retaining the ease of fit and simplicity of statistical inference. The performance of the discrete time analysis is demonstrated by application to several data sets.     

Estimation of the marginal survival time in the presence of dependent competing risks using inverse probability of censoring weighted (IPCW) methods

In medical studies, there is interest in inferring the marginal distribution of a survival time subject to competing risks. The Kyushu Lipid Intervention Study (KLIS) was a clinical study for hypercholesterolemia, where pravastatin treatment was compared with conventional treatment. The primary endpoint was time to events of coronary heart disease (CHD). In this study, however, some subjects died from causes other than CHD or were censored due to loss to follow-up. Because the treatments were targeted to reduce CHD events, the investigators were interested in the effect of the treatment on CHD events in the absence of causes of death or events other than CHD. In this paper, we present a method for estimating treatment group-specific marginal survival curves of time-to-event data in the presence of dependent competing risks. The proposed method is a straightforward extension of the Inverse Probability of Censoring Weighted (IPCW) method to settings with more than one reason for censoring. The results of our analysis showed that the IPCW marginal incidence for CHD was almost the same as the lower bound for which subjects with competing events were assumed to be censored at the end of all follow-up. This result provided reassurance that the results in KLIS were robust to competing risks.     

A New lifetime model for multivariate survival data with a surviving fraction

In this paper we propose a new lifetime model for multivariate survival data with a surviving fraction. We develop this model assuming that there are m types of unobservable competing risks, where each risk is related to a time of the occurrence of an event of interest. We explore the use of Markov chain Monte Carlo methods to develop a Bayesian analysis for the proposed model. We also perform a simulation study in order to analyse the frequentist coverage probabilities of credible interval derived from posteriors. Our modelling is illustrated through a real data set.     

A Multivariate Model for Repeated Failure Time Measurements

A parametric multivariate failure time distribution is derived from a frailty-type model with a particular frailty distribution. It covers as special cases certain distributions which have been used for multivariate survival data in recent years. Some properties of the distribution are derived: its marginal and conditional distributions lie within the parametric family, and association between the component variates can be positive or, to a limited extent, negative. The simple closed form of the survivor function is useful for right-censored data, as occur commonly in survival analysis, and for calculating uniform residuals. Also featured is the distribution of ratios of paired failure times. The model is applied to data from the literature     

Flexible parametric estimation technique for a competing risks model with unobserved heterogeneity: a Monte Carlo study

We analyse a flexible parametric estimation technique for a competing risks (CR) model with unobserved heterogeneity, by extending a local mixed proportional hazard single risk model for continuous duration time to a local mixture CR (LMCR) model for discrete duration time. The state-specific local hazard function for the LMCR model is per definition a valid density function if we have either one or two destination states. We conduct Monte Carlo experiments to compare the estimated parameters of the LMCR model, and to compare the estimated parameters of a CR model based on a Heckman–Singer-type (HS-type) technique, with the data-generating process parameters. The Monte Carlo results show that the LMCR model performs better or at least as good as the HS-type model with respect to the estimated structure parameters in most of the cases, but relatively poorer with respect to the estimated duration-dependence parameters.     

Methods for Bivariate Survival Data with Mismeasured Covariates Under an Accelerated Failure Time Model

Accelerated failure time models are useful in survival data analysis, but such models have received little attention in the context of measurement error. In this paper we discuss an accelerated failure time model for bivariate survival data with covariates subject to measurement error. In particular, methods based on the marginal and joint models are considered. Consistency and efficiency of the resultant estimators are investigated. Simulation studies are carried out to evaluate the performance of the estimators as well as the impact of ignoring the measurement error of covariates. As an illustration we apply the proposed methods to analyze a data set arising from the Busselton Health Study (Knuiman et al., 1994 Knuiman , M. W. , Cullent , K. J. , Bulsara , M. K. , Welborn , T. A. , Hobbs , M. S. T. ( 1994 ). Mortality trends, 1965 to 1989, in Busselton, the site of repeated health surveys and interventions . Austral. J. Public Health 18 : 129 – 135 . [CSA] [Crossref], [PubMed] , [Google Scholar]).     

Test for independence between time to failure and cause of failure in competing risks with k causes

In this paper, we develop a simple nonparametric test for testing the independence of time to failure and cause of failure in competing risks set up. We generalise the test to the situation where failure data is right censored. We obtain the asymptotic distribution of the test statistics for complete and censored data. The efficiency loss due to censoring is studied using Pitman efficiency. The performance of the proposed test is evaluated through simulations. Finally we illustrate our test procedure using three real data sets.     

Comparison Between Two Partial Likelihood Approaches for the Competing Risks Model with Missing Cause of Failure

In many clinical studies where time to failure is of primary interest, patients may fail or die from one of many causes where failure time can be right censored. In some circumstances, it might also be the case that patients are known to die but the cause of death information is not available for some patients. Under the assumption that cause of death is missing at random, we compare the Goetghebeur and Ryan (1995, Biometrika, 82, 821–833) partial likelihood approach with the Dewanji (1992, Biometrika, 79, 855–857)partial likelihood approach. We show that the estimator for the regression coefficients based on the Dewanji partial likelihood is not only consistent and asymptotically normal, but also semiparametric efficient. While the Goetghebeur and Ryan estimator is more robust than the Dewanji partial likelihood estimator against misspecification of proportional baseline hazards, the Dewanji partial likelihood estimator allows the probability of missing cause of failure to depend on covariate information without the need to model the missingness mechanism. Tests for proportional baseline hazards are also suggested and a robust variance estimator is derived.     

A Score Test for the Multivariate Burr and Other Weibull Mixture Distributions

The p -variate Burr distribution has been derived, developed, discussed and deployed by various authors. In this paper a score statistic for testing independence of the components, equivalent to testing for p independent Weibull against a p -variate Burr alternative, is obtained. Its null and non-null properties are investigated with and without nuisance parameters and including the possibility of censoring. Two applications to real data are described. The test is also discussed in the context of other Weibull mixture models.     

Analysis of repeated failures or durations, with application to shunt failures for patients with paediatric hydrocephalus

We consider studies involving the repeated occurrence of certain events, in which the emphasis is on the gaps or times between events. Interesting methodological issues arise in such situations, including the validity of semiparametric methods for multiplicative hazard-based models and the possibilities for marginal analysis of successive gap times. We discuss these and other points in conjunction with an examination of observational data on repeated shunt failures for a population of children with hydrocephalus.     

A polar coordinate transformation for estimating bivariate survival functions with randomly censored and truncated data

This paper proposes a new estimator for bivariate distribution functions under random truncation and random censoring. The new method is based on a polar coordinate transformation, which enables us to transform a bivariate survival function to a univariate survival function. A consistent estimator for the transformed univariate function is proposed. Then the univariate estimator is transformed back to a bivariate estimator. The estimator converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. Consistent truncation probability estimate is also provided. Numerical studies show that the distribution estimator and truncation probability estimator perform remarkably well.