Identifiability of masking probabilities in competing risks models with emphasis on Weibull models

Abstract

Lifetime data with masked failure causes arise in both reliability engineering and epidemiology. The phenomenon of masking occurs when a subject is exposed to multiple risks. A competing risks model with masking probabilities is widely used for the masked failure data. However, in many cases, the model suffers from an identification problem. We show that the identifiability of masking probabilities depends on both the structure of data and the cause-specific hazard functions. Motivated by this result, two existing solutions are reviewed and further improved.     

Vertical modeling: analysis of competing risks data with a cure fraction

In this paper, we extend the vertical modeling approach for the analysis of survival data with competing risks to incorporate a cure fraction in the population, that is, a proportion of the population for which none of the competing events can occur. The proposed method has three components: the proportion of cure, the risk of failure, irrespective of the cause, and the relative risk of a certain cause of failure, given a failure occurred. Covariates may affect each of these components. An appealing aspect of the method is that it is a natural extension to competing risks of the semi-parametric mixture cure model in ordinary survival analysis; thus, causes of failure are assigned only if a failure occurs. This contrasts with the existing mixture cure model for competing risks of Larson and Dinse, which conditions at the onset on the future status presumably attained. Regression parameter estimates are obtained using an EM-algorithm. The performance of the estimators is evaluated in a simulation study. The method is illustrated using a melanoma cancer data set.

     

Analysis of progressively type-I interval censored competing risks data for a class of an exponential distribution

ABSTRACT

In this paper, we consider some problems of point estimation and point prediction when the competing risks data from a class of exponential distribution are progressive type-I interval censored. The maximum likelihood estimation and mid-point approximation method are proposed for the estimations of parameters. Also several point predictors of censored units such as the maximum likelihood predictor, the best unbiased predictor and the conditional median predictor are obtained. The methods discussed here are applied when the lifetime distributions of the latent failure times are independent and Weibull-distributed. Finally a simulation study is given by using Monte-Carlo simulations to compare the performances of the different methods and one data analysis has been presented for illustrative purposes.     

Multiple imputation for competing risks regression with interval-censored data

ABSTRACT

We present here an extension of Pan's multiple imputation approach to Cox regression in the setting of interval-censored competing risks data. The idea is to convert interval-censored data into multiple sets of complete or right-censored data and to use partial likelihood methods to analyse them. The process is iterated, and at each step, the coefficient of interest, its variance–covariance matrix, and the baseline cumulative incidence function are updated from multiple posterior estimates derived from the Fine and Gray sub-distribution hazards regression given augmented data. Through simulation of patients at risks of failure from two causes, and following a prescheduled programme allowing for informative interval-censoring mechanisms, we show that the proposed method results in more accurate coefficient estimates as compared to the simple imputation approach. We have implemented the method in the MIICD R package, available on the CRAN website.     

Estimation of weibull parameters for grouped data with competing risks

We consider the problem of estimating Weibull parameters for grouped data when competing risks are present. We propose two simple methods of estimation and derive their asymptotic properties. A Monte Carlo study was carried out to evaluate the performance of these two methods.     

Parametric inference for quantile event times with adjustment for covariates on competing risks data

ABSTRACT

We propose parametric inferences for quantile event times with adjustment for covariates on competing risks data. We develop parametric quantile inferences using parametric regression modeling of the cumulative incidence function from the cause-specific hazard and direct approaches. Maximum likelihood inferences are developed for estimation of the cumulative incidence function and quantiles. We develop the construction of parametric confidence intervals for quantiles. Simulation studies show that the proposed methods perform well. We illustrate the methods using early stage breast cancer data.     

A dependent Dirichlet process model for survival data with competing risks

In this paper, we first propose a dependent Dirichlet process (DDP) model using a mixture of Weibull models with each mixture component resembling a Cox model for survival data. We then build a Dirichlet process mixture model for competing risks data without regression covariates. Next we extend this model to a DDP model for competing risks regression data by using a multiplicative covariate effect on subdistribution hazards in the mixture components. Though built on proportional hazards (or subdistribution hazards) models, the proposed nonparametric Bayesian regression models do not require the assumption of constant hazard (or subdistribution hazard) ratio. An external time-dependent covariate is also considered in the survival model. After describing the model, we discuss how both cause-specific and subdistribution hazard ratios can be estimated from the same nonparametric Bayesian model for competing risks regression. For use with the regression models proposed, we introduce an omnibus prior that is suitable when little external information is available about covariate effects. Finally we compare the models’ performance with existing methods through simulations. We also illustrate the proposed competing risks regression model with data from a breast cancer study. An R package “DPWeibull” implementing all of the proposed methods is available at CRAN.

     

An improper form of Weibull distribution for competing risks analysis with Bayesian approach

ABSTRACT

In survival analysis, individuals may fail due to multiple causes of failure called competing risks setting. Parametric models such as Weibull model are not improper that ignore the assumption of multiple failure times. In this study, a novel extension of Weibull distribution is proposed which is improper and then can incorporate to the competing risks framework. This model includes the original Weibull model before a pre-specified time point and an exponential form for the tail of the time axis. A Bayesian approach is used for parameter estimation. A simulation study is performed to evaluate the proposed model. The conducted simulation study showed identifiability and appropriate convergence of the proposed model. The proposed model and the 3-parameter Gompertz model, another improper parametric distribution, are fitted to the acute lymphoblastic leukemia dataset.     

Control chart for monitoring the Weibull shape parameter under two competing risks

ABSTRACT

In this paper, we propose a control chart to monitor the Weibull shape parameter where the observations are censored due to competing risks. We assume that the failure occurs due to two competing risks that are independent and follow Weibull distribution with different shape and scale parameters. The control charts are proposed to monitor one or both of the shape parameters of competing risk distributions and established based on the conditional expected values. The proposed control chart for both shape parameters is used in certain situations and allows to monitor both shape parameters in only one chart. The control limits depend on the sample size, number of failures due to each risk and the desired stable average run length (ARL). We also consider the estimation problem of the target parameters when the Phase I sample is incomplete. We assumed that some of the products that fail during the life testing have a cause of failure that is only known to belong to a certain subset of all possible failures. This case is known as masking. In the presence of masking, the expectation-maximization (EM) algorithm is proposed to estimate the parameters. For both cases, with and without masking, the behaviour of ARLs of charts is studied through the numerical methods. The influence of masking on the performance of proposed charts is also studied through a simulation study. An example illustrates the applicability of the proposed charts.     

Inference for cumulative incidence on discrete failure times with competing risks

In the competing risks analysis, most inferences have been developed based on continuous failure time data. However, failure times are sometimes observed as being discrete. We propose nonparametric inferences for the cumulative incidence function for pure discrete data with competing risks. When covariate information is available, we propose semiparametric inferences for direct regression modelling of the cumulative incidence function for grouped discrete failure time data with competing risks. Simulation studies show that the procedures perform well. The proposed methods are illustrated with a study of contraceptive use in Indonesia.     

Accelerated failure time models for the analysis of competing risks

Competing risks are common in clinical cancer research, as patients are subject to multiple potential failure outcomes, such as death from the cancer itself or from complications arising from the disease. In the analysis of competing risks, several regression methods are available for the evaluation of the relationship between covariates and cause-specific failures, many of which are based on Cox’s proportional hazards model. Although a great deal of research has been conducted on estimating competing risks, less attention has been devoted to linear regression modeling, which is often referred to as the accelerated failure time (AFT) model in survival literature. In this article, we address the use and interpretation of linear regression analysis with regard to the competing risks problem. We introduce two types of AFT modeling framework, where the influence of a covariate can be evaluated in relation to either a cause-specific hazard function, referred to as cause-specific AFT (CS-AFT) modeling in this study, or the cumulative incidence function of a particular failure type, referred to as crude-risk AFT (CR-AFT) modeling. Simulation studies illustrate that, as in hazard-based competing risks analysis, these two models can produce substantially different effects, depending on the relationship between the covariates and both the failure type of principal interest and competing failure types. We apply the AFT methods to data from non-Hodgkin lymphoma patients, where the dataset is characterized by two competing events, disease relapse and death without relapse, and non-proportionality. We demonstrate how the data can be analyzed and interpreted, using linear competing risks regression models.     

Bayesian Analysis of Masked Series System Lifetime Data

The problem of analyzing series system lifetime data with masked or partial information on cause of failure is recent, compared to that of the standard competing risks model. A generic Gibbs sampling scheme is developed in this article towards a Bayesian analysis for a general parametric competing risks model with masked cause of failure data. The masking probabilities are not subjected to the symmetry assumption and independent Dirichlet priors are used to marginalize these nuisance parameters. The developed methodology is illustrated for the case where the components of a series system have independent log-Normal life distributions by employing independent Normal-Gamma priors for these component lifetime parameters. The Gibbs sampling scheme developed for the required analysis can also be used to provide a Bayesian analysis of data arising from the conventional competing risks model of independent log-Normals, which interestingly has so far remained by and large neglected in the literature. The developed methodology is deployed to analyze a masked lifetime data of PS/2 computer systems.     

Generalized Supremum Tests for the Equality of Cause Specific Hazard Rates

In this paper we propose two new classes of asymptotically distribution-free Renyi-type tests for testing the equality of two risks in a competing risk model with possible censoring. This work extends the work of Aly, Kochar and McKeague [1994, Journal of American Statistical Association, 89, 994–999] and many of the existing tests for this problem belong to these newly proposed classes. The asymptotic properties of the proposed tests are investigated. Simulation studies are done to compare the performance with existing tests. A competing risks data set is analyzed to demonstrate the usefulness of the procedure.     

Identification of longitudinal biomarkers for survival by a score test derived from a joint model of longitudinal and competing risks data

In this paper, we consider joint modelling of repeated measurements and competing risks failure time data. For competing risks time data, a semiparametric mixture model in which proportional hazards model are specified for failure time models conditional on cause and a multinomial model for the marginal distribution of cause conditional on covariates. We also derive a score test based on joint modelling of repeated measurements and competing risks failure time data to identify longitudinal biomarkers or surrogates for a time to event outcome in competing risks data.     

A proportional hazards model for the analysis of doubly censored competing risks data

ABSTRACT

Competing risks data are common in medical research in which lifetime of individuals can be classified in terms of causes of failure. In survival or reliability studies, it is common that the patients (objects) are subjected to both left censoring and right censoring, which is refereed as double censoring. The analysis of doubly censored competing risks data in presence of covariates is the objective of this study. We propose a proportional hazards model for the analysis of doubly censored competing risks data, using the hazard rate functions of Gray (1988 Gray, R.J. (1988). A class of k-sample tests for comparing the cumulative incidence of a competing risk. Ann. Statist. 16:1141–1154.[Crossref], [Web of Science ®] , [Google Scholar]), while focusing upon one major cause of failure. We derive estimators for regression parameter vector and cumulative baseline cause specific hazard rate function. Asymptotic properties of the estimators are discussed. A simulation study is conducted to assess the finite sample behavior of the proposed estimators. We illustrate the method using a real life doubly censored competing risks data.     

On the identifiability of copulas in bivariate competing risks models

In competing risks models, the joint distribution of the event times is not identifiable even when the margins are fully known, which has been referred to as the “identifiability crisis in competing risks analysis” (Crowder, 1991). We model the dependence between the event times by an unknown copula and show that identification is actually possible within many frequently used families of copulas. The result is then extended to the case where one margin is unknown. The Canadian Journal of Statistics 41: 291–303; 2013 © 2013 Statistical Society of Canada     

Penalized variable selection in competing risks regression

Penalized variable selection methods have been extensively studied for standard time-to-event data. Such methods cannot be directly applied when subjects are at risk of multiple mutually exclusive events, known as competing risks. The proportional subdistribution hazard (PSH) model proposed by Fine and Gray (J Am Stat Assoc 94:496–509, 1999) has become a popular semi-parametric model for time-to-event data with competing risks. It allows for direct assessment of covariate effects on the cumulative incidence function. In this paper, we propose a general penalized variable selection strategy that simultaneously handles variable selection and parameter estimation in the PSH model. We rigorously establish the asymptotic properties of the proposed penalized estimators and modify the coordinate descent algorithm for implementation. Simulation studies are conducted to demonstrate the good performance of the proposed method. Data from deceased donor kidney transplants from the United Network of Organ Sharing illustrate the utility of the proposed method.     

Semiparametric analysis of mixture regression models with competing risks data

In the analysis of competing risks data, cumulative incidence function is a useful summary of the overall crude risk for a failure type of interest. Mixture regression modeling has served as a natural approach to performing covariate analysis based on this quantity. However, existing mixture regression methods with competing risks data either impose parametric assumptions on the conditional risks or require stringent censoring assumptions. In this article, we propose a new semiparametric regression approach for competing risks data under the usual conditional independent censoring mechanism. We establish the consistency and asymptotic normality of the resulting estimators. A simple resampling method is proposed to approximate the distribution of the estimated parameters and that of the predicted cumulative incidence functions. Simulation studies and an analysis of a breast cancer dataset demonstrate that our method performs well with realistic sample sizes and is appropriate for practical use.     

Testing equality of cause-specific hazard rates corresponding to m competing risks among K groups

In this paper, a class of tests is developed for comparing the cause-specific hazard rates of m competing risks simultaneously in K ( 2) groups. The data available for a unit are the failure time of the unit along with the identifier of the risk claiming the failure. In practice, the failure time data are generally right censored. The tests are based on the difference between the weighted averages of the cause-specific hazard rates corresponding to each risk. No assumption regarding the dependence of the competing risks is made. It is shown that the proposed test statistic has asymptotically chi-squared distribution. The proposed test is shown to be optimal for a specific type of local alternatives. The choice of weight function is also discussed. A simulation study is carried out using multivariate Gumbel distribution to compare the optimal weight function with a proposed weight function which is to be used in practice. Also, the proposed test is applied to real data on the termination of an intrauterine device.An erratum to this article can be found at     

Identification of Longitudinal Biomarkers in Survival Analysis for Competing Risks Data

In recent years, joint analysis of longitudinal measurements and survival data has received much attention. However, previous work has primarily focused on a single failure type for the event time. In this article, we consider joint modeling of repeated measurements and competing risks failure time data to allow for more than one distinct failure type in the survival endpoint so we fit a cause-specific hazards sub-model to allow for competing risks, with a separate latent association between longitudinal measurements and each cause of failure. Besides, previous work does not focus on the hypothesis to test a separate latent association between longitudinal measurements and each cause of failure. In this article, we derive a score test to identify longitudinal biomarkers or surrogates for a time to event outcome in competing risks data. With a carefully chosen definition of complete data, the maximum likelihood estimation of the cause-specific hazard functions is performed via an EM algorithm. We extend this work and allow random effects to be present in both the longitudinal biomarker and underlying survival function. The random effects in the biomarker are introduced via an explicit term while the random effect in the underlying survival function is introduced by the inclusion of frailty into the model.

We use simulations to explore how the number of individuals, the number of time points per individual and the functional form of the random effects from the longitudinal biomarkers considering heterogeneous baseline hazards in individuals influence the power to detect the association of a longitudinal biomarker and the survival time.