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.     

THE THEORY OF COMPETING RISKS1

The theory of competing risks is motivated, certain aspects are reviewed critically, and some extensions are indicated. A unified formulation of the theory is given covering dependent as well as independent risks. The relations between various functions useful in the theory are made explicit. In a historical note a valuable early result is put into modern notation. The currently controversial subject of identifiability when risks are dependent is discussed and it is indicated under what conditions some of the difficulties raised can be overcome. Consequences of assuming proportional hazard rates are set out. New conditions are provided under which this assumption holds and it is shown how the assumption may be tested. Some concluding remarks deal with limitations of the theory and point out areas needing further work.     

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.     

A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects

This article studies a general joint model for longitudinal measurements and competing risks survival data. The model consists of a linear mixed effects sub-model for the longitudinal outcome, a proportional cause-specific hazards frailty sub-model for the competing risks survival data, and a regression sub-model for the variance–covariance matrix of the multivariate latent random effects based on a modified Cholesky decomposition. The model provides a useful approach to adjust for non-ignorable missing data due to dropout for the longitudinal outcome, enables analysis of the survival outcome with informative censoring and intermittently measured time-dependent covariates, as well as joint analysis of the longitudinal and survival outcomes. Unlike previously studied joint models, our model allows for heterogeneous random covariance matrices. It also offers a framework to assess the homogeneous covariance assumption of existing joint models. A Bayesian MCMC procedure is developed for parameter estimation and inference. Its performances and frequentist properties are investigated using simulations. A real data example is used to illustrate the usefulness of the approach.     

On Non-parametric Estimation of the Survival Function with Competing Risks

The non-parametric maximum likelihood estimators (MLEs) are derived for survival functions associated with individual risks or system components in a reliability framework. Lifetimes are observed for systems that contain one or more of those components. Analogous to a competing risks model, the system is assumed to fail upon the first instance of any component failure; i.e. the system is configured in series. For any given risk or component type, the asymptotic distribution is shown to depend explicitly on the unknown survival function of the other risks, as well as the censoring distribution. Survival functions with increasing failure rate are investigated as a special case. The order restricted MLE is shown to be consistent under mild assumptions of the underlying component lifetime distributions.     

A Corrected Pseudo-score Approach for Additive Hazards Model with Longitudinal Covariates Measured with Error

In medical studies, it is often of interest to characterize the relationship between a time-to-event and covariates, not only time-independent but also time-dependent. Time-dependent covariates are generally measured intermittently and with error. Recent interests focus on the proportional hazards framework, with longitudinal data jointly modeled through a mixed effects model. However, approaches under this framework depend on the normality assumption of the error, and might encounter intractable numerical difficulties in practice. This motivates us to consider an alternative framework, that is, the additive hazards model, about which little research has been done when time-dependent covariates are measured with error. We propose a simple corrected pseudo-score approach for the regression parameters with no assumptions on the distribution of the random effects and the error beyond those for the variance structure of the latter. The estimator has an explicit form and is shown to be consistent and asymptotically normal. We illustrate the method via simulations and by application to data from an HIV clinical trial.     

Stratified proportional subdistribution hazards model with covariate-adjusted censoring weight for case-cohort studies

The case-cohort study design is widely used to reduce cost when collecting expensive covariates in large cohort studies with survival or competing risks outcomes. A case-cohort study dataset consists of two parts: (a) a random sample and (b) all cases or failures from a specific cause of interest. Clinicians often assess covariate effects on competing risks outcomes. The proportional subdistribution hazards model directly evaluates the effect of a covariate on the cumulative incidence function under the non-covariate-dependent censoring assumption for the full cohort study. However, the non-covariate-dependent censoring assumption is often violated in many biomedical studies. In this article, we propose a proportional subdistribution hazards model for case-cohort studies with stratified data with covariate-adjusted censoring weight. We further propose an efficient estimator when extra information from the other causes is available under case-cohort studies. The proposed estimators are shown to be consistent and asymptotically normal. Simulation studies show (a) the proposed estimator is unbiased when the censoring distribution depends on covariates and (b) the proposed efficient estimator gains estimation efficiency when using extra information from the other causes. We analyze a bone marrow transplant dataset and a coronary heart disease dataset using the proposed method.     

A Proportional Hazards Regression Model for the Subdistribution with Covariates‐adjusted Censoring Weight for Competing Risks Data

With competing risks data, one often needs to assess the treatment and covariate effects on the cumulative incidence function. Fine and Gray proposed a proportional hazards regression model for the subdistribution of a competing risk with the assumption that the censoring distribution and the covariates are independent. Covariate‐dependent censoring sometimes occurs in medical studies. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with proper adjustments for covariate‐dependent censoring. We consider a covariate‐adjusted weight function by fitting the Cox model for the censoring distribution and using the predictive probability for each individual. Our simulation study shows that the covariate‐adjusted weight estimator is basically unbiased when the censoring time depends on the covariates, and the covariate‐adjusted weight approach works well for the variance estimator as well. We illustrate our methods with bone marrow transplant data from the Center for International Blood and Marrow Transplant Research. Here, cancer relapse and death in complete remission are two competing risks.     

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.

     

Modelling Survival Events with Longitudinal Covariates Measured with Error

In survival analysis, time-dependent covariates are usually present as longitudinal data collected periodically and measured with error. The longitudinal data can be assumed to follow a linear mixed effect model and Cox regression models may be used for modelling of survival events. The hazard rate of survival times depends on the underlying time-dependent covariate measured with error, which may be described by random effects. Most existing methods proposed for such models assume a parametric distribution assumption on the random effects and specify a normally distributed error term for the linear mixed effect model. These assumptions may not be always valid in practice. In this article, we propose a new likelihood method for Cox regression models with error-contaminated time-dependent covariates. The proposed method does not require any parametric distribution assumption on random effects and random errors. Asymptotic properties for parameter estimators are provided. Simulation results show that under certain situations the proposed methods are more efficient than the existing methods.     

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.     

Tests for comparison of competing risks under the additive risk model

It is of interest that researchers study competing risks in which subjects may fail from any one of k causes. Comparing any two competing risks with covariate effects is very important in medical studies. In this paper, we develop tests for comparing cause-specific hazard rates and cumulative incidence functions at specified covariate levels under the additive risk model by a weighted difference of estimates of cumulative cause-specific hazard rates. Motivated by McKeague et al. (2001), we construct simultaneous confidence bands for the difference of two conditional cumulative incidence functions as a useful graphical tool. In addition, we conduct a simulation study, and the simulation result shows that the proposed procedure has a good finite sample performance. A melanoma data set in clinical trial is used for the purpose of illustration.     

Score tests for independence in semiparametric competing risks models

A popular model for competing risks postulates the existence of a latent unobserved failure time for each risk. Assuming that these underlying failure times are independent is attractive since it allows standard statistical tools for right-censored lifetime data to be used in the analysis. This paper proposes simple independence score tests for the validity of this assumption when the individual risks are modeled using semiparametric proportional hazards regressions. It assumes that covariates are available, making the model identifiable. The score tests are derived for alternatives that specify that copulas are responsible for a possible dependency between the competing risks. The test statistics are constructed by adding to the partial likelihoods for the individual risks an explanatory variable for the dependency between the risks. A variance estimator is derived by writing the score function and the Fisher information matrix for the marginal models as stochastic integrals. Pitman efficiencies are used to compare test statistics. A simulation study and a numerical example illustrate the methodology proposed in this paper.     

Non-identifiable parametric probability models and reparametrization

Identifiability is a primary assumption in virtually all classical statistical theory. However, such an assumption may be violated in a variety of statistical models. We consider parametric models where the assumption of identifiability is violated, but otherwise satisfy standard assumptions. We propose an analytic method for constructing new parameters under which the model will be at least locally identifiable. This method is based on solving a system of linear partial differential equations involving the Fisher information matrix. Some consequences and valid inference procedures under non-identifiability have been discussed. The method of reparametrization is illustrated with an example.     

Bounds for the hazard gradients in the competing risks set up

The joint survival function of the latent lifetimes in the dependent competing risks set up is nonidentifiable from the joint probability distribution of the observation (failure time, cause of failure). This paper concentrates on the hazard gradients associated with the joint survival function, for which we provide bounds in the general dependent case and improve these in case the type of dependence (e.g., RTI, RCSI, etc.) is known. This leads to bounds on net (marginal) hazard rates in terms of cause-specific hazard rates which are identifiable. The problem of estimation of these bounds is discussed at the end.     

A Semi‐parametric Transformation Frailty Model for Semi‐competing Risks Survival Data

In the analysis of semi‐competing risks data interest lies in estimation and inference with respect to a so‐called non‐terminal event, the observation of which is subject to a terminal event. Multi‐state models are commonly used to analyse such data, with covariate effects on the transition/intensity functions typically specified via the Cox model and dependence between the non‐terminal and terminal events specified, in part, by a unit‐specific shared frailty term. To ensure identifiability, the frailties are typically assumed to arise from a parametric distribution, specifically a Gamma distribution with mean 1.0 and variance, say, σ². When the frailty distribution is misspecified, however, the resulting estimator is not guaranteed to be consistent, with the extent of asymptotic bias depending on the discrepancy between the assumed and true frailty distributions. In this paper, we propose a novel class of transformation models for semi‐competing risks analysis that permit the non‐parametric specification of the frailty distribution. To ensure identifiability, the class restricts to parametric specifications of the transformation and the error distribution; the latter are flexible, however, and cover a broad range of possible specifications. We also derive the semi‐parametric efficient score under the complete data setting and propose a non‐parametric score imputation method to handle right censoring; consistency and asymptotic normality of the resulting estimators is derived and small‐sample operating characteristics evaluated via simulation. Although the proposed semi‐parametric transformation model and non‐parametric score imputation method are motivated by the analysis of semi‐competing risks data, they are broadly applicable to any analysis of multivariate time‐to‐event outcomes in which a unit‐specific shared frailty is used to account for correlation. Finally, the proposed model and estimation procedures are applied to a study of hospital readmission among patients diagnosed with pancreatic cancer.     

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.     

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.     

Reinforced urns and the subdistribution beta‐Stacy process prior for competing risks analysis

In this paper, we introduce the subdistribution beta‐Stacy process, a novel Bayesian nonparametric process prior for subdistribution functions useful for the analysis of competing risks data. In particular, we (i) characterize this process from a predictive perspective by means of an urn model with reinforcement, (ii) show that it is conjugate with respect to right‐censored data, and (iii) highlight its relations with other prior processes for competing risks data. Additionally, we consider the subdistribution beta‐Stacy process prior in a nonparametric regression model for competing risks data, which, contrary to most others available in the literature, is not based on the proportional hazards assumption.     

The competing risks Cox model with auxiliary case covariates under weaker missing-at-random cause of failure

In the analysis of time-to-event data with multiple causes using a competing risks Cox model, often the cause of failure is unknown for some of the cases. The probability of a missing cause is typically assumed to be independent of the cause given the time of the event and covariates measured before the event occurred. In practice, however, the underlying missing-at-random assumption does not necessarily hold. Motivated by colorectal cancer molecular pathological epidemiology analysis, we develop a method to conduct valid analysis when additional auxiliary variables are available for cases only. We consider a weaker missing-at-random assumption, with missing pattern depending on the observed quantities, which include the auxiliary covariates. We use an informative likelihood approach that will yield consistent estimates even when the underlying model for missing cause of failure is misspecified. The superiority of our method over naive methods in finite samples is demonstrated by simulation study results. We illustrate the use of our method in an analysis of colorectal cancer data from the Nurses’ Health Study cohort, where, apparently, the traditional missing-at-random assumption fails to hold.