Competing risks regression for stratified data

For competing risks data, the Fine-Gray proportional hazards model for subdistribution has gained popularity for its convenience in directly assessing the effect of covariates on the cumulative incidence function. However, in many important applications, proportional hazards may not be satisfied, including multicenter clinical trials, where the baseline subdistribution hazards may not be common due to varying patient populations. In this article, we consider a stratified competing risks regression, to allow the baseline hazard to vary across levels of the stratification covariate. According to the relative size of the number of strata and strata sizes, two stratification regimes are considered. Using partial likelihood and weighting techniques, we obtain consistent estimators of regression parameters. The corresponding asymptotic properties and resulting inferences are provided for the two regimes separately. Data from a breast cancer clinical trial and from a bone marrow transplantation registry illustrate the potential utility of the stratified Fine-Gray model.     

Covariate analysis of viral eradication studies

An important issue arising in therapeutic studies of hepatitis C and HIV is the identification of and adjustment for covariates associated with viral eradication and resistance. Analyses of such data are complicated by the fact that eradication is an occult event that is not directly observable, resulting in unique types of censored observations that do not arise in other competing risks settings. This paper proposes a semiparametric regression model to assess the association between multiple covariates and the eradication/resistance processes. The proposed methods are based on a piecewise proportional hazards model that allows parameters to vary between observation times. We illustrate the methods with data from recent hepatitis C clinical trials.     

Time-dependent covariates in the proportional subdistribution hazards model for competing risks

Separate Cox analyses of all cause-specific hazards are the standard technique of choice to study the effect of a covariate in competing risks, but a synopsis of these results in terms of cumulative event probabilities is challenging. This difficulty has led to the development of the proportional subdistribution hazards model. If the covariate is known at baseline, the model allows for a summarizing assessment in terms of the cumulative incidence function. black Mathematically, the model also allows for including random time-dependent covariates, but practical implementation has remained unclear due to a certain risk set peculiarity. We use the intimate relationship of discrete covariates and multistate models to naturally treat time-dependent covariates within the subdistribution hazards framework. The methodology then straightforwardly translates to real-valued time-dependent covariates. As with classical survival analysis, including time-dependent covariates does not result in a model for probability functions anymore. Nevertheless, the proposed methodology provides a useful synthesis of separate cause-specific hazards analyses. We illustrate this with hospital infection data, where time-dependent covariates and competing risks are essential to the subject research question.     

A semi-Markov model for survival data with covariates

Clinical trials are often concerned with the evaluation of two or more time-dependent stochastic events and their relationship. The information on covariates for individuals in the studies is valuable in assessing the survival function. This paper develops a multistate stochastic survival model which incorporates covariates. It is assumed that the underlying process follows a semi-Markov model. The proportional hazards techniques are applied to estimate the force of transition in the process. The maximum likelihood estimators are derived along with the survival function for competing risks problems. An application is given to analyzing the survival of patients in the Stanford Heart Transplant Program.     

Competing risks regression for clustered data

A population average regression model is proposed to assess the marginal effects of covariates on the cumulative incidence function when there is dependence across individuals within a cluster in the competing risks setting. This method extends the Fine-Gray proportional hazards model for the subdistribution to situations, where individuals within a cluster may be correlated due to unobserved shared factors. Estimators of the regression parameters in the marginal model are developed under an independence working assumption where the correlation across individuals within a cluster is completely unspecified. The estimators are consistent and asymptotically normal, and variance estimation may be achieved without specifying the form of the dependence across individuals. A simulation study evidences that the inferential procedures perform well with realistic sample sizes. The practical utility of the methods is illustrated with data from the European Bone Marrow Transplant Registry.     

A joint model for longitudinal measurements and survival data in the presence of multiple failure types

Summary .   In this article we study a joint model for longitudinal measurements and competing risks survival data. Our joint model provides a flexible approach to handle possible nonignorable missing data in the longitudinal measurements due to dropout. It is also an extension of previous joint models with a single failure type, offering a possible way to model informatively censored events as a competing risk. Our model consists of a linear mixed effects submodel for the longitudinal outcome and a proportional cause-specific hazards frailty submodel ( Prentice et al., 1978 , Biometrics 34, 541–554) for the competing risks survival data, linked together by some latent random effects. We propose to obtain the maximum likelihood estimates of the parameters by an expectation maximization (EM) algorithm and estimate their standard errors using a profile likelihood method. The developed method works well in our simulation studies and is applied to a clinical trial for the scleroderma lung disease.     

Regression modeling of competing risks data based on pseudovalues of the cumulative incidence function

Typically, regression models for competing risks outcomes are based on proportional hazards models for the crude hazard rates. These estimates often do not agree with impressions drawn from plots of cumulative incidence functions for each level of a risk factor. We present a technique which models the cumulative incidence functions directly. The method is based on the pseudovalues from a jackknife statistic constructed from the cumulative incidence curve. These pseudovalues are used in a generalized estimating equation to obtain estimates of model parameters. We study the properties of this estimator and apply the technique to a study of the effect of alternative donors on relapse for patients given a bone marrow transplant for leukemia.     

Analysis of a partially observed binary covariate process and a censored failure time in the presence of truncation and competing risks

We develop methods for assessing the association between a binary time-dependent covariate process and a failure time endpoint when the former is observed only at a single time point and the latter is right censored, and when the observations are subject to truncation and competing causes of failure. Using a proportional hazards model for the effect of the covariate process on the failure time of interest, we develop an approach utilizing EM algorithm and profile likelihood for estimating the relative risk parameter and cause-specific hazards for failure. The methods are extended to account for other covariates that can influence the time-dependent covariate process and cause-specific risks of failure. We illustrate the methods with data from a recent study on the association between loss of hepatitis B e antigen and the development of hepatocellular carcinoma in a population of chronic carriers of hepatitis B.     

Competing Risks and Time‐Dependent Covariates

Time‐dependent covariates are frequently encountered in regression analysis for event history data and competing risks. They are often essential predictors, which cannot be substituted by time‐fixed covariates. This study briefly recalls the different types of time‐dependent covariates, as classified by Kalbfleisch and Prentice [The Statistical Analysis of Failure Time Data, Wiley, New York, 2002] with the intent of clarifying their role and emphasizing the limitations in standard survival models and in the competing risks setting. If random (internal) time‐dependent covariates are to be included in the modeling process, then it is still possible to estimate cause‐specific hazards but prediction of the cumulative incidences and survival probabilities based on these is no longer feasible. This article aims at providing some possible strategies for dealing with these prediction problems. In a multi‐state framework, a first approach uses internal covariates to define additional (intermediate) transient states in the competing risks model. Another approach is to apply the landmark analysis as described by van Houwelingen [Scandinavian Journal of Statistics 2007, 34 , 70–85] in order to study cumulative incidences at different subintervals of the entire study period. The final strategy is to extend the competing risks model by considering all the possible combinations between internal covariate levels and cause‐specific events as final states. In all of those proposals, it is possible to estimate the changes/differences of the cumulative risks associated with simple internal covariates. An illustrative example based on bone marrow transplant data is presented in order to compare the different methods.     

Cox regression model with randomly censored covariates

This paper deals with a Cox proportional hazards regression model, where some covariates of interest are randomly right‐censored. While methods for censored outcomes have become ubiquitous in the literature, methods for censored covariates have thus far received little attention and, for the most part, dealt with the issue of limit‐of‐detection. For randomly censored covariates, an often‐used method is the inefficient complete‐case analysis (CCA) which consists in deleting censored observations in the data analysis. When censoring is not completely independent, the CCA leads to biased and spurious results. Methods for missing covariate data, including type I and type II covariate censoring as well as limit‐of‐detection do not readily apply due to the fundamentally different nature of randomly censored covariates. We develop a novel method for censored covariates using a conditional mean imputation based on either Kaplan–Meier estimates or a Cox proportional hazards model to estimate the effects of these covariates on a time‐to‐event outcome. We evaluate the performance of the proposed method through simulation studies and show that it provides good bias reduction and statistical efficiency. Finally, we illustrate the method using data from the Framingham Heart Study to assess the relationship between offspring and parental age of onset of cardiovascular events.