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.     

An additive subdistribution hazard model for competing risks data

The cumulative incidence function plays an important role in assessing its treatment and covariate effects with competing risks data. In this article, we consider an additive hazard model allowing the time-varying covariate effects for the subdistribution and propose the weighted estimating equation under the covariate-dependent censoring by fitting the Cox-type hazard model for the censoring distribution. When there exists some association between the censoring time and the covariates, the proposed coefficients’ estimations are unbiased and the large-sample properties are established. The finite-sample properties of the proposed estimators are examined in the simulation study. The proposed Cox-weighted method is applied to a competing risks dataset from a Hodgkin's disease study.     

Competing risk and the Cox proportional hazard model

We propose a heuristic for evaluating model adequacy for the Cox proportional hazard model by comparing the population cumulative hazard with the baseline cumulative hazard. We illustrate how recent results from the theory of competing risk can contribute to analysis of data with the Cox proportional hazard model. A classical theorem on independent competing risks allows us to assess model adequacy under the hypothesis of random right censoring, and a recent result on mixtures of exponentials predicts the patterns of the conditional subsurvival functions of random right censored data if the proportional hazard model holds.     

Flexible competing risks regression modeling and goodness-of-fit

In this paper we consider different approaches for estimation and assessment of covariate effects for the cumulative incidence curve in the competing risks model. The classic approach is to model all cause-specific hazards and then estimate the cumulative incidence curve based on these cause-specific hazards. Another recent approach is to directly model the cumulative incidence by a proportional model (Fine and Gray, J Am Stat Assoc 94:496–509, 1999), and then obtain direct estimates of how covariates influences the cumulative incidence curve. We consider a simple and flexible class of regression models that is easy to fit and contains the Fine–Gray model as a special case. One advantage of this approach is that our regression modeling allows for non-proportional hazards. This leads to a new simple goodness-of-fit procedure for the proportional subdistribution hazards assumption that is very easy to use. The test is constructive in the sense that it shows exactly where non-proportionality is present. We illustrate our methods to a bone marrow transplant data from the Center for International Blood and Marrow Transplant Research (CIBMTR). Through this data example we demonstrate the use of the flexible regression models to analyze competing risks data when non-proportionality is present in the data.     

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.     

Estimation of the cumulative incidence function under multiple dependent and independent censoring mechanisms

Competing risks occur in a time-to-event analysis in which a patient can experience one of several types of events. Traditional methods for handling competing risks data presuppose one censoring process, which is assumed to be independent. In a controlled clinical trial, censoring can occur for several reasons: some independent, others dependent. We propose an estimator of the cumulative incidence function in the presence of both independent and dependent censoring mechanisms. We rely on semi-parametric theory to derive an augmented inverse probability of censoring weighted (AIPCW) estimator. We demonstrate the efficiency gained when using the AIPCW estimator compared to a non-augmented estimator via simulations. We then apply our method to evaluate the safety and efficacy of three anti-HIV regimens in a randomized trial conducted by the AIDS Clinical Trial Group, ACTG A5095.     

Analysis of interval censored competing risk data with missing causes of failure using pseudo values approach

Competing risks often occur when subjects may fail from one of several mutually exclusive causes. For example, when a patient suffering a cancer may die from other cause, we are interested in the effect of a certain covariate on the probability of dying of cancer at a certain time. Several approaches have been suggested to analyse competing risk data in the presence of complete information of failure cause. In this paper, our interest is to consider the occurrence of missing causes as well as interval censored failure time. There exist no method to discuss this problem. We applied a Klein–Andersen's pseudo-value approach [Klein, JP Andersen PK. Regression modeling of competing risks data based on pseudovalues of the cumulative incidence function. Biometrics. 2005;61:223–229] based on the estimated cumulative incidence function and a regression coefficient is estimated through a multiple imputation. We evaluate the suggested method by comparing with a complete case analysis in several simulation settings.     

Competing risks with missing covariates: effect of haplotypematch on hematopoietic cell transplant patients

In this paper we consider a problem from hematopoietic cell transplant (HCT) studies where there is interest on assessing the effect of haplotype match for donor and patient on the cumulative incidence function for a right censored competing risks data. For the HCT study, donor??s and patient??s genotype are fully observed and matched but their haplotypes are missing. In this paper we describe how to deal with missing covariates of each individual for competing risks data. We suggest a procedure for estimating the cumulative incidence functions for a flexible class of regression models when there are missing data, and establish the large sample properties. Small sample properties are investigated using simulations in a setting that mimics the motivating haplotype matching problem. The proposed approach is then applied to the HCT study.     

Exact inference for competing risks model with generalized type-I hybrid censored exponential data

Recently, exact inference under hybrid censoring scheme has attracted extensive attention in the field of reliability analysis. However, most of the authors neglect the possibility of competing risks model. This paper mainly discusses the exact likelihood inference for the analysis of generalized type-I hybrid censoring data with exponential competing failure model. Based on the maximum likelihood estimates for unknown parameters, we establish the exact conditional distribution of parameters by conditional moment generating function, and then obtain moment properties as well as exact confidence intervals (CIs) for parameters. Furthermore, approximate CIs are constructed by asymptotic distribution and bootstrap method as well. We also compare their performances with exact method through the use of Monte Carlo simulations. And finally, a real data set is analysed to illustrate the validity of all the methods developed here.     

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.     

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.     

Product-limit Estimators and Cox Regression with Missing Censoring Information

The Kaplan–Meier estimator of a survival function requires that the censoring indicator is always observed. A method of survival function estimation is developed when the censoring indicators are missing completely at random (MCAR). The resulting estimator is a smooth functional of the Nelson–Aalen estimators of certain cumulative transition intensities. The asymptotic properties of this estimator are derived. A simulation study shows that the proposed estimator has greater efficiency than competing MCAR-based estimators. The approach is extended to the Cox model setting for the estimation of a conditional survival function given a covariate.     

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.     

Semiparametric estimation of the cumulative incidence function under competing risks and left-truncated sampling

In this article, we propose semiparametric methods to estimate the cumulative incidence function of two dependent competing risks for left-truncated and right-censored data. The proposed method is based on work by Huang and Wang (1995). We extend previous model by allowing for a general parametric truncation distribution and a third competing risk before recruitment. Based on work by Vardi (1989), several iterative algorithms are proposed to obtain the semiparametric estimates of cumulative incidence functions. The asymptotic properties of the semiparametric estimators are derived. Simulation results show that a semiparametric approach assuming the parametric truncation distribution is correctly specified produces estimates with smaller mean squared error than those obtained in a fully nonparametric model.     

Estimation of conditional cumulative incidence functions under generalized semiparametric regression models with missing covariates,with application to analysis of biomarker correlates in vaccine trials

This article presents generalized semiparametric regression models for conditional cumulative incidence functions with competing risks data when covariates are missing by sampling design or happenstance. A doubly robust augmented inverse probability weighted (AIPW) complete-case approach to estimation and inference is investigated. This approach modifies IPW complete-case estimating equations by exploiting the key features in the relationship between the missing covariates and the phase-one data to improve efficiency. An iterative numerical procedure is derived to solve the nonlinear estimating equations. The asymptotic properties of the proposed estimators are established. A simulation study examining the finite-sample performances of the proposed estimators shows that the AIPW estimators are more efficient than the IPW estimators. The developed method is applied to the RV144 HIV-1 vaccine efficacy trial to investigate vaccine-induced IgG binding antibodies to HIV-1 as correlates of acquisition of HIV-1 infection while taking account of whether the HIV-1 sequences are near or far from the HIV-1 sequences represented in the vaccine construct.     

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.     

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.     

Asymptotic efficiency of a competing risks model: A two-sample case

A generalized Cox regression model is studied for the covariance analysis of competing risks data subject to independent random censoring. The information of the maximum partial likelihood estimates is compared with that of maximum likelihood estimates assuming a log linear hazard function.The method of generalized variance is used to define the efficiency of estimation between the two models. This is then applied to two-sample problems with two exponentially censoring rates. Numerical results are summarized ane presented graphically.The detailed results indicate that the semi-parametric model wrks well for a higher rate of censoring. A method of generalizing the result to type 1 censoring and the efficiency of estimating the coefficient of the covariate are discussecd. A brief account of using the results to help design experiments is also given.     

Conditional maximum likelihood estimation in semiparametric transformation model with LTRC data

Left-truncated data often arise in epidemiology and individual follow-up studies due to a biased sampling plan since subjects with shorter survival times tend to be excluded from the sample. Moreover, the survival time of recruited subjects are often subject to right censoring. In this article, a general class of semiparametric transformation models that include proportional hazards model and proportional odds model as special cases is studied for the analysis of left-truncated and right-censored data. We propose a conditional likelihood approach and develop the conditional maximum likelihood estimators (cMLE) for the regression parameters and cumulative hazard function of these models. The derived score equations for regression parameter and infinite-dimensional function suggest an iterative algorithm for cMLE. The cMLE is shown to be consistent and asymptotically normal. The limiting variances for the estimators can be consistently estimated using the inverse of negative Hessian matrix. Intensive simulation studies are conducted to investigate the performance of the cMLE. An application to the Channing House data is given to illustrate the methodology.     

Semiparametric analysis of longitudinal data with informative observation times and censoring times

We focus on regression analysis of irregularly observed longitudinal data which often occur in medical follow-up studies and observational investigations. The model for such data involves two processes: a longitudinal response process of interest and an observation process controlling observation times. Restrictive models and questionable assumptions, such as Poisson assumption and independent censoring time assumption, were posed in previous works for analysing longitudinal data. In this paper, we propose a more general model together with a robust estimation approach for longitudinal data with informative observation times and censoring times, and the asymptotic normalities of the proposed estimators are established. Both simulation studies and real data application indicate that the proposed method is promising.