Cause‐Specific Cumulative Incidence Estimation and the Fine and Gray Model Under Both Left Truncation and Right Censoring

Summary The standard estimator for the cause‐specific cumulative incidence function in a competing risks setting with left truncated and/or right censored data can be written in two alternative forms. One is a weighted empirical cumulative distribution function and the other a product‐limit estimator. This equivalence suggests an alternative view of the analysis of time‐to‐event data with left truncation and right censoring: individuals who are still at risk or experienced an earlier competing event receive weights from the censoring and truncation mechanisms. As a consequence, inference on the cumulative scale can be performed using weighted versions of standard procedures. This holds for estimation of the cause‐specific cumulative incidence function as well as for estimation of the regression parameters in the Fine and Gray proportional subdistribution hazards model. We show that, with the appropriate filtration, a martingale property holds that allows deriving asymptotic results for the proportional subdistribution hazards model in the same way as for the standard Cox proportional hazards model. Estimation of the cause‐specific cumulative incidence function and regression on the subdistribution hazard can be performed using standard software for survival analysis if the software allows for inclusion of time‐dependent weights. We show the implementation in the R statistical package. The proportional subdistribution hazards model is used to investigate the effect of calendar period as a deterministic external time varying covariate, which can be seen as a special case of left truncation, on AIDS related and non‐AIDS related cumulative mortality.     

Double Inverse‐Weighted Estimation of Cumulative Treatment Effects Under Nonproportional Hazards and Dependent Censoring

Summary In medical studies of time‐to‐event data, nonproportional hazards and dependent censoring are very common issues when estimating the treatment effect. A traditional method for dealing with time‐dependent treatment effects is to model the time‐dependence parametrically. Limitations of this approach include the difficulty to verify the correctness of the specified functional form and the fact that, in the presence of a treatment effect that varies over time, investigators are usually interested in the cumulative as opposed to instantaneous treatment effect. In many applications, censoring time is not independent of event time. Therefore, we propose methods for estimating the cumulative treatment effect in the presence of nonproportional hazards and dependent censoring. Three measures are proposed, including the ratio of cumulative hazards, relative risk, and difference in restricted mean lifetime. For each measure, we propose a double inverse‐weighted estimator, constructed by first using inverse probability of treatment weighting (IPTW) to balance the treatment‐specific covariate distributions, then using inverse probability of censoring weighting (IPCW) to overcome the dependent censoring. The proposed estimators are shown to be consistent and asymptotically normal. We study their finite‐sample properties through simulation. The proposed methods are used to compare kidney wait‐list mortality by race.     

Estimating Subject‐Specific Dependent Competing Risk Profile with Censored Event Time Observations

Summary In a longitudinal study, suppose that the primary endpoint is the time to a specific event. This response variable, however, may be censored by an independent censoring variable or by the occurrence of one of several dependent competing events. For each study subject, a set of baseline covariates is collected. The question is how to construct a reliable prediction rule for the future subject's profile of all competing risks of interest at a specific time point for risk‐benefit decision making. In this article, we propose a two‐stage procedure to make inferences about such subject‐specific profiles. For the first step, we use a parametric model to obtain a univariate risk index score system. We then estimate consistently the average competing risks for subjects who have the same parametric index score via a nonparametric function estimation procedure. We illustrate this new proposal with the data from a randomized clinical trial for evaluating the efficacy of a treatment for prostate cancer. The primary endpoint for this study was the time to prostate cancer death, but had two types of dependent competing events, one from cardiovascular death and the other from death of other causes.     

Multiple imputation for estimating the risk of developing dementia and its impact on survival

Dementia, Alzheimer's disease in particular, is one of the major causes of disability and decreased quality of life among the elderly and a leading obstacle to successful aging. Given the profound impact on public health, much research has focused on the age-specific risk of developing dementia and the impact on survival. Early work has discussed various methods of estimating age-specific incidence of dementia, among which the illness-death model is popular for modeling disease progression. In this article we use multiple imputation to fit multi-state models for survival data with interval censoring and left truncation. This approach allows semi-Markov models in which survival after dementia depends on onset age. Such models can be used to estimate the cumulative risk of developing dementia in the presence of the competing risk of dementia-free death. Simulations are carried out to examine the performance of the proposed method. Data from the Honolulu Asia Aging Study are analyzed to estimate the age-specific and cumulative risks of dementia and to examine the effect of major risk factors on dementia onset and death.     

A measure of explained variation for event history data

Summary There is no shortage of proposed measures of prognostic value of survival models in the statistical literature. They come under different names, including explained variation, correlation, explained randomness, and information gain, but their goal is common: to define something analogous to the coefficient of determination R² in linear regression. None however have been uniformly accepted, none have been extended to general event history data, including recurrent events, and many cannot incorporate time‐varying effects or covariates. We present here a measure specifically tailored for use with general dynamic event history regression models. The measure is applicable and interpretable in discrete or continuous time; with tied data or otherwise; with time‐varying, time‐fixed, or dynamic covariates; with time‐varying or time‐constant effects; with single or multiple event times; with parametric or semiparametric models; and under general independent censoring/observation. For single‐event survival data with neither censoring nor time dependency it reduces to the concordance index. We give expressions for its population value and the variance of the estimator and explore its use in simulations and applications. A web link to R software is provided.     

Nonparametric inference for competing risks current status data with continuous, discrete or grouped observation times

New methods and theory have recently been developed to nonparametrically estimate cumulative incidence functions for competing risks survival data subject to current status censoring. In particular, the limiting distribution of the nonparametric maximum likelihood estimator and a simplified naive estimator have been established under certain smoothness conditions. In this paper, we establish the large-sample behaviour of these estimators in two additional models, namely when the observation time distribution has discrete support and when the observation times are grouped. These asymptotic results are applied to the construction of confidence intervals in the three different models. The methods are illustrated on two datasets regarding the cumulative incidence of different types of menopause from a cross-sectional sample of women in the United States and subtype-specific HIV infection from a sero-prevalence study in injecting drug users in Thailand.     

Dynamic clinical prediction models for discrete time‐to‐event data with competing risks—A case study on the OUTCOMEREA database

The development of clinical prediction models requires the selection of suitable predictor variables. Techniques to perform objective Bayesian variable selection in the linear model are well developed and have been extended to the generalized linear model setting as well as to the Cox proportional hazards model. Here, we consider discrete time‐to‐event data with competing risks and propose methodology to develop a clinical prediction model for the daily risk of acquiring a ventilator‐associated pneumonia (VAP) attributed to P. aeruginosa (PA) in intensive care units. The competing events for a PA VAP are extubation, death, and VAP due to other bacteria. Baseline variables are potentially important to predict the outcome at the start of ventilation, but may lose some of their predictive power after a certain time. Therefore, we use a landmark approach for dynamic Bayesian variable selection where the set of relevant predictors depends on the time already spent at risk. We finally determine the direct impact of a variable on each competing event through cause‐specific variable selection.     

Marginal Models for Clustered Time‐to‐Event Data with Competing Risks Using Pseudovalues

Summary Many time‐to‐event studies are complicated by the presence of competing risks and by nesting of individuals within a cluster, such as patients in the same center in a multicenter study. Several methods have been proposed for modeling the cumulative incidence function with independent observations. However, when subjects are clustered, one needs to account for the presence of a cluster effect either through frailty modeling of the hazard or subdistribution hazard, or by adjusting for the within‐cluster correlation in a marginal model. We propose a method for modeling the marginal cumulative incidence function directly. We compute leave‐one‐out pseudo‐observations from the cumulative incidence function at several time points. These are used in a generalized estimating equation to model the marginal cumulative incidence curve, and obtain consistent estimates of the model parameters. A sandwich variance estimator is derived to adjust for the within‐cluster correlation. The method is easy to implement using standard software once the pseudovalues are obtained, and is a generalization of several existing models. Simulation studies show that the method works well to adjust the SE for the within‐cluster correlation. We illustrate the method on a dataset looking at outcomes after bone marrow transplantation.     

Confidence bands for multiplicative hazards models: Flexible resampling approaches

We propose new resampling‐based approaches to construct asymptotically valid time‐simultaneous confidence bands for cumulative hazard functions in multistate Cox models. In particular, we exemplify the methodology in detail for the simple Cox model with time‐dependent covariates, where the data may be subject to independent right‐censoring or left‐truncation. We use simulations to investigate their finite sample behavior. Finally, the methods are utilized to analyze two empirical examples with survival and competing risks data.     

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.     

Nonparametric association analysis of bivariate left‐truncated competing risks data

We develop time‐varying association analyses for onset ages of two lung infections to address the statistical challenges in utilizing registry data where onset ages are left‐truncated by ages of entry and competing‐risk censored by deaths. Two types of association estimators are proposed based on conditional cause‐specific hazard function and cumulative incidence function that are adapted from unconditional quantities to handle left truncation. Asymptotic properties of the estimators are established by using the empirical process techniques. Our simulation study shows that the estimators perform well with moderate sample sizes. We apply our methods to the Cystic Fibrosis Foundation Registry data to study the relationship between onset ages of Pseudomonas aeruginosa and Staphylococcus aureus infections.     

Improved confidence intervals for a difference of two cause-specific cumulative incidence functions estimated in the presence of competing risks and random censoring

A cause-specific cumulative incidence function (CIF) is the probability of failure from a specific cause as a function of time. In randomized trials, a difference of cause-specific CIFs (treatment minus control) represents a treatment effect. Cause-specific CIF in each intervention arm can be estimated based on the usual non-parametric Aalen–Johansen estimator which generalizes the Kaplan–Meier estimator of CIF in the presence of competing risks. Under random censoring, asymptotically valid Wald-type confidence intervals (CIs) for a difference of cause-specific CIFs at a specific time point can be constructed using one of the published variance estimators. Unfortunately, these intervals can suffer from substantial under-coverage when the outcome of interest is a rare event, as may be the case for example in the analysis of uncommon adverse events. We propose two new approximate interval estimators for a difference of cause-specific CIFs estimated in the presence of competing risks and random censoring. Theoretical analysis and simulations indicate that the new interval estimators are superior to the Wald CIs in the sense of avoiding substantial under-coverage with rare events, while being equivalent to the Wald CIs asymptotically. In the absence of censoring, one of the two proposed interval estimators reduces to the well-known Agresti–Caffo CI for a difference of two binomial parameters. The new methods can be easily implemented with any software package producing point and variance estimates for the Aalen–Johansen estimator, as illustrated in a real data example.     

Nonparametric maximum likelihood estimation for competing risks survival data subject to interval censoring and truncation

We derive the nonparametric maximum likelihood estimate (NPMLE) of the cumulative incidence functions for competing risks survival data subject to interval censoring and truncation. Since the cumulative incidence function NPMLEs give rise to an estimate of the survival distribution which can be undefined over a potentially larger set of regions than the NPMLE of the survival function obtained ignoring failure type, we consider an alternative pseudolikelihood estimator. The methods are then applied to data from a cohort of injecting drug users in Thailand susceptible to infection from HIV-1 subtypes B and E.     

A simulation approach for power calculation in large cohort studies based on multistate models

Realistic power calculations for large cohort studies and nested case control studies are essential for successfully answering important and complex research questions in epidemiology and clinical medicine. For this, we provide a methodical framework for general realistic power calculations via simulations that we put into practice by means of an R‐based template. We consider staggered recruitment and individual hazard rates, competing risks, interaction effects, and the misclassification of covariates. The study cohort is assembled with respect to given age‐, gender‐, and community distributions. Nested case‐control analyses with a varying number of controls enable comparisons of power with a full cohort analysis. Time‐to‐event generation under competing risks, including delayed study‐entry times, is realized on the basis of a six‐state Markov model. Incidence rates, prevalence of risk factors and prefixed hazard ratios allow for the assignment of age‐dependent transition rates given in the form of Cox models. These provide the basis for a central simulation‐algorithm, which is used for the generation of sample paths of the underlying time‐inhomogeneous Markov processes. With the inclusion of frailty terms into the Cox models the Markov property is specifically biased. An “individual Markov process given frailty” creates some unobserved heterogeneity between individuals. Different left‐truncation‐ and right‐censoring patterns call for the use of Cox models for data analysis. p‐values are recorded over repeated simulation runs to allow for the desired power calculations. For illustration, we consider scenarios with a “testing” character as well as realistic scenarios. This enables the validation of a correct implementation of theoretical concepts and concrete sample size recommendations against an actual epidemiological background, here given with possible substudy designs within the German National Cohort.     

Confidence bands for cumulative incidence curves under the additive risk model

In the context of competing risks, the cumulative incidence function is often used to summarize the cause-specific failure-time data. As an alternative to the proportional hazards model, the additive risk model is used to investigate covariate effects by specifying that the subject-specific hazard function is the sum of a baseline hazard function and a regression function of covariates. Based on such a formulation, we present an approach to constructing simultaneous confidence intervals for the cause-specific cumulative incidence function of patients with given risk factors. A melanoma data set is used for the purpose of illustration.     

Estimation in the progressive illness‐death model: A nonexhaustive review

Multistate models can be successfully used for describing complex event history data, for example, describing stages in the disease progression of a patient. The so‐called “illness‐death” model plays a central role in the theory and practice of these models. Many time‐to‐event datasets from medical studies with multiple end points can be reduced to this generic structure. In these models one important goal is the modeling of transition rates but biomedical researchers are also interested in reporting interpretable results in a simple and summarized manner. These include estimates of predictive probabilities, such as the transition probabilities, occupation probabilities, cumulative incidence functions, and the sojourn time distributions. We will give a review of some of the available methods for estimating such quantities in the progressive illness‐death model conditionally (or not) on covariate measures. For some of these quantities estimators based on subsampling are employed. Subsampling, also referred to as landmarking, leads to small sample sizes and usually to heavily censored data leading to estimators with higher variability. To overcome this issue estimators based on a preliminary estimation (presmoothing) of the probability of censoring may be used. Among these, the presmoothed estimators for the cumulative incidences are new. We also introduce feasible estimation methods for the cumulative incidence function conditionally on covariate measures. The proposed methods are illustrated using real data. A comparative simulation study of several estimation approaches is performed and existing software in the form of R packages is discussed.     

Nonparametric inference for the cumulative incidence function of a competing risk,with an emphasis on confidence bands in the presence of left‐truncation

Motivated by a study on pregnancy outcome, a computationally simple resampling procedure for nonparametric analysis of the cumulative incidence function of a competing risk is investigated for left‐truncated data. We also modify the original procedure to producing the more desirable Greenwood‐type variance estimates. These approaches are used to construct simultaneous confidence bands of the cumulative incidence functions which is otherwise hampered by the complicated nature of the covariance process. Simulation results and a real data example are provided.     

Nonparametric inference on cause‐specific quantile residual life

In medical research, investigators are often interested in inferring time‐to‐event distributions under competing risks. It is well known, however, that the naive approach based on the Kaplan–Meier method to estimate the proportion of cause‐specific events overestimates the true quantity. In this paper, we show that the quantile residual life function, a natural and popular summary measure of survival data, could be also seriously affected by the competing events. An existing two‐sample test statistic for inference on median residual life is modified for competing risks data, which does not involve estimation of the improper probability density function of the subdistribution of cause‐specific events under censoring. Simulation results demonstrate that the test statistic controls the type 1 error probabilities reasonably well. The proposed method is applied to a real data example from a large‐scale phase III breast cancer study.     

Constrained parametric model for simultaneous inference of two cumulative incidence functions

We propose a parametric regression model for the cumulative incidence functions (CIFs) commonly used for competing risks data. The model adopts a modified logistic model as the baseline CIF and a generalized odds‐rate model for covariate effects, and it explicitly takes into account the constraint that a subject with any given prognostic factors should eventually fail from one of the causes such that the asymptotes of the CIFs should add up to one. This constraint intrinsically holds in a nonparametric analysis without covariates, but is easily overlooked in a semiparametric or parametric regression setting. We hence model the CIF from the primary cause assuming the generalized odds‐rate transformation and the modified logistic function as the baseline CIF. Under the additivity constraint, the covariate effects on the competing cause are modeled by a function of the asymptote of the baseline distribution and the covariate effects on the primary cause. The inference procedure is straightforward by using the standard maximum likelihood theory. We demonstrate desirable finite‐sample performance of our model by simulation studies in comparison with existing methods. Its practical utility is illustrated in an analysis of a breast cancer dataset to assess the treatment effect of tamoxifen, adjusting for age and initial pathological tumor size, on breast cancer recurrence that is subject to dependent censoring by second primary cancers and deaths.     

Competing Risk Analysis of Censored Survival Data

A competing risk model is developed to accommodate both planned Type I censoring and random withdrawals. MLE's, their properties, confidence regions for parameters and mean lifetimes are obtained for a model regarding random censoring as a competing risk and compared to those obtained for the model in which withdrawals are regarded as random censoring. Estimated net and crude probabilities are calculated and compared for the two models. The model is developed for two competing risks, one following a Weibull distribution and the other a Rayleigh distribution, and random withdrawals following a Weibull distribution.