Testing treatment effects in the presence of competing risks

Competing risks are often encountered in clinical research. In the presence of multiple failure types, the time to the first failure of any type is typically used as an overall measure of the clinical impact for the patients. On the other hand, use of endpoints based on the type of failure directly related to the treatment mechanism of action allows one to focus on the aspect of the disease targeted by treatment. We review the methodology commonly used for testing failure specific treatment effects. Simulation results demonstrate that the cause-specific log-rank test is robust (in the sense of preserving the nominal level of the test) and has good power properties for testing for differences in the marginal latent failure-time distributions, whereas the use of a popular cumulative incidence based approach may be problematic for this aim.     

Analysing and interpreting competing risk data

When competing risks are present, two types of analysis can be performed: modelling the cause specific hazard and modelling the hazard of the subdistribution. This paper contrasts these two methods and presents the benefits of each. The interpretation is specific to the analysis performed. When modelling the cause specific hazard, one performs the analysis under the assumption that the competing risks do not exist. This could be beneficial when, for example, the main interest is whether the treatment works in general. In modelling the hazard of the subdistribution, one incorporates the competing risks in the analysis. This analysis compares the observed incidence of the event of interest between groups. The latter analysis is specific to the structure of the observed data and it can be generalized only to another population with similar competing risks.     

Parametric mixture models to evaluate and summarize hazard ratios in the presence of competing risks with time-dependent hazards and delayed entry

In the analysis of survival data, there are often competing events that preclude an event of interest from occurring. Regression analysis with competing risks is typically undertaken using a cause-specific proportional hazards model. However, modern alternative methods exist for the analysis of the subdistribution hazard with a corresponding subdistribution proportional hazards model. In this paper, we introduce a flexible parametric mixture model as a unifying method to obtain estimates of the cause-specific and subdistribution hazards and hazard-ratio functions. We describe how these estimates can be summarized over time to give a single number comparable to the hazard ratio that is obtained from a corresponding cause-specific or subdistribution proportional hazards model. An application to the Women's Interagency HIV Study is provided to investigate injection drug use and the time to either the initiation of effective antiretroviral therapy, or clinical disease progression as a competing event.     

Linear and nonlinear variable selection in competing risks data

Subdistribution hazard model for competing risks data has been applied extensively in clinical researches. Variable selection methods of linear effects for competing risks data have been studied in the past decade. There is no existing work on selection of potential nonlinear effects for subdistribution hazard model. We propose a two‐stage procedure to select the linear and nonlinear covariate(s) simultaneously and estimate the selected covariate effect(s). We use spectral decomposition approach to distinguish the linear and nonlinear parts of each covariate and adaptive LASSO to select each of the 2 components. Extensive numerical studies are conducted to demonstrate that the proposed procedure can achieve good selection accuracy in the first stage and small estimation biases in the second stage. The proposed method is applied to analyze a cardiovascular disease data set with competing death causes.     

Hierarchical likelihood inference on clustered competing risks data

The frailty model, an extension of the proportional hazards model, is often used to model clustered survival data. However, some extension of the ordinary frailty model is required when there exist competing risks within a cluster. Under competing risks, the underlying processes affecting the events of interest and competing events could be different but correlated. In this paper, the hierarchical likelihood method is proposed to infer the cause‐specific hazard frailty model for clustered competing risks data. The hierarchical likelihood incorporates fixed effects as well as random effects into an extended likelihood function, so that the method does not require intensive numerical methods to find the marginal distribution. Simulation studies are performed to assess the behavior of the estimators for the regression coefficients and the correlation structure among the bivariate frailty distribution for competing events. The proposed method is illustrated with a breast cancer dataset. Copyright © 2015 John Wiley & Sons, Ltd.     

Variable selection in subdistribution hazard frailty models with competing risks data

The proportional subdistribution hazards model (i.e. Fine‐Gray model) has been widely used for analyzing univariate competing risks data. Recently, this model has been extended to clustered competing risks data via frailty. To the best of our knowledge, however, there has been no literature on variable selection method for such competing risks frailty models. In this paper, we propose a simple but unified procedure via a penalized h‐likelihood (HL) for variable selection of fixed effects in a general class of subdistribution hazard frailty models, in which random effects may be shared or correlated. We consider three penalty functions, least absolute shrinkage and selection operator (LASSO), smoothly clipped absolute deviation (SCAD) and HL, in our variable selection procedure. We show that the proposed method can be easily implemented using a slight modification to existing h‐likelihood estimation approaches. Numerical studies demonstrate that the proposed procedure using the HL penalty performs well, providing a higher probability of choosing the true model than LASSO and SCAD methods without losing prediction accuracy. The usefulness of the new method is illustrated using two actual datasets from multi‐center clinical trials. Copyright © 2014 John Wiley & Sons, Ltd.     

Misspecified regression model for the subdistribution hazard of a competing risk

We consider a competing risks setting, when evaluating the prognostic influence of an exposure on a specific cause of failure. Two main regression models are used in such analyses, the Cox cause-specific proportional hazards model and the subdistribution proportional hazards model. They are exemplified in a real data example focusing on relapse-free interval in acute leukaemia patients. We examine the properties of the estimator based on the latter model when the true model is the former. An explicit relationship between subdistribution hazards ratio and cause-specific hazards ratio is derived, assuming a flexible parametric distribution for latent failure times.     

Patient death as a censoring event or competing risk event in models of nursing home placement

Participant death is often observed in studies that examine predictors of events, such as hospitalization or institutionalization, in older adult populations. The Cox proportional hazards modeling of the target event, whereby death is treated as a censoring event, is the standard analysis in this competing risks situation. However, the assumption of noninformative censoring applied to a frequently occurring competing event like death may be invalid and complicate interpretation in terms of the probability of the event. Multiple cause‐specific hazard (CSH) models can be estimated, but ambiguities may arise when interpreting covariate effects across multiple CSH models and in terms of the cumulative incidence function (CIF). Alternatively, one can model the proportional hazards of the subdistribution of the CIF and evaluate the covariate effects on the CIF directly. We examine and compare these two approaches with nursing home (NH) placement data from a randomized controlled trial of a counseling and support intervention for spouse‐caregivers of patients with Alzheimer's disease. CSHs for NH placement (where death is treated as a censoring event) and death (where NH placement is treated as a censoring event) and subdistribution hazards of the CIF for NH placement are modeled separately. In the presence of multiple covariates, the intervention effect is significant in both approaches, but the interpretation of the covariate effects requires joint evaluation of all estimated models. Copyright © 2009 John Wiley & Sons, Ltd.     

Flexible modeling of competing risks in survival analysis

Prognostic studies often involve modeling competing risks, where an individual can experience only one of alternative events, and the goal is to estimate hazard functions and covariate effects associated with each event type. Lunn and McNeil proposed data manipulation that permits extending the Cox's proportional hazards model to estimate covariate effects on the hazard of each competing events. However, the hazard functions for competing events are assumed to remain proportional over the entire follow‐up period, implying the same shape of all event‐specific hazards, and covariate effects are restricted to also remain constant over time, even if such assumptions are often questionable. To avoid such limitations, we propose a flexible model to (i) obtain distinct estimates of the baseline hazard functions for each event type, and (ii) allow estimating time‐dependent covariate effects in a parsimonious model. Our flexible competing risks regression model uses smooth cubic regression splines to model the time‐dependent changes in (i) the ratio of event‐specific baseline hazards, and (ii) the covariate effects. In simulations, we evaluate the performance of the proposed estimators and likelihood ratio tests, under different assumptions. We apply the proposed flexible model in a prognostic study of colorectal cancer mortality, with two competing events: ‘death from colorectal cancer’ and ‘death from other causes’. Copyright © 2010 John Wiley & Sons, Ltd.     

Sample size formula for proportional hazards modelling of competing risks

To test the effect of a therapeutic or prognostic factor on the occurrence of a particular cause of failure in the presence of other causes, the interest has shifted in some studies from the modelling of the cause-specific hazard to that of the subdistribution hazard. We present approximate sample size formulas for the proportional hazards modelling of competing risk subdistribution, considering either independent or correlated covariates. The validity of these approximate formulas is investigated through numerical simulations. Two illustrations are provided, a randomized clinical trial, and a prospective prognostic study.     

Bias in progression‐free survival analysis due to intermittent assessment of progression

Cancer clinical trials are routinely designed to assess the effect of treatment on disease progression and death, often in terms of a composite endpoint called progression‐free survival. When progression status is known only at periodic assessment times, the progression time is interval censored, and complications arise in the analysis of progression‐free survival. Despite the advances in methods for dealing with interval‐censored data, naive methods such as right‐endpoint imputation are widely adopted in this setting. We examine the asymptotic and empirical properties of estimators of the marginal progression‐free survival functions and associated treatment effects under this scheme. Specifically, we explore the determinants of the asymptotic bias and point out that there is typically a loss in power of tests for treatment effects. Copyright © 2015 John Wiley & Sons, Ltd.     

Analyzing semi‐competing risks data with missing cause of informative terminal event

Cancer studies frequently yield multiple event times that correspond to landmarks in disease progression, including non‐terminal events (i.e., cancer recurrence) and an informative terminal event (i.e., cancer‐related death). Hence, we often observe semi‐competing risks data. Work on such data has focused on scenarios in which the cause of the terminal event is known. However, in some circumstances, the information on cause for patients who experience the terminal event is missing; consequently, we are not able to differentiate an informative terminal event from a non‐informative terminal event. In this article, we propose a method to handle missing data regarding the cause of an informative terminal event when analyzing the semi‐competing risks data. We first consider the nonparametric estimation of the survival function for the terminal event time given missing cause‐of‐failure data via the expectation–maximization algorithm. We then develop an estimation method for semi‐competing risks data with missing cause of the terminal event, under a pre‐specified semiparametric copula model. We conduct simulation studies to investigate the performance of the proposed method. We illustrate our methodology using data from a study of early‐stage breast cancer. Copyright © 2016 John Wiley & Sons, Ltd.     

Association measures for clustered competing risks

We propose a semiparameteric model for multivariate clustered competing risks data when the cause-specific failure times and the occurrence of competing risk events among subjects within the same cluster are of interest. The cause-specific hazard functions are assumed to follow Cox proportional hazard models, and the associations between failure times given the same or different cause events and the associations between occurrences of competing risk events within the same cluster are investigated through copula models. A cross-odds ratio measure is explored under our proposed models. Two-stage estimation procedure is proposed in which the marginal models are estimated in the first stage, and the dependence parameters are estimated via an expectation-maximization algorithm in the second stage. The proposed estimators are shown to yield consistent and asymptotically normal under mild regularity conditions. Simulation studies are conducted to assess finite sample performance of the proposed method. The proposed technique is demonstrated through an application to a multicenter Bone Marrow transplantation dataset.     

A proportional hazards regression model for the subdistribution with right-censored and left-truncated competing risks data

With competing risks failure time data, one often needs to assess the covariate effects on the cumulative incidence probabilities. Fine and Gray proposed a proportional hazards regression model to directly model the subdistribution of a competing risk. They developed the estimating procedure for right-censored competing risks data, based on the inverse probability of censoring weighting. Right-censored and left-truncated competing risks data sometimes occur in biomedical researches. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with right-censored and left-truncated data. We adopt a new weighting technique to estimate the parameters in this model. We have derived the large sample properties of the proposed estimators. To illustrate the application of the new method, we analyze the failure time data for children with acute leukemia. In this example, the failure times for children who had bone marrow transplants were left truncated.     

Practical recommendations for reporting Fine‐Gray model analyses for competing risk data

In survival analysis, a competing risk is an event whose occurrence precludes the occurrence of the primary event of interest. Outcomes in medical research are frequently subject to competing risks. In survival analysis, there are 2 key questions that can be addressed using competing risk regression models: first, which covariates affect the rate at which events occur, and second, which covariates affect the probability of an event occurring over time. The cause‐specific hazard model estimates the effect of covariates on the rate at which events occur in subjects who are currently event‐free. Subdistribution hazard ratios obtained from the Fine‐Gray model describe the relative effect of covariates on the subdistribution hazard function. Hence, the covariates in this model can also be interpreted as having an effect on the cumulative incidence function or on the probability of events occurring over time. We conducted a review of the use and interpretation of the Fine‐Gray subdistribution hazard model in articles published in the medical literature in 2015. We found that many authors provided an unclear or incorrect interpretation of the regression coefficients associated with this model. An incorrect and inconsistent interpretation of regression coefficients may lead to confusion when comparing results across different studies. Furthermore, an incorrect interpretation of estimated regression coefficients can result in an incorrect understanding about the magnitude of the association between exposure and the incidence of the outcome. The objective of this article is to clarify how these regression coefficients should be reported and to propose suggestions for interpreting these coefficients.     

Simulating biologically plausible complex survival data

Simulation studies are conducted to assess the performance of current and novel statistical models in pre‐defined scenarios. It is often desirable that chosen simulation scenarios accurately reflect a biologically plausible underlying distribution. This is particularly important in the framework of survival analysis, where simulated distributions are chosen for both the event time and the censoring time. This paper develops methods for using complex distributions when generating survival times to assess methods in practice. We describe a general algorithm involving numerical integration and root‐finding techniques to generate survival times from a variety of complex parametric distributions, incorporating any combination of time‐dependent effects, time‐varying covariates, delayed entry, random effects and covariates measured with error. User‐friendly Stata software is provided. Copyright © 2013 John Wiley & Sons, Ltd.     

The analysis of case cohort design in the presence of competing risks with application to estimate the risk of delayed cardiac toxicity among Hodgkin Lymphoma survivors

The case‐cohort design is an economical solution to studying the association between an exposure and a rare disease. When the disease of interest has a delayed occurrence, then other types of event may preclude observation of the disease of interest giving rise to a competing risk situation. In this paper, we introduce a modification of the pseudolikelihood proposed by Prentice (Biometrika 1986; 73 :1–11) for the analysis of case‐cohort design, to accommodate the existence of competing risks. The modification is based on the Fine and Gray (J. Amer. Statist. Assoc. 1999; 94 :496–509) approach to enable the modeling of the hazard of subdistribution. We show through simulations that the estimate that maximizes this modified pseudolikelihood is almost unbiased. The predictive probabilities based on the model are close to the theoretical probabilities. The variance for the estimates can be calculated using the jackknife approach. An application of this method on the analysis of late cardiac morbidity among Hodgkin Lymphoma survivors is presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Adjustment for competing risk in kin-cohort estimation

Kin-cohort design can be used to study the effect of a genetic mutation on the risk of multiple events, using the same study. In this design, the outcome data consist of the event history of the relatives of a sample of genotyped subjects. Existing methods for kin-cohort estimation allow estimation of the risk of one event at a time with the assumption that the censoring events are unrelated to the genetic mutation under study. These methods, however, may produce biased estimates of risk when multiple events are related to the genetic mutation, and follow-up of some of the events may be censored by the onset of other events. Using a competing risk framework to address this problem, we show that cause-specific hazard functions for carriers and noncarriers are identifiable from kin-cohort data. For estimation, we propose an extension of a composite-likelihood approach we described previously. We illustrate the use of the proposed method for estimation of the risk of ovarian cancer from BRCA1/2 mutations in the absence of breast cancer, based on data from the Washington Ashkenazi Kin-Cohort Study. We also evaluate the performance of the proposed estimation method, based on simulated data that were generated following the setup of the Washington Ashkenazi Study.     

Semiparametric accelerated failure time cure rate mixture models with competing risks

Modern medical treatments have substantially improved survival rates for many chronic diseases and have generated considerable interest in developing cure fraction models for survival data with a non‐ignorable cured proportion. Statistical analysis of such data may be further complicated by competing risks that involve multiple types of endpoints. Regression analysis of competing risks is typically undertaken via a proportional hazards model adapted on cause‐specific hazard or subdistribution hazard. In this article, we propose an alternative approach that treats competing events as distinct outcomes in a mixture. We consider semiparametric accelerated failure time models for the cause‐conditional survival function that are combined through a multinomial logistic model within the cure‐mixture modeling framework. The cure‐mixture approach to competing risks provides a means to determine the overall effect of a treatment and insights into how this treatment modifies the components of the mixture in the presence of a cure fraction. The regression and nonparametric parameters are estimated by a nonparametric kernel‐based maximum likelihood estimation method. Variance estimation is achieved through resampling methods for the kernel‐smoothed likelihood function. Simulation studies show that the procedures work well in practical settings. Application to a sarcoma study demonstrates the use of the proposed method for competing risk data with a cure fraction.