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.     

Local influence for the subdistribution of a competing risk

Local influence measures have been shown to be a useful tool in identifying influential individuals, and assessing model behaviour towards small perturbations. In the setting of competing risks, we developed a measure of local influence for the Fine and Gray model for the subdistribution hazard. The plot of local influence showed some relationship with time failure rank. This was illustrated on a real data set from a randomized clinical trial in acute promyelocytic leukaemia and on a simulation study.     

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.     

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.     

Dealing with competing risks: testing covariates and calculating sample size

It is universally agreed that Kaplan-Meier estimates overestimate the probability of the event of interest in the presence of competing risks. Kalbfleisch and Prentice recommend using the cumulative incidence as an estimate of the probability of an event of interest. However, there is no consensus on how to test the effect of a covariate in the presence of competing risks. Using simulations, this paper illustrates that the Cox proportional hazards model gives valid results when employed in testing the effect of a covariate on the hazard rate and when estimating the hazard ratio. A method to calculate the sample size for testing the effect of a covariate on outcome in the presence of competing risks is also provided.     

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.     

Sample size determination for jointly testing a cause‐specific hazard and the all‐cause hazard in the presence of competing risks

This article considers sample size determination for jointly testing a cause‐specific hazard and the all‐cause hazard for competing risks data. The cause‐specific hazard and the all‐cause hazard jointly characterize important study end points such as the disease‐specific survival and overall survival, which are commonly used as coprimary end points in clinical trials. Specifically, we derive sample size calculation methods for 2‐group comparisons based on an asymptotic chi‐square joint test and a maximum joint test of the aforementioned quantities, taking into account censoring due to lost to follow‐up as well as staggered entry and administrative censoring. We illustrate the application of the proposed methods using the Die Deutsche Diabetes Dialyse Studies clinical trial. An R package “powerCompRisk” has been developed and made available at the CRAN R library.     

Estimating twin concordance for bivariate competing risks twin data

For twin time‐to‐event data, we consider different concordance probabilities, such as the casewise concordance that are routinely computed as a measure of the lifetime dependence/correlation for specific diseases. The concordance probability here is the probability that both twins have experienced the event of interest. Under the assumption that both twins are censored at the same time, we show how to estimate this probability in the presence of right censoring, and as a consequence, we can then estimate the casewise twin concordance. In addition, we can model the magnitude of within pair dependence over time, and covariates may be further influential on the marginal risk and dependence structure. We establish the estimators large sample properties and suggest various tests, for example, for inferring familial influence. The method is demonstrated and motivated by specific twin data on cancer events with the competing risk death. We thus aim to quantify the degree of dependence through the casewise concordance function and show a significant genetic component. Copyright © 2013 John Wiley & Sons, Ltd.     

Model selection in competing risks regression

In the analysis of time‐to‐event data, the problem of competing risks occurs when an individual may experience one, and only one, of m different types of events. The presence of competing risks complicates the analysis of time‐to‐event data, and standard survival analysis techniques such as Kaplan–Meier estimation, log‐rank test and Cox modeling are not always appropriate and should be applied with caution. Fine and Gray developed a method for regression analysis that models the hazard that corresponds to the cumulative incidence function. This model is becoming widely used by clinical researchers and is now available in all the major software environments. Although model selection methods for Cox proportional hazards models have been developed, few methods exist for competing risks data. We have developed stepwise regression procedures, both forward and backward, based on AIC, BIC, and BICcr (a newly proposed criteria that is a modified BIC for competing risks data subject to right censoring) as selection criteria for the Fine and Gray model. We evaluated the performance of these model selection procedures in a large simulation study and found them to perform well. We also applied our procedures to assess the importance of bone mineral density in predicting the absolute risk of hip fracture in the Women's Health Initiative–Observational Study, where mortality was the competing risk. We have implemented our method as a freely available R package called crrstep. Copyright © 2013 John Wiley & Sons, Ltd.     

The graft-versus-leukaemia effect after allogeneic bone-marrow transplantation: assessment through competing risks approaches

In failure time studies involving a chronic disease such as cancer, data often focus on one or more non-fatal events, in addition to survival, to describe the course of the disease. In the example of allogeneic bone marrow transplantation in leukaemic patients, acute graft-versus host disease (aGvHD), relapse and death were taken as the reported events, and we focused on testing the existence of graft-versus-leukaemia (GvL) effect, i.e. that the occurrence of aGvHD modifies the probability of relapse. One of the weaknesses of the standard competing risks models is their inability to model secondary relapses. We thus derived, from two competing risks models, two estimators of cumulative incidence functions of primary and secondary relapses, as well as statistics to test the GvL effect. The approach is illustrated by application to a data set from a cohort of 442 children with acute leukaemia who received an unrelated transplant.     

Sample size calculation for the proportional hazards cure model

In clinical trials with time‐to‐event endpoints, it is not uncommon to see a significant proportion of patients being cured (or long‐term survivors), such as trials for the non‐Hodgkins lymphoma disease. The popularly used sample size formula derived under the proportional hazards (PH) model may not be proper to design a survival trial with a cure fraction, because the PH model assumption may be violated. To account for a cure fraction, the PH cure model is widely used in practice, where a PH model is used for survival times of uncured patients and a logistic distribution is used for the probability of patients being cured. In this paper, we develop a sample size formula on the basis of the PH cure model by investigating the asymptotic distributions of the standard weighted log‐rank statistics under the null and local alternative hypotheses. The derived sample size formula under the PH cure model is more flexible because it can be used to test the differences in the short‐term survival and/or cure fraction. Furthermore, we also investigate as numerical examples the impacts of accrual methods and durations of accrual and follow‐up periods on sample size calculation. The results show that ignoring the cure rate in sample size calculation can lead to either underpowered or overpowered studies. We evaluate the performance of the proposed formula by simulation studies and provide an example to illustrate its application with the use of data from a melanoma trial. Copyright © 2012 John Wiley & Sons, Ltd.     

Direct likelihood inference on the cause‐specific cumulative incidence function: A flexible parametric regression modelling approach

In a competing risks analysis, interest lies in the cause‐specific cumulative incidence function (CIF) that can be calculated by either (1) transforming on the cause‐specific hazard or (2) through its direct relationship with the subdistribution hazard. We expand on current competing risks methodology from within the flexible parametric survival modelling framework (FPM) and focus on approach (2). This models all cause‐specific CIFs simultaneously and is more useful when we look to questions on prognosis. We also extend cure models using a similar approach described by Andersson et al for flexible parametric relative survival models. Using SEER public use colorectal data, we compare and contrast our approach with standard methods such as the Fine & Gray model and show that many useful out‐of‐sample predictions can be made after modelling the cause‐specific CIFs using an FPM approach. Alternative link functions may also be incorporated such as the logit link. Models can also be easily extended for time‐dependent effects.     

Semiparametric regression on cumulative incidence function with interval‐censored competing risks data

Many biomedical and clinical studies with time‐to‐event outcomes involve competing risks data. These data are frequently subject to interval censoring. This means that the failure time is not precisely observed but is only known to lie between two observation times such as clinical visits in a cohort study. Not taking into account the interval censoring may result in biased estimation of the cause‐specific cumulative incidence function, an important quantity in the competing risks framework, used for evaluating interventions in populations, for studying the prognosis of various diseases, and for prediction and implementation science purposes. In this work, we consider the class of semiparametric generalized odds rate transformation models in the context of sieve maximum likelihood estimation based on B‐splines. This large class of models includes both the proportional odds and the proportional subdistribution hazard models (i.e., the Fine–Gray model) as special cases. The estimator for the regression parameter is shown to be consistent, asymptotically normal and semiparametrically efficient. Simulation studies suggest that the method performs well even with small sample sizes. As an illustration, we use the proposed method to analyze data from HIV‐infected individuals obtained from a large cohort study in sub‐Saharan Africa. We also provide the R function ciregic that implements the proposed method and present an illustrative example. Copyright © 2017 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.     

A review of the use of time-varying covariates in the Fine-Gray subdistribution hazard competing risk regression model

In survival analysis, time-varying covariates are covariates whose value can change during follow-up. Outcomes in medical research are frequently subject to competing risks (events precluding the occurrence of the primary outcome). We review the types of time-varying covariates and highlight the effect of their inclusion in the subdistribution hazard model. External time-dependent covariates are external to the subject, can effect the failure process, but are not otherwise involved in the failure mechanism. Internal time-varying covariates are measured on the subject, can effect the failure process directly, and may also be impacted by the failure mechanism. In the absence of competing risks, a consequence of including internal time-dependent covariates in the Cox model is that one cannot estimate the survival function or the effect of covariates on the survival function. In the presence of competing risks, the inclusion of internal time-varying covariates in a subdistribution hazard model results in the loss of the ability to estimate the cumulative incidence function (CIF) or the effect of covariates on the CIF. Furthermore, the definition of the risk set for the subdistribution hazard function can make defining internal time-varying covariates difficult or impossible. We conducted a review of the use of time-varying covariates in subdistribution hazard models in articles published in the medical literature in 2015 and in the first 5 months of 2019. Seven percent of articles published included a time-varying covariate. Several inappropriately described a time-varying covariate as having an association with the risk of the outcome.     

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.     

Sequential trials in the context of competing risks: Concepts and case study,with R and SAS code

Sequential designs and competing risks methodology are both well established. Their combined use has recently received some attention from a theoretical perspective, but their joint application in practice has been discussed less. The aim of this paper is to provide the applied statistician with a basic understanding of both sequential design theory and competing risks methodology and how to combine them in practice. Relevant references to more detailed theoretical discussions are provided, and all discussions are illustrated using a real case study. Extensive R and SAS code is provided in the online Supplementary Material.     

Goodness‐of‐fit test for proportional subdistribution hazards model

This paper concerns using modified weighted Schoenfeld residuals to test the proportionality of subdistribution hazards for the Fine–Gray model, similar to the tests proposed by Grambsch and Therneau for independently censored data. We develop a score test for the time‐varying coefficients based on the modified Schoenfeld residuals derived assuming a certain form of non‐proportionality. The methods perform well in simulations and a real data analysis of breast cancer data, where the treatment effect exhibits non‐proportional hazards. Copyright © 2013 John Wiley & Sons, Ltd.     

Flexible parametric modelling of the cause‐specific cumulative incidence function

Competing risks arise with time‐to‐event data when individuals are at risk of more than one type of event and the occurrence of one event precludes the occurrence of all other events. A useful measure with competing risks is the cause‐specific cumulative incidence function (CIF), which gives the probability of experiencing a particular event as a function of follow‐up time, accounting for the fact that some individuals may have a competing event. When modelling the cause‐specific CIF, the most common model is a semi‐parametric proportional subhazards model. In this paper, we propose the use of flexible parametric survival models to directly model the cause‐specific CIF where the effect of follow‐up time is modelled using restricted cubic splines. The models provide smooth estimates of the cause‐specific CIF with the important advantage that the approach is easily extended to model time‐dependent effects. The models can be fitted using standard survival analysis tools by a combination of data expansion and introducing time‐dependent weights. Various link functions are available that allow modelling on different scales and have proportional subhazards, proportional odds and relative absolute risks as particular cases. We conduct a simulation study to evaluate how well the spline functions approximate subhazard functions with complex shapes. The methods are illustrated using data from the European Blood and Marrow Transplantation Registry showing excellent agreement between parametric estimates of the cause‐specific CIF and those obtained from a semi‐parametric model. We also fit models relaxing the proportional subhazards assumption using alternative link functions and/or including time‐dependent effects. Copyright © 2016 John Wiley & Sons, Ltd.     

Assessing treatment benefit with competing risks not affected by the randomized treatment

The comparison of overall survival curves between treatment arms will always be of interest in a randomized clinical trial involving a life‐shortening disease. In some settings, the experimental treatment is only expected to affect the deaths caused by the disease, and the proportion of deaths caused by the disease is relatively low. In these settings, the ability to assess treatment‐effect differences between Kaplan–Meier survival curves can be hampered by the large proportion of deaths in both arms that are unrelated to the disease. To address this problem, frequently displayed are cause‐specific survival curves or cumulative incidence curves, which respectively censor and immortalize events (deaths) not caused by the disease. However, the differences between the experimental and control treatment arms for these curves overestimate the difference between the overall survival curves for the treatment arms and thus could result in overestimation of the benefit of the experimental treatment for the patients. To address this issue, we propose new estimators of overall survival for the treatment arms that are appropriate when the treatment does not affect the non‐disease‐related deaths. These new estimators give a more precise estimate of the treatment benefit, potentially enabling future patients to make a more informed decision concerning treatment choice. We also consider the case where an exponential assumption allows the simple presentation of mortality rates as the outcome measures. Applications are given for estimating overall survival in a prostate‐cancer treatment randomized clinical trial, and for estimating the overall mortality rates in a prostate‐cancer screening trial. Copyright © 2014 John Wiley & Sons, Ltd.