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.     

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.     

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.     

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.     

Evaluating competing adverse and beneficial outcomes using a mixture model

A competing risk framework occurs when individuals have the potential to experience only one of the several mutually exclusive outcomes. Standard survival methods often overestimate the cumulative incidence of events when competing events are censored. Mixture distributions have been previously applied to the competing risk framework to obtain inferences regarding the subdistribution of an event of interest. Often the competing event is treated as a nuisance, but it may be of interest to compare adverse events against the beneficial outcome when dealing with an intervention. In this paper, methods for using a mixture model to estimate an adverse-benefit ratio curve (ratio of the cumulative incidence curves for the two competing events) and the ratio of the subhazards for the two competing events are presented. A parametric approach is described with some remarks for extending the model to include uncertainty in the event type that occurred, left truncation in order to allow for time-dependent analyses, and uncertainty in the timing of the event resulting in interval censoring. The methods are illustrated with data from an HIV clinical cohort examining whether individuals initiating effective antiretroviral therapy have a greater risk of antiretroviral discontinuation or switching compared with HIV RNA suppression.     

Assessing cumulative incidence functions under the semiparametric additive risk model

In analyzing competing risks data, a quantity of considerable interest is the cumulative incidence function. Often, the effect of covariates on the cumulative incidence function is modeled via the proportional hazards model for the cause‐specific hazard function. As the proportionality assumption may be too restrictive in practice, we consider an alternative more flexible semiparametric additive hazards model of (Biometrika 1994; 81 :501–514) for the cause‐specific hazard. This model specifies the effect of covariates on the cause‐specific hazard to be additive as well as allows the effect of some covariates to be fixed and that of others to be time varying. We present an approach for constructing confidence intervals as well as confidence bands for the cause‐specific cumulative incidence function of subjects with given values of the covariates. Furthermore, we also present an approach for constructing confidence intervals and confidence bands for comparing two cumulative incidence functions given values of the covariates. The finite sample property of the proposed estimators is investigated through simulations. We conclude our paper with an analysis of the well‐known malignant melanoma data using our method. Published in 2009 by John Wiley & Sons, Ltd.     

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.     

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.     

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.     

Instrumental variable with competing risk model

In this paper, we discuss causal inference on the efficacy of a treatment or medication on a time‐to‐event outcome with competing risks. Although the treatment group can be randomized, there can be confoundings between the compliance and the outcome. Unmeasured confoundings may exist even after adjustment for measured covariates. Instrumental variable methods are commonly used to yield consistent estimations of causal parameters in the presence of unmeasured confoundings. On the basis of a semiparametric additive hazard model for the subdistribution hazard, we propose an instrumental variable estimator to yield consistent estimation of efficacy in the presence of unmeasured confoundings for competing risk settings. We derived the asymptotic properties for the proposed estimator. The estimator is shown to be well performed under finite sample size according to simulation results. We applied our method to a real transplant data example and showed that the unmeasured confoundings lead to significant bias in the estimation of the effect (about 50% attenuated). Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

Semiparametric mixture cure model analysis with competing risks data: Application to vascular access thrombosis data

The article is motivated by a nephrology study in Taiwan, which enrolled hemodialysis patients who suffered from vascular access thrombosis. After treatment, some patients were cured of thrombosis, while some may experience recurrence of either type (acute or nonacute) of vascular access thrombosis. Our major interest is to estimate the cumulative incidence probability of time to the first recurrence of acute thrombosis after therapy. Since the occurrence of one type of vascular access thrombosis precludes occurrence of the other type, patients are subject to competing risks. To account for the presence of competing risks and cured patients, we develop a mixture model approach to the regression analysis of competing-risks data with a cure fraction. We make inference about the effects of factors on both the cure rate and cumulative incidence function (CIF) for a failure of interest, which are separately specified in the logistic regression model and semiparametric regression model with time-varying and time-invariant effects. Based on two-stage method, we develop novel estimation equations using the inverse probability censoring weight techniques. The asymptotic properties of the estimators are rigorously studied and the plug-in variance estimators can be obtained for constructing interval estimators. We also propose a lack-of-fit test for assessing the adequacy of the proposed model and several tests for time-varying effects. The simulation studies and vascular access thrombosis data analysis are conducted to illustrate the proposed method.     

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.     

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.     

Predicting the absolute risk of dying from colorectal cancer and from other causes using population-based cancer registry data

This paper describes how population cancer registry data from the Surveillance, Epidemiology, and End Results program of the National Cancer Institute can be used to develop a prognostic model to predict the absolute risk of mortality from cancer and from other causes for an individual with specific covariates. It incorporates previously developed methods for competing risk modeling along with an imputation method to address missing cause of death information. We illustrate these approaches with colorectal cancer and evaluate the model discriminatory and calibration accuracy by time-dependent area under the receiver operating characteristic curve and calibration plot.     

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.     

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.     

A new generalization of Weibull distribution with application to a breast cancer data set

In this article, we propose a new generalization of the Weibull distribution, which incorporates the exponentiated Weibull distribution introduced by Mudholkar and Srivastava (IEEE Trans. Reliab. 1993; 42 :299–302) as a special case. We refer to the new family of distributions as the beta‐Weibull distribution. We investigate the potential usefulness of the beta‐Weibull distribution for modeling censored survival data from biomedical studies. Several other generalizations of the standard two‐parameter Weibull distribution are compared with regards to maximum likelihood inference of the cumulative incidence function, under the setting of competing risks. These Weibull‐based parametric models are fit to a breast cancer data set from the National Surgical Adjuvant Breast and Bowel Project. In terms of statistical significance of the treatment effect and model adequacy, all generalized models lead to similar conclusions, suggesting that the beta‐Weibull family is a reasonable candidate for modeling survival data. Copyright © 2009 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.