Dynamic prediction by landmarking in competing risks

We propose an extension of the landmark model for ordinary survival data as a new approach to the problem of dynamic prediction in competing risks with time‐dependent covariates. We fix a set of landmark time points t_LM within the follow‐up interval. For each of these landmark time points t_LM, we create a landmark data set by selecting individuals at risk at t_LM; we fix the value of the time‐dependent covariate in each landmark data set at t_LM. We assume Cox proportional hazard models for the cause‐specific hazards and consider smoothing the (possibly) time‐dependent effect of the covariate for the different landmark data sets. Fitting this model is possible within the standard statistical software. We illustrate the features of the landmark modelling on a real data set on bone marrow transplantation. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

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.     

A random forest approach for competing risks based on pseudo‐values

Random forest is a supervised learning method that combines many classification or regression trees for prediction. Here we describe an extension of the random forest method for building event risk prediction models in survival analysis with competing risks. In case of right‐censored data, the event status at the prediction horizon is unknown for some subjects. We propose to replace the censored event status by a jackknife pseudo‐value, and then to apply an implementation of random forests for uncensored data. Because the pseudo‐responses take on values on a continuous scale, the node variance is chosen as split criterion for growing regression trees. In a simulation study, the pseudo split criterion is compared with the Gini split criterion when the latter is applied to the uncensored event status. To investigate the resulting pseudo random forest method for building risk prediction models, we analyze it in a simulation study of predictive performance where we compare it to Cox regression and random survival forest. The method is further illustrated in two real data sets. Copyright © 2013 John Wiley & Sons, Ltd.     

Tutorial in biostatistics: competing risks and multi-state models

Standard survival data measure the time span from some time origin until the occurrence of one type of event. If several types of events occur, a model describing progression to each of these competing risks is needed. Multi-state models generalize competing risks models by also describing transitions to intermediate events. Methods to analyze such models have been developed over the last two decades. Fortunately, most of the analyzes can be performed within the standard statistical packages, but may require some extra effort with respect to data preparation and programming. This tutorial aims to review statistical methods for the analysis of competing risks and multi-state models. Although some conceptual issues are covered, the emphasis is on practical issues like data preparation, estimation of the effect of covariates, and estimation of cumulative incidence functions and state and transition probabilities. Examples of analysis with standard software are shown.     

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.     

Calibration plots for risk prediction models in the presence of competing risks

A predicted risk of 17% can be called reliable if it can be expected that the event will occur to about 17 of 100 patients who all received a predicted risk of 17%. Statistical models can predict the absolute risk of an event such as cardiovascular death in the presence of competing risks such as death due to other causes. For personalized medicine and patient counseling, it is necessary to check that the model is calibrated in the sense that it provides reliable predictions for all subjects. There are three often encountered practical problems when the aim is to display or test if a risk prediction model is well calibrated. The first is lack of independent validation data, the second is right censoring, and the third is that when the risk scale is continuous, the estimation problem is as difficult as density estimation. To deal with these problems, we propose to estimate calibration curves for competing risks models based on jackknife pseudo‐values that are combined with a nearest neighborhood smoother and a cross‐validation approach to deal with all three problems. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

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.     

Comparing predictions among competing risks models with time‐dependent covariates

Prediction of cumulative incidences is often a primary goal in clinical studies with several endpoints. We compare predictions among competing risks models with time‐dependent covariates. For a series of landmark time points, we study the predictive accuracy of a multi‐state regression model, where the time‐dependent covariate represents an intermediate state, and two alternative landmark approaches. At each landmark time point, the prediction performance is measured as the t‐year expected Brier score where pseudovalues are constructed in order to deal with right‐censored event times. We apply the methods to data from a bone marrow transplant study where graft versus host disease is considered a time‐dependent covariate for predicting relapse and death in remission. Copyright © 2013 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 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.     

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.     

Number needed to treat for time‐to‐event data with competing risks

The number needed to treat is a tool often used in clinical settings to illustrate the effect of a treatment. It has been widely adopted in the communication of risks to both clinicians and non‐clinicians, such as patients, who are better able to understand this measure than absolute risk or rate reductions. The concept was introduced by Laupacis, Sackett, and Roberts in 1988 for binary data, and extended to time‐to‐event data by Altman and Andersen in 1999. However, up to the present, there is no definition of the number needed to treat for time‐to‐event data with competing risks. This paper introduces such a definition using the cumulative incidence function and suggests non‐parametric and semi‐parametric inferential methods for right‐censored time‐to‐event data in the presence of competing risks. The procedures are illustrated using the data from a breast cancer clinical trial. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Joint modeling of longitudinal ordinal data and competing risks survival times and analysis of the NINDS rt‐PA stroke trial

Existing joint models for longitudinal and survival data are not applicable for longitudinal ordinal outcomes with possible non‐ignorable missing values caused by multiple reasons. We propose a joint model for longitudinal ordinal measurements and competing risks failure time data, in which a partial proportional odds model for the longitudinal ordinal outcome is linked to the event times by latent random variables. At the survival endpoint, our model adopts the competing risks framework to model multiple failure types at the same time. The partial proportional odds model, as an extension of the popular proportional odds model for ordinal outcomes, is more flexible and at the same time provides a tool to test the proportional odds assumption. We use a likelihood approach and derive an EM algorithm to obtain the maximum likelihood estimates of the parameters. We further show that all the parameters at the survival endpoint are identifiable from the data. Our joint model enables one to make inference for both the longitudinal ordinal outcome and the failure times simultaneously. In addition, the inference at the longitudinal endpoint is adjusted for possible non‐ignorable missing data caused by the failure times. We apply the method to the NINDS rt‐PA stroke trial. Our study considers the modified Rankin Scale only. Other ordinal outcomes in the trial, such as the Barthel and Glasgow scales, can be treated in the same way. Copyright © 2009 John Wiley & Sons, Ltd.     

Parametric multistate survival models: Flexible modelling allowing transition‐specific distributions with application to estimating clinically useful measures of effect differences

Multistate models are increasingly being used to model complex disease profiles. By modelling transitions between disease states, accounting for competing events at each transition, we can gain a much richer understanding of patient trajectories and how risk factors impact over the entire disease pathway. In this article, we concentrate on parametric multistate models, both Markov and semi‐Markov, and develop a flexible framework where each transition can be specified by a variety of parametric models including exponential, Weibull, Gompertz, Royston‐Parmar proportional hazards models or log‐logistic, log‐normal, generalised gamma accelerated failure time models, possibly sharing parameters across transitions. We also extend the framework to allow time‐dependent effects. We then use an efficient and generalisable simulation method to calculate transition probabilities from any fitted multistate model, and show how it facilitates the simple calculation of clinically useful measures, such as expected length of stay in each state, and differences and ratios of proportion within each state as a function of time, for specific covariate patterns. We illustrate our methods using a dataset of patients with primary breast cancer. User‐friendly Stata software is provided.     

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.     

Quantifying and estimating the predictive accuracy for censored time‐to‐event data with competing risks

This paper focuses on quantifying and estimating the predictive accuracy of prognostic models for time‐to‐event outcomes with competing events. We consider the time‐dependent discrimination and calibration metrics, including the receiver operating characteristics curve and the Brier score, in the context of competing risks. To address censoring, we propose a unified nonparametric estimation framework for both discrimination and calibration measures, by weighting the censored subjects with the conditional probability of the event of interest given the observed data. The proposed method can be extended to time‐dependent predictive accuracy metrics constructed from a general class of loss functions. We apply the methodology to a data set from the African American Study of Kidney Disease and Hypertension to evaluate the predictive accuracy of a prognostic risk score in predicting end‐stage renal disease, accounting for the competing risk of pre–end‐stage renal disease death, and evaluate its numerical performance in extensive simulation studies.