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.     

Evaluating hospital performance based on excess cause‐specific incidence

Formal evaluation of hospital performance in specific types of care is becoming an indispensable tool for quality assurance in the health care system. When the prime concern lies in reducing the risk of a cause‐specific event, we propose to evaluate performance in terms of an average excess cumulative incidence, referring to the center's observed patient mix. Its intuitive interpretation helps give meaning to the evaluation results and facilitates the determination of important benchmarks for hospital performance. We apply it to the evaluation of cerebrovascular deaths after stroke in Swedish stroke centers, using data from Riksstroke, the Swedish stroke registry. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Estimation of the adjusted cause-specific cumulative probability using flexible regression models for the cause-specific hazards

In competing risks setting, we account for death according to a specific cause and the quantities of interest are usually the cause-specific hazards (CSHs) and the cause-specific cumulative probabilities. A cause-specific cumulative probability can be obtained with a combination of the CSHs or via the subdistribution hazard. Here, we modeled the CSH with flexible hazard-based regression models using B-splines for the baseline hazard and time-dependent (TD) effects. We derived the variance of the cause-specific cumulative probabilities at the population level using the multivariate delta method and showed how we could easily quantify the impact of a covariate on the cumulative probability scale using covariate-adjusted cause-specific cumulative probabilities and their difference. We conducted a simulation study to evaluate the performance of this approach in its ability to estimate the cumulative probabilities using different functions for the cause-specific log baseline hazard and with or without a TD effect. In the scenario with TD effect, we tested both well-specified and misspecified models. We showed that the flexible regression models perform nearly as well as the nonparametric method, if we allow enough flexibility for the baseline hazards. Moreover, neglecting the TD effect hardly affects the cumulative probabilities estimates of the whole population but impacts them in the various subgroups. We illustrated our approach using data from people diagnosed with monoclonal gammopathy of undetermined significance and provided the R-code to derive those quantities, as an extension of the R-package mexhaz .     

Inference for cumulative incidence quantiles via parametric and nonparametric approaches

In survival analysis, a point estimate and confidence interval for median survival time have been frequently used to summarize the survival curve. However, such quantile analyses on competing risks data have not been widely investigated. In this paper, we propose parametric inferences for quantiles from the cumulative incidence function and develop parametric confidence intervals for quantiles. In addition, we study a simplified method of inference for the nonparametric approach. We compare the parametric and nonparametric inferences in empirical studies. Simulation studies show that the procedures perform well, with parametric analyses yielding smaller mean square error when the model is not too badly misspecified. We illustrate the methods with data from a breast cancer clinical trial.     

Regression modeling of the cumulative incidence function with missing causes of failure using pseudo‐values

Competing risks arise when patients may fail from several causes. Strategies for modeling event‐specific quantities often assume that the cause of failure is known for all patients, but this is seldom the case. Several authors have addressed the problem of modeling the cause‐specific hazard rates with missing causes of failure. In contrast, direct modeling of the cumulative incidence function has received little attention. We provide a general framework for regression modeling of this function in the missing cause setting, encompassing key models such as the Fine and Gray and additive models, by considering two extensions of the Andersen–Klein pseudo‐value approach. The first extension is a novel inverse probability weighting method, whereas the second extension is based on a previously proposed multiple imputation procedure. We evaluated the gain in using these approaches with small samples in an extensive simulation study. We analyzed the data from an Eastern Cooperative Oncology Group breast cancer treatment clinical trial to illustrate the practical value and ease of implementation of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Smooth semi‐nonparametric (SNP) estimation of the cumulative incidence function

This paper presents a novel approach to estimation of the cumulative incidence function in the presence of competing risks. The underlying statistical model is specified via a mixture factorization of the joint distribution of the event type and the time to the event. The time to event distributions conditional on the event type are modeled using smooth semi‐nonparametric densities. One strength of this approach is that it can handle arbitrary censoring and truncation while relying on mild parametric assumptions. A stepwise forward algorithm for model estimation and adaptive selection of smooth semi‐nonparametric polynomial degrees is presented, implemented in the statistical software R, evaluated in a sequence of simulation studies, and applied to data from a clinical trial in cryptococcal meningitis. The simulations demonstrate that the proposed method frequently outperforms both parametric and nonparametric alternatives. They also support the use of ‘ad hoc’ asymptotic inference to derive confidence intervals. An extension to regression modeling is also presented, and its potential and challenges are discussed. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Nonparametric estimation of the cumulative incidence function under outcome misclassification using external validation data

Misclassification of outcomes or event types is common in health sciences research and can lead to serious bias when estimating the cumulative incidence functions in settings with competing risks. Recent work has shown how to estimate nonparametric cumulative incidence functions in the presence of nondifferential outcome misclassification when the misclassification probabilities are known. Here, we extend this approach to account for misclassification that is differential with respect to important predictors of the outcome using misclassification probabilities estimated from external validation data. Moreover, we propose a bootstrap approach in which the observations from both the main study data and the external validation study are resampled to allow the uncertainty in the misclassification probabilities to propagate through the analysis into the final confidence intervals, ensuring appropriate confidence interval coverage probabilities. The proposed estimator is shown to be uniformly consistent and simulation studies indicate that both the estimator and the standard error estimation approach perform well in finite samples. The methodology is applied to estimate the cumulative incidence of death and disengagement from HIV care in a large cohort of HIV infected individuals in sub-Saharan Africa, where a significant death underreporting issue leads to outcome misclassification. This analysis uses external validation data from a separate study conducted in the same country.     

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.     

Multiple imputation methods for nonparametric inference on cumulative incidence with missing cause of failure

We propose a nonparametric approach for cumulative incidence estimation when causes of failure are unknown or missing for some subjects. Under the missing at random assumption, we estimate the cumulative incidence function using multiple imputation methods. We develop asymptotic theory for the cumulative incidence estimators obtained from multiple imputation methods. We also discuss how to construct confidence intervals for the cumulative incidence function and perform a test for comparing the cumulative incidence functions in two samples with missing cause of failure. Through simulation studies, we show that the proposed methods perform well. The methods are illustrated with data from a randomized clinical trial in early stage breast cancer. Copyright © 2014 John Wiley & Sons, Ltd.     

Excess cumulative incidence estimation for matched cohort survival studies

We suggest a regression approach to estimate the excess cumulative incidence function (CIF) when matched data are available. In a competing risk setting, we define the excess risk as the difference between the CIF in the exposed group and the background CIF observed in the unexposed group. We show that the excess risk can be estimated through an extended binomial regression model that actively uses the matched structure of the data, avoiding further estimation of both the exposed and the unexposed CIFs. The method naturally deals with two time scales, age and time since exposure and simplifies how to deal with the left truncation on the age time-scale. The model makes it easy to predict individual excess risk scenarios and allows for a direct interpretation of the covariate effects on the cumulative incidence scale. After introducing the model and some theory to justify the approach, we show via simulations that our model works well in practice. We conclude by applying the excess risk model to data from the ALiCCS study to investigate the excess risk of late events in childhood cancer survivors.     

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.     

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.     

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.     

Estimation of life‐years gained and cost effectiveness based on cause‐specific mortality

Cost‐effectiveness analysis is usually based on life‐years gained estimated from all‐cause mortality. When an intervention affects only a few causes of death accounting for a small fraction of all deaths, this approach may lack precision. We develop a novel technique for cost‐effectiveness analysis when life‐years gained are estimated from cause‐specific mortality, allowing for competing causes of death. In the context of randomised trial data, we adjust for other‐cause mortality combined across randomised groups. This method yields a greater precision than analysis based on total mortality, and we show application to life‐years gained, quality‐adjusted life‐years gained, incremental costs, and cost effectiveness. In multi‐state health economic models, however, mortality from competing causes is commonly derived from national statistics and is assumed to be known and equal across intervention groups. In such models, our method based on cause‐specific mortality and standard methods using total mortality give essentially identical estimates and precision. The methods are applied to a randomised trial and a health economic model, both of screening for abdominal aortic aneurysm. A gain in precision for cost‐effectiveness estimates is clearly helpful for decision making, but it is important to ensure that ‘cause‐specific mortality’ is defined to include all causes of death potentially affected by the intervention. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

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 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.     

A competing-risks nomogram for sarcoma-specific death following local recurrence

The majority of staging systems focus on the definition of stage, and, therefore, prediction of prognosis. In the current era of clinical trial research, it has become apparent that the clinical stage alone is not sufficient to assess patient risk of treatment failure. As the number of biological markers increases, our ability to partition the traditional disease classification system improves, and our ability to predict patient success continues to increase. One approach to quantifying individual patient risk is through the nomogram. Nomograms are graphical representations of statistical models, which provide the probability of treatment outcome based on patient-specific covariates. We will focus on the use of the nomogram when the response variable is time to failure and there are multiple, possibly dependent, competing causes of failure. In this setting, estimation of the failure probability through direct application of the Cox proportional hazards model provides the probability of failure (for example, death from cancer) assuming failure from a dependent competing cause will not occur. In many clinical settings this is an unrealistic assumption. The purpose of this study is to illustrate the use of the conditional cumulative incidence function for providing a patient-specific prediction of the probability of failure in the setting of competing risks. A competing risks nomogram is produced to estimate the probability of death due to sarcoma for patients who have already developed a local recurrence of their initially treated soft-tissue sarcoma.     

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.