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.     

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

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

Correcting for the dependent competing risk of treatment using inverse probability of censoring weighting and copulas in the estimation of natural conception chances

When estimating the probability of natural conception from observational data on couples with an unfulfilled child wish, the start of assisted reproductive therapy (ART) is a competing event that cannot be assumed to be independent of natural conception. In clinical practice, interest lies in the probability of natural conception in the absence of ART, as this probability determines the need for therapy. We thus want to estimate the marginal cumulative pregnancy distribution. Without assumptions on the dependence structure between the two competing events, this marginal distribution is not identifiable. We first use inverse probability of censoring weighting assuming that the factors influencing the choice to start ART are known. Then, we parameterize the event distributions for conception and for start of ART and use copulas to account for the dependency between both events. By using these two ways of correcting for the dependent risk of treatment, we obtain a plausible estimation region for the cumulative pregnancy curve and for the prognostic effect of tubal tests. Copyright © 2014 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.     

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.     

A causal framework for classical statistical estimands in failure-time settings with competing events

In failure-time settings, a competing event is any event that makes it impossible for the event of interest to occur. For example, cardiovascular disease death is a competing event for prostate cancer death because an individual cannot die of prostate cancer once he has died of cardiovascular disease. Various statistical estimands have been defined as possible targets of inference in the classical competing risks literature. Many reviews have described these statistical estimands and their estimating procedures with recommendations about their use. However, this previous work has not used a formal framework for characterizing causal effects and their identifying conditions, which makes it difficult to interpret effect estimates and assess recommendations regarding analytic choices. Here we use a counterfactual framework to explicitly define each of these classical estimands. We clarify that, depending on whether competing events are defined as censoring events, contrasts of risks can define a total effect of the treatment on the event of interest or a direct effect of the treatment on the event of interest not mediated by the competing event. In contrast, regardless of whether competing events are defined as censoring events, counterfactual hazard contrasts cannot generally be interpreted as causal effects. We illustrate how identifying assumptions for all of these counterfactual estimands can be represented in causal diagrams, in which competing events are depicted as time-varying covariates. We present an application of these ideas to data from a randomized trial designed to estimate the effect of estrogen therapy on prostate cancer mortality.     

Methods for comparing center‐specific survival outcomes using direct standardization

The evaluation of center‐specific outcomes is often through survival analysis methods. Such evaluations must account for differences in the distribution of patient characteristics across centers. In the context of censored event times, it is also important that the measure chosen to evaluate centers not be influenced by imbalances in the center‐specific censoring distributions. The practice of using center indicators in a hazard regression model is often invalid, inconvenient, or undesirable to carry out. We propose a semiparametric version of the standardized rate ratio (SRR) useful for the evaluation of centers with respect to a right‐censored event time. The SRR for center j can be interpreted as the ratio of the expected number of deaths in the total population (if the total population were in fact subject to the center j mortality hazard) to the observed number of events. The proposed measure is not affected by differences in center‐specific covariate or censoring distributions. Asymptotic properties of the proposed estimators are derived, with finite‐sample properties examined through simulation studies. The proposed methods are applied to national kidney transplant data. Copyright © 2014 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.     

Simulating survival data with predefined censoring rates for proportional hazards models

The proportional hazard model is one of the most important statistical models used in medical research involving time‐to‐event data. Simulation studies are routinely used to evaluate the performance and properties of the model and other alternative statistical models for time‐to‐event outcomes under a variety of situations. Complex simulations that examine multiple situations with different censoring rates demand approaches that can accommodate this variety. In this paper, we propose a general framework for simulating right‐censored survival data for proportional hazards models by simultaneously incorporating a baseline hazard function from a known survival distribution, a known censoring time distribution, and a set of baseline covariates. Specifically, we present scenarios in which time to event is generated from exponential or Weibull distributions and censoring time has a uniform or Weibull distribution. The proposed framework incorporates any combination of covariate distributions. We describe the steps involved in nested numerical integration and using a root‐finding algorithm to choose the censoring parameter that achieves predefined censoring rates in simulated survival data. We conducted simulation studies to assess the performance of the proposed framework. We demonstrated the application of the new framework in a comprehensively designed simulation study. We investigated the effect of censoring rate on potential bias in estimating the conditional treatment effect using the proportional hazard model in the presence of unmeasured confounding variables. Copyright © 2016 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.     

Incorporation of the time aspect into the liability‐threshold model for case‐control‐family data

Familial aggregation and the role of genetic and environmental factors can be investigated through family studies analysed using the liability‐threshold model. The liability‐threshold model ignores the timing of events including the age of disease onset and right censoring, which can lead to estimates that are difficult to interpret and are potentially biased. We incorporate the time aspect into the liability‐threshold model for case‐control‐family data following the same approach that has been applied in the twin setting. Thus, the data are considered as arising from a competing risks setting and inverse probability of censoring weights are used to adjust for right censoring. In the case‐control‐family setting, recognising the existence of competing events is highly relevant to the sampling of control probands. Because of the presence of multiple family members who may be censored at different ages, the estimation of inverse probability of censoring weights is not as straightforward as in the twin setting but requires consideration. We propose to employ a composite likelihood conditioning on proband status that markedly simplifies adjustment for right censoring. We assess the proposed approach using simulation studies and apply it in the analysis of two Danish register‐based case‐control‐family studies: one on cancer diagnosed in childhood and adolescence, and one on early‐onset breast cancer. Copyright © 2017 John Wiley & Sons, Ltd.     

Sizing clinical trials when comparing bivariate time‐to‐event outcomes

Clinical trials with multiple primary time‐to‐event outcomes are common. Use of multiple endpoints creates challenges in the evaluation of power and the calculation of sample size during trial design particularly for time‐to‐event outcomes. We present methods for calculating the power and sample size for randomized superiority clinical trials with two correlated time‐to‐event outcomes. We do this for independent and dependent censoring for three censoring scenarios: (i) the two events are non‐fatal; (ii) one event is fatal (semi‐competing risk); and (iii) both are fatal (competing risk). We derive the bivariate log‐rank test in all three censoring scenarios and investigate the behavior of power and the required sample sizes. Separate evaluations are conducted for two inferential goals, evaluation of whether the test intervention is superior to the control on: (1) all of the endpoints (multiple co‐primary) or (2) at least one endpoint (multiple primary). Copyright © 2017 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.     

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.     

A full competing risk analysis of hospital-acquired infections can easily be performed by a case-cohort approach

ObjectivesWe provide a case-cohort approach and show that a full competing risk analysis is feasible even in a reduced data set. Competing events for hospital-acquired infections are death or discharge from the hospital because they preclude the observation of such infections.Study Design and SettingUsing surveillance data of 6,568 patient admissions (full cohort) from two Spanish intensive care units, we propose a case-cohort approach which uses only data from a random sample of the full cohort and all infected patients (the cases). We combine established methodology to study following measures: event-specific as well as subdistribution hazard ratios for all three events (infection, death, and discharge), cumulative hazards as well as incidence functions by risk factor, and also for all three events.ResultsCompared with the values from the full cohort, all measures are well approximated with the case-cohort design. For the event of interest (infection), event-specific and subdistribution hazards can be estimated with the full efficiency of the case-cohort design. So, standard errors are only slightly increased, whereas the precision of estimated hazards of the competing events is inflated according to the size of the subcohort.ConclusionThe case-cohort design provides an appropriate sampling design for studying hospital-acquired infections in a reduced data set. Potential effects of risk factors on the competing events (death and discharge) can be evaluated.     

A missing data approach to semi-competing risks problems

For event time data involving multiple mutually exclusive competing causes of failure, classic competing risks results show that marginal survival distributions are not identifiable. In a related instance, one or more failure modes may be observed provided that the failure events occur in a specific order. In such situations, sometimes referred to as semi-competing risks problems, the observations may under realistic assumptions lend information about parameters of interest that would be nonidentifiable in the strict competing risks case. Here, we present an approach that makes use of partially observable multiple modes of failures to obtain an estimate of the marginal distribution of one event type that may occur prior to the occurrence of another event type or be precluded by it. We apply the proposed method to the problem of estimating the distribution of time to tumour recurrence at specific sites among breast cancer patients participating in randomized clinical trials.     

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.     

Analysis of accelerated failure time data with dependent censoring using auxiliary variables via nonparametric multiple imputation

We consider the situation of estimating the marginal survival distribution from censored data subject to dependent censoring using auxiliary variables. We had previously developed a nonparametric multiple imputation approach. The method used two working proportional hazards (PH) models, one for the event times and the other for the censoring times, to define a nearest neighbor imputing risk set. This risk set was then used to impute failure times for censored observations. Here, we adapt the method to the situation where the event and censoring times follow accelerated failure time models and propose to use the Buckley–James estimator as the two working models. Besides studying the performances of the proposed method, we also compare the proposed method with two popular methods for handling dependent censoring through the use of auxiliary variables, inverse probability of censoring weighted and parametric multiple imputation methods, to shed light on the use of them. In a simulation study with time‐independent auxiliary variables, we show that all approaches can reduce bias due to dependent censoring. The proposed method is robust to misspecification of either one of the two working models and their link function. This indicates that a working proportional hazards model is preferred because it is more cumbersome to fit an accelerated failure time model. In contrast, the inverse probability of censoring weighted method is not robust to misspecification of the link function of the censoring time model. The parametric imputation methods rely on the specification of the event time model. The approaches are applied to a prostate cancer dataset. Copyright © 2015 John Wiley & Sons, Ltd.     

Limitation of inverse probability-of-censoring weights in estimating survival in the presence of strong selection bias

In time-to-event analyses, artificial censoring with correction for induced selection bias using inverse probability-of-censoring weights can be used to 1) examine the natural history of a disease after effective interventions are widely available, 2) correct bias due to noncompliance with fixed or dynamic treatment regimens, and 3) estimate survival in the presence of competing risks. Artificial censoring entails censoring participants when they meet a predefined study criterion, such as exposure to an intervention, failure to comply, or the occurrence of a competing outcome. Inverse probability-of-censoring weights use measured common predictors of the artificial censoring mechanism and the outcome of interest to determine what the survival experience of the artificially censored participants would be had they never been exposed to the intervention, complied with their treatment regimen, or not developed the competing outcome. Even if all common predictors are appropriately measured and taken into account, in the context of small sample size and strong selection bias, inverse probability-of-censoring weights could fail because of violations in assumptions necessary to correct selection bias. The authors used an example from the Multicenter AIDS Cohort Study, 1984-2008, regarding estimation of long-term acquired immunodeficiency syndrome-free survival to demonstrate the impact of violations in necessary assumptions. Approaches to improve correction methods are discussed.     

Survival analysis using auxiliary variables via non-parametric multiple imputation

We develop an approach, based on multiple imputation, that estimates the marginal survival distribution in survival analysis using auxiliary variables to recover information for censored observations. To conduct the imputation, we use two working survival models to define a nearest neighbour imputing risk set. One model is for the event times and the other for the censoring times. Based on the imputing risk set, two non-parametric multiple imputation methods are considered: risk set imputation, and Kaplan-Meier imputation. For both methods a future event or censoring time is imputed for each censored observation. With a categorical auxiliary variable, we show that with a large number of imputes the estimates from the Kaplan-Meier imputation method correspond to the weighted Kaplan-Meier estimator. We also show that the Kaplan-Meier imputation method is robust to mis-specification of either one of the two working models. In a simulation study with time independent and time-dependent auxiliary variables, we compare the multiple imputation approaches with an inverse probability of censoring weighted method. We show that all approaches can reduce bias due to dependent censoring and improve the efficiency. We apply the approaches to AIDS clinical trial data comparing ZDV and placebo, in which CD4 count is the time-dependent auxiliary variable.