Estimation of the marginal survival time in the presence of dependent competing risks using inverse probability of censoring weighted (IPCW) methods

In medical studies, there is interest in inferring the marginal distribution of a survival time subject to competing risks. The Kyushu Lipid Intervention Study (KLIS) was a clinical study for hypercholesterolemia, where pravastatin treatment was compared with conventional treatment. The primary endpoint was time to events of coronary heart disease (CHD). In this study, however, some subjects died from causes other than CHD or were censored due to loss to follow-up. Because the treatments were targeted to reduce CHD events, the investigators were interested in the effect of the treatment on CHD events in the absence of causes of death or events other than CHD. In this paper, we present a method for estimating treatment group-specific marginal survival curves of time-to-event data in the presence of dependent competing risks. The proposed method is a straightforward extension of the Inverse Probability of Censoring Weighted (IPCW) method to settings with more than one reason for censoring. The results of our analysis showed that the IPCW marginal incidence for CHD was almost the same as the lower bound for which subjects with competing events were assumed to be censored at the end of all follow-up. This result provided reassurance that the results in KLIS were robust to competing risks.     

Direct parametric inference for the cumulative incidence function

Summary.  In survival data that are collected from phase III clinical trials on breast cancer, a patient may experience more than one event, including recurrence of the original cancer, new primary cancer and death. Radiation oncologists are often interested in comparing patterns of local or regional recurrences alone as first events to identify a subgroup of patients who need to be treated by radiation therapy after surgery. The cumulative incidence function provides estimates of the cumulative probability of locoregional recurrences in the presence of other competing events. A simple version of the Gompertz distribution is proposed to parameterize the cumulative incidence function directly. The model interpretation for the cumulative incidence function is more natural than it is with the usual cause-specific hazard parameterization. Maximum likelihood analysis is used to estimate simultaneously parametric models for cumulative incidence functions of all causes. The parametric cumulative incidence approach is applied to a data set from the National Surgical Adjuvant Breast and Bowel Project and compared with analyses that are based on parametric cause-specific hazard models and nonparametric cumulative incidence estimation.     

Assessing the treatment effect in a randomized controlled trial with extensive non‐adherence: the EVOLVE trial

Intention‐to‐treat (ITT) analysis is widely used to establish efficacy in randomized clinical trials. However, in a long‐term outcomes study where non‐adherence to study drug is substantial, the on‐treatment effect of the study drug may be underestimated using the ITT analysis. The analyses presented herein are from the EVOLVE trial, a double‐blind, placebo‐controlled, event‐driven cardiovascular outcomes study conducted to assess whether a treatment regimen including cinacalcet compared with placebo in addition to other conventional therapies reduces the risk of mortality and major cardiovascular events in patients receiving hemodialysis with secondary hyperparathyroidism. Pre‐specified sensitivity analyses were performed to assess the impact of non‐adherence on the estimated effect of cinacalcet. These analyses included lag‐censoring, inverse probability of censoring weights (IPCW), rank preserving structural failure time model (RPSFTM) and iterative parameter estimation (IPE). The relative hazard (cinacalcet versus placebo) of mortality and major cardiovascular events was 0.93 (95% confidence interval 0.85, 1.02) using the ITT analysis; 0.85 (0.76, 0.95) using lag‐censoring analysis; 0.81 (0.70, 0.92) using IPCW; 0.85 (0.66, 1.04) using RPSFTM and 0.85 (0.75, 0.96) using IPE. These analyses, while not providing definitive evidence, suggest that the intervention may have an effect while subjects are receiving treatment. The ITT method remains the established method to evaluate efficacy of a new treatment; however, additional analyses should be considered to assess the on‐treatment effect when substantial non‐adherence to study drug is expected or observed. Copyright © 2015 John Wiley & Sons, Ltd.     

A Proportional Hazards Regression Model for the Subdistribution with Covariates‐adjusted Censoring Weight for Competing Risks Data

With competing risks data, one often needs to assess the treatment and covariate effects on the cumulative incidence function. Fine and Gray proposed a proportional hazards regression model for the subdistribution of a competing risk with the assumption that the censoring distribution and the covariates are independent. Covariate‐dependent censoring sometimes occurs in medical studies. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with proper adjustments for covariate‐dependent censoring. We consider a covariate‐adjusted weight function by fitting the Cox model for the censoring distribution and using the predictive probability for each individual. Our simulation study shows that the covariate‐adjusted weight estimator is basically unbiased when the censoring time depends on the covariates, and the covariate‐adjusted weight approach works well for the variance estimator as well. We illustrate our methods with bone marrow transplant data from the Center for International Blood and Marrow Transplant Research. Here, cancer relapse and death in complete remission are two competing risks.     

A joint model of cancer incidence,metastasis, and mortality

Many diseases, especially cancer, are not static, but rather can be summarized by a series of events or stages (e.g. diagnosis, remission, recurrence, metastasis, death). Most available methods to analyze multi-stage data ignore intermediate events and focus on the terminal event or consider (time to) multiple events as independent. Competing-risk or semi-competing-risk models are often deficient in describing the complex relationship between disease progression events which are driven by a shared progression stochastic process. A multi-stage model can only examine two stages at a time and thus fails to capture the effect of one stage on the time spent between other stages. Moreover, most models do not account for latent stages. We propose a semi-parametric joint model of diagnosis, latent metastasis, and cancer death and use nonparametric maximum likelihood to estimate covariate effects on the risks of intermediate events and death and the dependence between them. We illustrate the model with Monte Carlo simulations and analysis of real data on prostate cancer from the SEER database.     

Analysis of interval censored competing risk data with missing causes of failure using pseudo values approach

Competing risks often occur when subjects may fail from one of several mutually exclusive causes. For example, when a patient suffering a cancer may die from other cause, we are interested in the effect of a certain covariate on the probability of dying of cancer at a certain time. Several approaches have been suggested to analyse competing risk data in the presence of complete information of failure cause. In this paper, our interest is to consider the occurrence of missing causes as well as interval censored failure time. There exist no method to discuss this problem. We applied a Klein–Andersen's pseudo-value approach [Klein, JP Andersen PK. Regression modeling of competing risks data based on pseudovalues of the cumulative incidence function. Biometrics. 2005;61:223–229] based on the estimated cumulative incidence function and a regression coefficient is estimated through a multiple imputation. We evaluate the suggested method by comparing with a complete case analysis in several simulation settings.     

Estimation of survival quantiles in two-stage randomization designs

In two-stage randomization designs, patients are randomized to one or more available therapies upon entry into the study. Depending on the response to the initial treatment (such as complete remission or shrinkage of tumor), patients are then randomized to receive maintenance treatments to maintain the response or salvage treatment to induce response. One goal of such trials is to compare the combinations of initial and maintenance or salvage therapies in the form of treatment strategies. In cases where the endpoint is defined as overall survival, Lunceford et al. [2002. Estimation of survival distributions of treatment policies in two-stage and randomization designs in clinical trials. Biometrics 58, 48–57] used mean survival time and pointwise survival probability to compare treatment strategies. But, mean survival time or survival probability at a specific time may not be a good summary representative of the overall distribution when the data are skewed or contain influential tail observations. In this article, we propose consistent and asymptotic normal estimators for percentiles of survival curves under various treatment strategies and demonstrate the use of percentiles for comparing treatment strategies. Small sample properties of these estimators are investigated using simulation. We demonstrate our methods by applying them to a leukemia clinical trial data set that motivated this research.     

Applying a marginalized frailty model to competing risks

The marginalized frailty model is often used for the analysis of correlated times in survival data. When only two correlated times are analyzed, this model is often referred to as the Clayton–Oakes model [7,22]. With time-to-event data, there may exist multiple end points (competing risks) suggesting that an analysis focusing on all available outcomes is of interest. The purpose of this work is to extend the single risk marginalized frailty model to the multiple risk setting via cause-specific hazards (CSH). The methods herein make use of the marginalized frailty model described by Pipper and Martinussen [24]. As such, this work uses the martingale theory to develop a likelihood based on estimating equations and observed histories. The proposed multivariate CSH model yields marginal regression parameter estimates while accommodating the clustering of outcomes. The multivariate CSH model can be fitted using a data augmentation algorithm described by Lunn and McNeil [21] or by fitting a series of single risk models for each of the competing risks. An example of the application of the multivariate CSH model is provided through the analysis of a family-based follow-up study of breast cancer with death in absence of breast cancer as a competing risk.     

Estimating and comparing adverse event probabilities in the presence of varying follow-up times and competing events

Safety analyses of adverse events (AEs) are important in assessing benefit–risk of therapies but are often rather simplistic compared to efficacy analyses. AE probabilities are typically estimated by incidence proportions, sometimes incidence densities or Kaplan–Meier estimation are proposed. These analyses either do not account for censoring, rely on a too restrictive parametric model, or ignore competing events. With the non-parametric Aalen-Johansen estimator as the “gold standard”, that is, reference estimator, potential sources of bias are investigated in an example from oncology and in simulations, for both one-sample and two-sample scenarios. The Aalen-Johansen estimator serves as a reference, because it is the proper non-parametric generalization of the Kaplan–Meier estimator to multiple outcomes. Because of potential large variances at the end of follow-up, comparisons also consider further quantiles of the observed times. To date, consequences for safety comparisons have hardly been investigated, the impact of using different estimators for group comparisons being unclear. For example, the ratio of two both underestimating or overestimating estimators may not be comparable to the ratio of the reference, and our investigation also considers the ratio of AE probabilities. We find that ignoring competing events is more of a problem than falsely assuming constant hazards by the use of the incidence density and that the choice of the AE probability estimator is crucial for group comparisons.     

Simultaneous inference of a misclassified outcome and competing risks failure time data

Ipsilateral breast tumor relapse (IBTR) often occurs in breast cancer patients after their breast conservation therapy. The IBTR status' classification (true local recurrence versus new ipsilateral primary tumor) is subject to error and there is no widely accepted gold standard. Time to IBTR is likely informative for IBTR classification because new primary tumor tends to have a longer mean time to IBTR and is associated with improved survival as compared with the true local recurrence tumor. Moreover, some patients may die from breast cancer or other causes in a competing risk scenario during the follow-up period. Because the time to death can be correlated to the unobserved true IBTR status and time to IBTR (if relapse occurs), this terminal mechanism is non-ignorable. In this paper, we propose a unified framework that addresses these issues simultaneously by modeling the misclassified binary outcome without a gold standard and the correlated time to IBTR, subject to dependent competing terminal events. We evaluate the proposed framework by a simulation study and apply it to a real data set consisting of 4477 breast cancer patients. The adaptive Gaussian quadrature tools in SAS procedure NLMIXED can be conveniently used to fit the proposed model. We expect to see broad applications of our model in other studies with a similar data structure.     

Estimating the concordance probability in a survival analysis with a discrete number of risk groups

A clinical risk classification system is an important component of a treatment decision algorithm. A measure used to assess the strength of a risk classification system is discrimination, and when the outcome is survival time, the most commonly applied global measure of discrimination is the concordance probability. The concordance probability represents the pairwise probability of lower patient risk given longer survival time. The c-index and the concordance probability estimate have been used to estimate the concordance probability when patient-specific risk scores are continuous. In the current paper, the concordance probability estimate and an inverse probability censoring weighted c-index are modified to account for discrete risk scores. Simulations are generated to assess the finite sample properties of the concordance probability estimate and the weighted c-index. An application of these measures of discriminatory power to a metastatic prostate cancer risk classification system is examined.     

Statistical methods for dependent competing risks

Many biological and medical studies have as a response of interest the time to occurrence of some event,X, such as the occurrence of cessation of smoking, conception, a particular symptom or disease, remission, relapse, death due to some specific disease, or simply death. Often it is impossible to measureX due to the occurrence of some other competing event, usually termed a competing risk. This competing event may be the withdrawal of the subject from the study (for whatever reason), death from some cause other than the one of interest, or any eventuality that precludes the main event of interest from occurring. Usually the assumption is made that all such censoring times and lifetimes are independent. In this case one uses either the Kaplan-Meier estimator or the Nelson-Aalen estimator to estimate the survival function. However, if the competing risk or censoring times are not independent ofX, then there is no generally acceptable way to estimate the survival function. There has been considerable work devoted to this problem of dependent competing risks scattered throughout the statistical literature in the past several years and this paper presents a survey of such work.     

Reduced rank proportional hazards model for competing risks: An application to a breast cancer trial

In many cancer trials patients are at risk of recurrence and death after the appearance and the successful treatment of the first diagnosed tumour. In this situation competing risks models that model several competing causes of therapy or surgery failure are a natural framework to describe the evolution of the disease.Typically, regression models for competing risks outcomes are based on proportional hazards model for each of the cause-specific hazard rates. An immediate practical problem is then how to deal with the abundance of regression parameters. The aim of reduced rank proportional hazards models is to reduce the number of parameters that need to be estimated while at the same time keeping the distinction between different transitions. They have the advantage of describing the competing risks model in fewer parameters, cope with transitions where few events are present and facilitate the interpretation of these estimates.We shall illustrate the use of this technique on 2795 patients from a breast cancer trial (EORTC 10854).     

Evaluation of Viable Dynamic Treatment Regimes in a Sequentially Randomized Trial of Advanced Prostate Cancer

We present new statistical analyses of data arising from a clinical trial designed to compare two-stage dynamic treatment regimes (DTRs) for advanced prostate cancer. The trial protocol mandated that patients were to be initially randomized among four chemotherapies, and that those who responded poorly were to be rerandomized to one of the remaining candidate therapies. The primary aim was to compare the DTRs' overall success rates, with success defined by the occurrence of successful responses in each of two consecutive courses of the patient's therapy. Of the one hundred and fifty study participants, forty seven did not complete their therapy per the algorithm. However, thirty five of them did so for reasons that precluded further chemotherapy; i.e. toxicity and/or progressive disease. Consequently, rather than comparing the overall success rates of the DTRs in the unrealistic event that these patients had remained on their assigned chemotherapies, we conducted an analysis that compared viable switch rules defined by the per-protocol rules but with the additional provision that patients who developed toxicity or progressive disease switch to a non-prespecified therapeutic or palliative strategy. This modification involved consideration of bivariate per-course outcomes encoding both efficacy and toxicity. We used numerical scores elicited from the trial's Principal Investigator to quantify the clinical desirability of each bivariate per-course outcome, and defined one endpoint as their average over all courses of treatment. Two other simpler sets of scores as well as log survival time also were used as endpoints. Estimation of each DTR-specific mean score was conducted using inverse probability weighted methods that assumed that missingness in the twelve remaining drop-outs was informative but explainable in that it only depended on past recorded data. We conducted additional worst-best case analyses to evaluate sensitivity of our findings to extreme departures from the explainable drop-out assumption.     

Estimating the turning point of the log-logistic hazard function in the presence of long-term survivors with an application for uterine cervical cancer data

The hazard function plays an important role in cancer patient survival studies, as it quantifies the instantaneous risk of death of a patient at any given time. Often in cancer clinical trials, unimodal hazard functions are observed, and it is of interest to detect (estimate) the turning point (mode) of hazard function, as this may be an important measure in patient treatment strategies with cancer. Moreover, when patient cure is a possibility, estimating cure rates at different stages of cancer, in addition to their proportions, may provide a better summary of the effects of stages on survival rates. Therefore, the main objective of this paper is to consider the problem of estimating the mode of hazard function of patients at different stages of cervical cancer in the presence of long-term survivors. To this end, a mixture cure rate model is proposed using the log-logistic distribution. The model is conveniently parameterized through the mode of the hazard function, in which cancer stages can affect both the cured fraction and the mode. In addition, we discuss aspects of model inference through the maximum likelihood estimation method. A Monte Carlo simulation study assesses the coverage probability of asymptotic confidence intervals.     

Multistate Survival Models as Transient Electrical Networks

In multistate survival analysis, the sojourn of a patient through various clinical states is shown to correspond to the diffusion of 1 C of electrical charge through an electrical network. The essential comparison has differentials of probability for the patient to correspond to differentials of charge, and it equates clinical states to electrical nodes. Indeed, if the death state of the patient corresponds to the sink node of the circuit, then the transient current that would be seen on an oscilloscope as the sink output is a plot of the probability density for the survival time of the patient. This electrical circuit analogy is further explored by considering the simplest possible survival model with two clinical states, alive and dead (sink), that incorporates censoring and truncation. The sink output seen on an oscilloscope is a plot of the Kaplan–Meier mass function. Thus, the Kaplan–Meier estimator finds motivation from the dynamics of current flow, as a fundamental physical law, rather than as a nonparametric maximum likelihood estimate (MLE). Generalization to competing risks settings with multiple death states (sinks) leads to cause‐specific Kaplan–Meier submass functions as outputs at sink nodes. With covariates present, the electrical analogy provides for an intuitive understanding of partial likelihood and various baseline hazard estimates often used with the proportional hazards model.     

Isotonic estimation of survival under a misattribution of cause of death

Several authors have indicated that incorrectly classified cause of death for prostate cancer survivors may have played a role in the observed recent peak and decline of prostate cancer mortality. Motivated by the suggestion we studied a competing risks model where other cause of death may be misattributed as a death of interest. We first consider a na?ve approach using unconstrained nonparametric maximum likelihood estimation (NPMLE), and then present the constrained NPMLE where the survival function is forced to be monotonic. Surprising observations were made as we studied their small-sample and asymptotic properties in continuous and discrete situations. Contrary to the common belief that the non-monotonicity of a survival function NPMLE is a small-sample problem, the constrained NPMLE is asymptotically biased in the continuous setting. Other isotonic approaches, the supremum (SUP) method and the Pooled-Adjacent-Violators (PAV) algorithm, and the EM algorithm are also considered. We found that the EM algorithm is equivalent to the constrained NPMLE. Both SUP method and PAV algorithm deliver consistent and asymptotically unbiased estimator. All methods behave well asymptotically in the discrete time setting. Data from the Surveillance, Epidemiology and End Results (SEER) database are used to illustrate the proposed estimators.     

Population-based absolute risk estimation with survey data

Absolute risk is the probability that a cause-specific event occurs in a given time interval in the presence of competing events. We present methods to estimate population-based absolute risk from a complex survey cohort that can accommodate multiple exposure-specific competing risks. The hazard function for each event type consists of an individualized relative risk multiplied by a baseline hazard function, which is modeled nonparametrically or parametrically with a piecewise exponential model. An influence method is used to derive a Taylor-linearized variance estimate for the absolute risk estimates. We introduce novel measures of the cause-specific influences that can guide modeling choices for the competing event components of the model. To illustrate our methodology, we build and validate cause-specific absolute risk models for cardiovascular and cancer deaths using data from the National Health and Nutrition Examination Survey. Our applications demonstrate the usefulness of survey-based risk prediction models for predicting health outcomes and quantifying the potential impact of disease prevention programs at the population level.     

Bayesian analysis of bivariate competing risks models with covariates

Bivariate exponential models have often been used for the analysis of competing risks data involving two correlated risk components. Competing risks data consist only of the time to failure and cause of failure. In situations where there is positive probability of simultaneous failure, possibly the most widely used model is the Marshall–Olkin (J. Amer. Statist. Assoc. 62 (1967) 30) bivariate lifetime model. This distribution is not absolutely continuous as it involves a singularity component. However, the likelihood function based on the competing risks data is then identifiable, and any inference, Bayesian or frequentist, can be carried out in a straightforward manner. For the analysis of absolutely continuous bivariate exponential models, standard approaches often run into difficulty due to the lack of a fully identifiable likelihood (Basu and Ghosh; Commun. Statist. Theory Methods 9 (1980) 1515). To overcome the nonidentifiability, the usual frequentist approach is based on an integrated likelihood. Such an approach is implicit in Wada et al. (Calcutta Statist. Assoc. Bull. 46 (1996) 197) who proved some related asymptotic results. We offer in this paper an alternative Bayesian approach. Since systematic prior elicitation is often difficult, the present study focuses on Bayesian analysis with noninformative priors. It turns out that with an appropriate reparameterization, standard noninformative priors such as Jeffreys’ prior and its variants can be applied directly even though the likelihood is not fully identifiable. Two noninformative priors are developed that consist of Laplace's prior for nonidentifiable parameters and Laplace's and Jeffreys's priors for identifiable parameters. The resulting Bayesian procedures possess some frequentist optimality properties as well. Finally, these Bayesian methods are illustrated with analyses of a data set originating out of a lung cancer clinical trial conducted by the Eastern Cooperative Oncology Group.     

Prognostic score matching methods for estimating the average effect of a non-reversible binary time-dependent treatment on the survival function

In evaluating the benefit of a treatment on survival, it is often of interest to compare post-treatment survival with the survival function that would have been observed in the absence of treatment. In many practical settings, treatment is time-dependent in the sense that subjects typically begin follow-up untreated, with some going on to receive treatment at some later time point. In observational studies, treatment is not assigned at random and, therefore, may depend on various patient characteristics. We have developed semi-parametric matching methods to estimate the average treatment effect on the treated (ATT) with respect to survival probability and restricted mean survival time. Matching is based on a prognostic score which reflects each patient’s death hazard in the absence of treatment. Specifically, each treated patient is matched with multiple as-yet-untreated patients with similar prognostic scores. The matched sets do not need to be of equal size, since each matched control is weighted in order to preserve risk score balancing across treated and untreated groups. After matching, we estimate the ATT non-parametrically by contrasting pre- and post-treatment weighted Nelson–Aalen survival curves. A closed-form variance is proposed and shown to work well in simulation studies. The proposed methods are applied to national organ transplant registry data.