Removing Cancer when it is Correlated with other Causes of Death

This paper presents an identifiability theorem in the theory of dependent competing risks and it applies the result by examining the effect of removing cancer from the United States population when cancer is correlated with the other causes of death. The paper shows how dependence can be modeled with copula functions and it shows that calculating the survival probabilities after cancer is removed is equivalent to solving a system of nonlinear differential equations.     

Multiple Decrement Analysis with Dependent Risks

In a competing risks problem where a well-defined population is exposed simultaneously to several causes of death, interest has centered on the estimation of the probability of death from a given cause when one or more other causes have been eliminated. A basic component of all available procedures for estimating these probabilities is the assumption that the several causes of death act independently—an unrealistic assumption in the context of human and animal populations. This article considers the estimation of these probabilities assuming the existence ofinterdependencies among the various causes of death. A general formula is derived based on a given set of crude probabilities of death as well as the characteristics of the joint distribution of random variables indicating death from the various causes. This formula identifies alternative assumptions, less restrictive than that of independent risks, which may he used for estimation purposes.     

Analysis of cause of death: Competing risks or progressive illness‐death model?

The analysis of cause of death is increasingly becoming a topic in oncology. It is usually distinguished between disease‐related and disease‐unrelated death. A frequently used approach is to define death as disease‐related when a progression to advanced phases has occurred before, otherwise as disease‐unrelated. The data are often analyzed as competing risks, while a progressive illness‐death model might in fact describe the situation more precisely. In this study, we investigated under which circumstances this misspecification leads to biased estimations of the state occupation probabilities. We simulated data according to the progressive illness‐death model in various settings, analyzed them with a competing risks model and with a progressive illness‐death model and compared them to the true state occupation probabilities. Censoring was either added independently of the status or based on the patients' status. The simulations showed that the censoring mechanism was decisive for the bias while neither the progression hazard nor the Markov property was important. Further, we found a slightly increased standard deviation for the competing risk estimator when censoring was independent of the patients' status. For illustration, both methods were applied to two practical examples of chronic myeloid leukemia (CML): one randomized controlled trial and one registry data set. While in the first case both estimators yielded almost identical results, in the latter case, visible differences were found between both methods.     

Regression modeling of competing crude failure probabilities

In a randomized trial of tamoxifen therapy for breast cancer, women can experience tumor recurrence or die from competing causes. One goal of analysis is to describe the effect of tamoxifen on the probabilities of recurrence or death from other causes. To this end, we propose a semi-parametric transformation model for the crude failure probabilities of a competing risk, conditional on covariates. The model is developed as an extension of the standard approach to survival data with independent right censoring. Estimation of the regression coefficients is achieved with a rank-based least squares criterion. Simulations show that the procedure works well with practical sample sizes. A separate estimating function is developed for the baseline parameter. Prediction of covariate-adjusted failure probabilities is considered. The methodology is motivated and illustrated with data from the tamoxifen trial.     

Competing risks analysis in mountain pine beetle dynamics

This paper presents an application and evaluation of competing risks analysis (Chiang , 1968) of mountain pine beetle life tables. Three known and one group of unknown risks are used. Interpretation of the results imply that only the crude probability of death from a specific cause is applicable to this situation; net and partial crude probabilities are yet incomplete. None of the risks (factors of mortality) exerted sufficient influence upon the population to be considered factors of regulation or reduction. Evidence remains that the mountain pine beetle is food-regulated at optimum temperature conditions and temperature-regulated at optimum food conditions.     

A Bayesian multi-risks survival (MRS) model in the presence of double censorings

Semi-competing risks data include the time to a nonterminating event and the time to a terminating event, while competing risks data include the time to more than one terminating event. Our work is motivated by a prostate cancer study, which has one nonterminating event and two terminating events with both semi-competing risks and competing risks present as well as two censoring times. In this paper, we propose a new multi-risks survival (MRS) model for this type of data. In addition, the proposed MRS model can accommodate noninformative right-censoring times for nonterminating and terminating events. Properties of the proposed MRS model are examined in detail. Theoretical and empirical results show that the estimates of the cumulative incidence function for a nonterminating event may be biased if the information on a terminating event is ignored. A Markov chain Monte Carlo sampling algorithm is also developed. Our methodology is further assessed using simulations and also an analysis of the real data from a prostate cancer study. As a result, a prostate-specific antigen velocity greater than 2.0 ng/mL per year and higher biopsy Gleason scores are positively associated with a shorter time to death due to prostate cancer.     

Nonparametric inference on cause‐specific quantile residual life

In medical research, investigators are often interested in inferring time‐to‐event distributions under competing risks. It is well known, however, that the naive approach based on the Kaplan–Meier method to estimate the proportion of cause‐specific events overestimates the true quantity. In this paper, we show that the quantile residual life function, a natural and popular summary measure of survival data, could be also seriously affected by the competing events. An existing two‐sample test statistic for inference on median residual life is modified for competing risks data, which does not involve estimation of the improper probability density function of the subdistribution of cause‐specific events under censoring. Simulation results demonstrate that the test statistic controls the type 1 error probabilities reasonably well. The proposed method is applied to a real data example from a large‐scale phase III breast cancer study.     

High-dimensional feature selection in competing risks modeling: A stable approach using a split-and-merge ensemble algorithm

Variable selection is critical in competing risks regression with high-dimensional data. Although penalized variable selection methods and other machine learning-based approaches have been developed, many of these methods often suffer from instability in practice. This paper proposes a novel method named Random Approximate Elastic Net (RAEN). Under the proportional subdistribution hazards model, RAEN provides a stable and generalizable solution to the large-p-small-n variable selection problem for competing risks data. Our general framework allows the proposed algorithm to be applicable to other time-to-event regression models, including competing risks quantile regression and accelerated failure time models. We show that variable selection and parameter estimation improved markedly using the new computationally intensive algorithm through extensive simulations. A user-friendly R package RAEN is developed for public use. We also apply our method to a cancer study to identify influential genes associated with the death or progression from bladder cancer.     

The analysis of failure times in the presence of competing risks.

Distinct problems in the analysis of failure times with competing causes of failure include the estimation of treatment or exposure effects on specific failure types, the study of interrelations among failure types, and the estimation of failure rates for some causes given the removal of certain other failure types. The usual formation of these problems is in terms of conceptual or latent failure times for each failure type. This approach is criticized on the basis of unwarranted assumptions, lack of physical interpretation and identifiability problems. An alternative approach utilizing cause-specific hazard functions for observable quantities, including time-dependent covariates, is proposed. Cause-specific hazard functions are shown to be the basic estimable quantities in the competing risks framework. A method, involving the estimation of parameters that relate time-dependent risk indicators for some causes to cause-specific hazard functions for other causes, is proposed for the study of interrelations among failure types. Further, it is argued that the problem of estimation of failure rates under the removal of certain causes is not well posed until a mechanism for cause removal is specified. Following such a specification, one will sometimes be in a position to make sensible extrapolations from available data to situations involving cause removal. A clinical program in bone marrow transplantation for leukemia provides a setting for discussion and illustration of each of these ideas. Failure due to censoring in a survivorship study leads to further discussion.     

The impact of under and over-recording of cancer on death certificates in a competing risks analysis: A simulation study

Background: With linked register and cause of death data becoming more accessible than ever, competing risks methodology is being increasingly used as a way of obtaining “real world” probabilities of death broken down by specific causes. It is important, in terms of the validity of these studies, to have accurate cause of death information. However, it is well documented that cause of death information taken from death certificates is often lacking in accuracy and completeness. Methods: We assess through use of a simulation study the effect of under and over-recording of cancer on death certificates in a competing risks analysis consisting of three competing causes of death: cancer, heart disease and other causes. Using realistic levels of misclassification, we consider 24 scenarios and examine the bias in the cause-specific hazard ratios and the cumulative incidence function. Results: The bias in the cumulative incidence function was highest in the oldest age group reaching values as high as 2.6 percentage units for the “good” cancer prognosis scenario and 9.7 percentage units for the “poor” prognosis scenario. Conclusion: The bias resulting from the chosen levels of misclassification in this study accentuate concerns that unreliable cause of death information may be providing misleading results. The results of this simulation study convey an important message to applied epidemiological researchers.     

Estimation of competing risks with general missing pattern in failure types

In competing risks data, missing failure types (causes) is a very common phenomenon. In this work, we consider a general missing pattern in which, if a failure type is not observed, one observes a set of possible types containing the true type, along with the failure time. We first consider maximum likelihood estimation with missing-at-random assumption via the expectation maximization (EM) algorithm. We then propose a Nelson-Aalen type estimator for situations when certain information on the conditional probability of the true type given a set of possible failure types is available from the experimentalists. This is based on a least-squares type method using the relationships between hazards for different types and hazards for different combinations of missing types. We conduct a simulation study to investigate the performance of this method, which indicates that bias may be small, even for high proportion of missing data, for sufficiently large number of observations. The estimates are somewhat sensitive to misspecification of the conditional probabilities of the true types when the missing proportion is high. We also consider an example from an animal experiment to illustrate our methodology.     

Joint Modeling of Multivariate Longitudinal Data and Competing Risks Using Multiphase Sub-models

In many clinical studies that involve follow-up, it is common to observe one or more sequences of longitudinal measurements, as well as one or more time to event outcomes. A competing risks situation arises when the probability of occurrence of one event is altered/hindered by another time to event. Recently, there has been much attention paid to the joint analysis of a single longitudinal response and a single time to event outcome, when the missing data mechanism in the longitudinal process is non-ignorable. We, in this paper, propose an extension where multiple longitudinal responses are jointly modeled with competing risks (multiple time to events). Our shared parameter joint model consists of a system of multiphase non-linear mixed effects sub-models for the multiple longitudinal responses, and a system of cause-specific non-proportional hazards frailty sub-models for competing risks, with associations among multiple longitudinal responses and competing risks modeled using latent parameters. The joint model is applied to a data set of patients who are on mechanical circulatory support and are awaiting heart transplant, using readily available software. While on the mechanical circulatory support, patient liver and renal functions may worsen and these in turn may influence one of the two possible competing outcomes: (i) death before transplant; (ii) transplant. In one application, we propose a system of multiphase cause-specific non-proportional hazard sub-model where frailty can be time varying. Performance under different scenarios was assessed using simulation studies. By using the proposed joint modeling of the multiphase sub-models, one can identify: (i) non-linear trends in multiple longitudinal outcomes; (ii) time-varying hazards and cumulative incidence functions of the competing risks; (iii) identify risk factors for the both types of outcomes, where the effect may or may not change with time; and (iv) assess the association between multiple longitudinal and competing risks outcomes, where the association may or may not change with time.     

Competing Risk Analysis of Censored Survival Data

A competing risk model is developed to accommodate both planned Type I censoring and random withdrawals. MLE's, their properties, confidence regions for parameters and mean lifetimes are obtained for a model regarding random censoring as a competing risk and compared to those obtained for the model in which withdrawals are regarded as random censoring. Estimated net and crude probabilities are calculated and compared for the two models. The model is developed for two competing risks, one following a Weibull distribution and the other a Rayleigh distribution, and random withdrawals following a Weibull distribution.     

On testing dependence between time to failure and cause of failure when causes of failure are missing

The hypothesis of independence between the failure time and the cause of failure is studied by using the conditional probabilities of failure due to a specific cause given that there is no failure up to certain fixed time. In practice, there are situations when the failure times are available for all units but the causes of failures might be missing for some units. We propose tests based on U-statistics to test for independence of the failure time and the cause of failure in the competing risks model when all the causes of failure cannot be observed. The asymptotic distribution is normal in each case. Simulation studies look at power comparisons for the proposed tests for two families of distributions. The one-sided and the two-sided tests based on Kendall type statistic perform exceedingly well in detecting departures from independence.     

Joint inference for competing risks data using multiple endpoints

Competing risks data are commonly encountered in randomized clinical trials and observational studies. This paper considers the situation where the ending statuses of competing events have different clinical interpretations and/or are of simultaneous interest. In clinical trials, often more than one competing event has meaningful clinical interpretations even though the trial effects of different events could be different or even opposite to each other. In this paper, we develop estimation procedures and inferential properties for the joint use of multiple cumulative incidence functions (CIFs). Additionally, by incorporating longitudinal marker information, we develop estimation and inference procedures for weighted CIFs and related metrics. The proposed methods are applied to a COVID-19 in-patient treatment clinical trial, where the outcomes of COVID-19 hospitalization are either death or discharge from the hospital, two competing events with completely different clinical implications.     

Estimating age‐dependent survival from age‐aggregated ringing data—extending the use of historical records

Bird ring‐recovery data have been widely used to estimate demographic parameters such as survival probabilities since the mid‐20th century. However, while the total number of birds ringed each year is usually known, historical information on age at ringing is often not available. A standard ring‐recovery model, for which information on age at ringing is required, cannot be used when historical data are incomplete. We develop a new model to estimate age‐dependent survival probabilities from such historical data when age at ringing is not recorded; we call this the historical data model. This new model provides an extension to the model of Robinson, 2010, Ibis, 152, 651–795 by estimating the proportion of the ringed birds marked as juveniles as an additional parameter. We conduct a simulation study to examine the performance of the historical data model and compare it with other models including the standard and conditional ring‐recovery models. Simulation studies show that the approach of Robinson, 2010, Ibis, 152, 651–795 can cause bias in parameter estimates. In contrast, the historical data model yields similar parameter estimates to the standard model. Parameter redundancy results show that the newly developed historical data model is comparable to the standard ring‐recovery model, in terms of which parameters can be estimated, and has fewer identifiability issues than the conditional model. We illustrate the new proposed model using Blackbird and Sandwich Tern data. The new historical data model allows us to make full use of historical data and estimate the same parameters as the standard model with incomplete data, and in doing so, detect potential changes in demographic parameters further back in time.     

Dependent competing risks: a stochastic process model

Analyses of human mortality data classified according to cause of death frequently are based on competing risk theory. In particular, the times to death for different causes often are assumed to be independent. In this paper, a competing risk model with a weaker assumption of conditional independence of the times to death, given an assumed stochastic covariate process, is developed and applied to cause specific mortality data from the Framingham Heart Study. The results generated under this conditional independence model are compared with analogous results under the standard marginal independence model. Under the assumption that this conditional independence model is valid, the comparison suggests that the standard model overestimates by 4% the effect on life expectancy at age 30 due to the hypothetical elimination of cancer and by 7% the effect for cardiovascular/cerebrovascular disease. By age 80 the overestimates were 11% for cancer and 16% for heart disease. These results suggest the importance of avoiding the marginal independence assumption when appropriate data are available — especially when focusing on mortality at advanced ages.     

A review and critique of some models used in competing risk analysis.

We have introduced a notation which allows one to define competing risk models easily and to examine underlying assumptions. We have treated the actuarial model for competing risk in detail, comparing it with other models and giving useful variance formulae both for the case when times of death are available and for the case when they are not. The generality of these methods is illustrated by an example treating two dependent competing risks.     

Competing risk analysis for life table data with known observation times

"Several models have been proposed for the analysis of cohort mortality in the presence of competing risks.... This paper describes a maximum likelihood approach to the analysis of follow up data in life table format for the case of two competing risks--a specific cause and its competing complement. The model developed uses a robust survivorship assumption--the piecewise exponential--and takes into account information on time to death and time to withdrawal." (summary in GER)     

Estimating Subject‐Specific Dependent Competing Risk Profile with Censored Event Time Observations

Summary In a longitudinal study, suppose that the primary endpoint is the time to a specific event. This response variable, however, may be censored by an independent censoring variable or by the occurrence of one of several dependent competing events. For each study subject, a set of baseline covariates is collected. The question is how to construct a reliable prediction rule for the future subject's profile of all competing risks of interest at a specific time point for risk‐benefit decision making. In this article, we propose a two‐stage procedure to make inferences about such subject‐specific profiles. For the first step, we use a parametric model to obtain a univariate risk index score system. We then estimate consistently the average competing risks for subjects who have the same parametric index score via a nonparametric function estimation procedure. We illustrate this new proposal with the data from a randomized clinical trial for evaluating the efficacy of a treatment for prostate cancer. The primary endpoint for this study was the time to prostate cancer death, but had two types of dependent competing events, one from cardiovascular death and the other from death of other causes.