Independent Censoring Research Articles

Two-stage pseudo maximum likelihood estimation of semiparametric copula-based regression models for semi-competing risks data.

We propose a two-stage estimation procedure for a copula-based model with semi-competing risks data, where the non-terminal event is subject to dependent censoring by the terminal event, and both events are subject to independent censoring. With a copula-based model, the marginal survival functions of individual event times are specified by semiparametric transformation models, and the dependence between the bivariate event times is specified by a parametric copula function. For the estimation procedure, in the first stage, the parameters associated with the marginal of the terminal event are estimated using only the corresponding observed outcomes, and in the second stage, the marginal parameters for the non-terminal event time and the copula parameter are estimated together via maximizing a pseudo-likelihood function based on the joint distribution of the bivariate event times. We derived the asymptotic properties of the proposed estimator and provided an analytic variance estimator for inference. Through simulation studies, we showed that our approach leads to consistent estimates with less computational cost and more robustness than the one-stage procedure developed in Chen YH (Lifetime Data Anal 18:36-57, 2012), where all parameters were estimated simultaneously. In addition, our approach demonstrates more desirable finite-sample performances over another existing two-stage estimation method proposed in Zhu H et al., (Commu Statistics-Theory Methods 51(22):7830-7845, 2021) . An R package PMLE4SCR is developed to implement our proposed method.

Effect of a correlated competing risk on marginal survival estimation in an accelerated failure time model

In problems with a time to event outcome, subjects may experience competing events, which censor the outcome of interest. Cox’s partial likelihood estimator treating competing events as independent censoring is commonly used to examine group differences in clinical trials but fails to adjust for omitted covariates and can bias the assessment of marginal benefit. A bivariate normal linear model generating latent data with dependent censoring is used to assess this bias. Our R-package bnc provides maximum penalized likelihood (MPL) parameter estimation using a novel EM algorithm. Using bnc, we study the properties of such MPL estimation. Simulation results for two-sample survival comparisons of time to an event of interest, with independent censoring accompanied by censoring from a correlated competing risk, are presented. Key parameters—means, hazard ratios, and correlation—are estimated. These results demonstrated that, despite ill-conditioning in models generating correlated competing risks, estimates of marginal effects are reliable. Bivariate normal models were fitted in a trial of head and neck cancer. Model fits help with clinical interpretation while also supplementing other standard methods for follow-up that are terminated by intervening risks.

Graphical criteria for the identification of marginal causal effects in continuous-time survival and event-history analyses

Abstract We consider continuous-time survival and event-history settings, where our aim is to graphically represent causal structures allowing us to characterize when a causal parameter is identified from observational data. This causal parameter is formalized as the effect on an outcome event of a (possibly hypothetical) intervention on the intensity of a treatment process. To establish identifiability, we propose novel graphical rules indicating whether the observed information is sufficient to obtain the desired causal effect by suitable reweighting. This requires a different type of graph than in discrete time. We formally define causal semantics for the corresponding dynamic graphs that represent local independence models for multivariate counting processes. Importantly, our work highlights that causal inference from censored data relies on subtle structural assumptions on the censoring process beyond independent censoring; these can be verified graphically. Put together, our results are the first to establish graphical rules for nonparametric causal identifiability in event processes in this generality for the continuous-time case, not relying on particular parametric survival models. We conclude with a data example on Human papillomavirus (HPV) testing for cervical cancer screening, where the assumptions are illustrated graphically and the desired effect is estimated by reweighted cumulative incidence curves.

A conditional approach for regression analysis of case K interval-censored failure time data with informative censoring

This paper discusses regression analysis of case K interval-censored failure time data, a general type of failure time data, in the presence of informative censoring with the focus on simultaneous variable selection and estimation. Although many authors have considered the challenging variable selection problem for interval-censored data, most of the existing methods assume independent or non-informative censoring. More importantly, the existing methods that allow for informative censoring are frailty model-based approaches and cannot directly assess the degree of informative censoring among other shortcomings. To address these, we propose a conditional approach and develop a penalized sieve maximum likelihood procedure for the simultaneous variable selection and estimation of covariate effects. Furthermore, we establish the oracle property of the proposed method and illustrate the appropriateness and usefulness of the approach using a simulation study. Finally we apply the proposed method to a set of real data on Alzheimer's disease and provide some new insights.

Proximal survival analysis to handle dependent right censoring

Abstract Many epidemiological and clinical studies aim to analyse a time-to-event endpoint. A common complication is right censoring. In some cases, right censoring occurs when subjects are still surviving after the study terminates or move out of the study area. In such cases, right censoring is typically treated as independent or noninformative. This assumption can be further relaxed to conditional independent censoring by leveraging possibly time-varying covariate information, if available, and assuming censoring and failure time are independent within covariate strata. In yet other instances, events may be censored by other competing events like death and are associated with censoring possibly through prognoses. Realistically, measured covariates can rarely capture all such associations with absolute certainty. In cases of dependent censoring, covariate measurements are often, at best, proxies of underlying prognoses. In this article, we establish a nonparametric identification framework by formally admitting that conditional independent censoring may fail in practice and accounting for covariate measurements as imperfect proxies of underlying association. The framework suggests adaptive estimators, and we provide generic assumptions under which they are consistent, asymptotically normal, and doubly robust. We examine the finite-sample performance of our proposed estimators via a Monte Carlo simulation and apply them to the SEER-Medicare dataset.

Calibration plots for multistate risk predictions models.

There is currently no guidance on how to assess the calibration of multistate models used for risk prediction. We introduce several techniques that can be used to produce calibration plots for the transition probabilities of a multistate model, before assessing their performance in the presence of random and independent censoring through a simulation. We studied pseudo-values based on the Aalen-Johansen estimator, binary logistic regression with inverse probability of censoring weights (BLR-IPCW), and multinomial logistic regression with inverse probability of censoring weights (MLR-IPCW). The MLR-IPCW approach results in a calibration scatter plot, providing extra insight about the calibration. We simulated data with varying levels of censoring and evaluated the ability of each method to estimate the calibration curve for a set of predicted transition probabilities. We also developed evaluated the calibration of a model predicting the incidence of cardiovascular disease, type 2 diabetes and chronic kidney disease among a cohort of patients derived from linked primary and secondary healthcare records. The pseudo-value, BLR-IPCW, and MLR-IPCW approaches give unbiased estimates of the calibration curves under random censoring. These methods remained predominately unbiased in the presence of independent censoring, even if the censoring mechanism was strongly associated with the outcome, with bias concentrated in low-density regions of predicted transition probability. We recommend implementing either the pseudo-value or BLR-IPCW approaches to produce a calibration curve, combined with the MLR-IPCW approach to produce a calibration scatter plot. The methods have been incorporated into the "calibmsm" R package available on CRAN.

A joint frailty model for recurrent and competing terminal events: Application to delirium in the ICU.

Joint models linking longitudinal biomarkers or recurrent event processes with a terminal event, for example, mortality, have been studied extensively. Motivated by studies of recurrent delirium events in patients receiving care in an intensive care unit (ICU), we devise a joint model for a recurrent event process and multiple terminal events. Being discharged alive from the ICU or experiencing mortality may be associated with a patient's hazard of delirium, violating the assumption of independent censoring. Moreover, the direction of the association between the hazards of delirium and mortality may be opposite of the direction of association between the hazards of delirium and ICU discharge. Hence treating either terminal event as independent censoring may bias inferences. We propose a competing joint model that uses a latent frailty to link a patient's recurrent and competing terminal event processes. We fit our model to data from a completed placebo-controlled clinical trial, which studied whether Haloperidol could prevent death and delirium among ICU patients. The clinical trial served as a foundation for a simulation study, in which we evaluate the properties, for example, bias and confidence interval coverage, of the competing joint model. As part of the simulation study, we demonstrate the shortcomings of using a joint model with a recurrent delirium process and a single terminal event to study delirium in the ICU. Lastly, we discuss limitations and possible extensions for the competing joint model. The competing joint model has been added to frailtypack, an R package for fitting an assortment of joint models.

Regression models for average hazard.

Limitations of using the traditional Cox's hazard ratio for summarizing the magnitude of the treatment effect on time-to-event outcomes have been widely discussed, and alternative measures that do not have such limitations are gaining attention. One of the alternative methods recently proposed, in a simple 2-sample comparison setting, uses the average hazard with survival weight (AH), which can be interpreted as the general censoring-free person-time incidence rate on a given time window. In this paper, we propose a new regression analysis approach for the AH with a truncation time τ. We investigate 3 versions of AH regression analysis, assuming (1) independent censoring, (2) group-specific censoring, and (3) covariate-dependent censoring. The proposed AH regression methods are closely related to robust Poisson regression. While the new approach needs to require a truncation time τ explicitly, it can be more robust than Poisson regression in the presence of censoring. With the AH regression approach, one can summarize the between-group treatment difference in both absolute difference and relative terms, adjusting for covariates that are associated with the outcome. This property will increase the likelihood that the treatment effect magnitude is correctly interpreted. The AH regression approach can be a useful alternative to the traditional Cox's hazard ratio approach for estimating and reporting the magnitude of the treatment effect on time-to-event outcomes.

Abstract 3565: The Cox proportional hazards mixed model enables accurate estimation of the total effect somatic SNVs have on treatment outcomes

Abstract Heritability is the variation in disease susceptibility attributed to genetic variation and is a fundamental parameter for interpreting disease architecture and making predictions. In the context of germline genetics, variance component models are commonly used to estimate heritability for continuous or case-control phenotypes (under the liability threshold model, LTM). However, models for time-to-event (TTE) outcomes have been largely unstudied. This has limited the characterization of age-of-onset and treatment response phenotypes, the latter particularly relevant for clinical studies. Here, we propose and evaluate a Cox proportional hazard Mixed Model (COXMM) to enable accurate estimation for TTE phenotypes with censoring. We apply these methods to Real World data from the Profile cohort from Dana-Farber Cancer Institute, a large prospectively collected tumor sequencing cohort with detailed clinical annotation. We implemented an efficient COXMM with a random effect modeled by the genetic relatedness across individuals, and benchmarked its performance in extensive simulations. We simulated phenotypes according to generative models reflecting either LTM or TTE models. For TTE phenotypes, data followed various Weibull distributions with independent censoring. Across TTE simulations, COXMM produced unbiased estimates while the classic case-control heritability estimator showed significant downward bias. Likewise, restricting only to cases and estimating the heritability of age-of-onset as a normalized continuous phenotype produced biased estimates, due to the artificial exclusion of controls. For example, for an age-of-onset with heritability of 0.8 and 40% of individuals in the study being cases, COXMM correctly inferred the heritability to be 0.80 +/- 0.007, compared to 0.32 +/- 0.004 for a conventional case-control model, and 0.09 +/- 0.02 for a conventional case-only age-of-onset model. In the Profile cohort, we analyzed the impact of somatic SNVs on progression free survival (PFS) and overall survival (OS) in patients with non-small cell lung cancer (N=954 and N=1796, respectively). We restricted to somatic SNVs present in at least 5% of patients. For OS, we observed that the COXMM heritability estimate was significantly larger than those computed by case-control status and using a normalized case-only age/duration (COXMM: 0.19 +/- 0.003; Case-Control: 0.07 +/- 0.2; Duration: 0.04 +/- 0.01). We observed a similar phenomenon for PFS as well (COXMM: 0.11 +/- 0.01; Case-Control: 0.03 +/- 0.4; Duration: 0.03 +/- 0.02). These results demonstrate that somatic features are most strongly associated with TTE and can be accurately estimated with COXMM. COXMM enables the efficient quantification of TTE heritability for diverse clinical phenotypes and outperforms existing methods. Citation Format: Kodi Taraszka, Alexander Gusev. The Cox proportional hazards mixed model enables accurate estimation of the total effect somatic SNVs have on treatment outcomes [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 3565.

Estimation of accelerated hazards models based on case K informatively interval-censored failure time data

The accelerated hazards model is one of the most commonly used models for regression analysis of failure time data and this is especially the case when, for example, the hazard functions may have monotonicity property. Correspondingly a large literature has been established for its estimation or inference when right-censored data are observed. Although several methods have also been developed for its inference based on interval-censored data, they apply only to limited situations or rely on some assumptions such as independent censoring. In this paper, we consider the situation where one observes case K interval-censored data, the type of failure time data that occur most in, for example, medical research such as clinical trials or periodical follow-up studies. For inference, we propose a sieve borrow-strength method and in particular, it allows for informative censoring. The asymptotic properties of the proposed estimators are established. Simulation studies demonstrate that the proposed inference procedure performs well. The method is applied to a set of real data set arising from an AIDS clinical trial.

Copula graphic estimation of the survival function with dependent censoring and its application to analysis of pancreatic cancer clinical trial.

In this article, we consider a survival function estimation method that may be suitable for analyses of clinical trials of cancer treatments whose prognosis is known to be poor such as pancreatic cancer treatment. Typically, these kinds of trials are not double-blind, and patients in the control group may drop out in more significant numbers than in the treatment group if their disease progresses (DP). If disease progression is associated with a higher risk of death, then censoring becomes dependent. To estimate the survival function with dependent censoring, we use copula-graphic estimation, where a parametric copula function is used to model the dependence in the joint survival function of the event and censoring time. In this article, we propose a novel method that one can use in choosing the copula parameter. As an application example, we estimate the survival function of the overall survival time of the KG4/2015 study, the phase 3 clinical trial of the efficacy of GV1001 as a treatment for pancreatic cancer. We provide both statistical and clinical pieces of evidence that support the violation of independent censoring. Applying the estimation method with dependent censoring, we obtain that the estimates of the median survival times are 339 days in the treatment group and 225.5 days in the control group. We also find that the estimated difference of the medians is 113.5 days, and the difference is statistically significant at the one-sided level with size 2.5.

Estimation of the Modified Weibull Additive Hazards Regression Model under Competing Risks

The additive hazard regression model plays an important role when the excess risk is the quantity of interest compared to the relative risks, where the proportional hazard model is better. This paper discusses parametric regression analysis of survival data using the additive hazards model with competing risks in the presence of independent right censoring. In this paper, the baseline hazard function is parameterized using a modified Weibull distribution as a lifetime model. The model parameters are estimated using maximum likelihood and Bayesian estimation methods. We also derive the asymptotic confidence interval and the Bayes credible interval of the unknown parameters. The finite sample behaviour of the proposed estimators is investigated through a Monte Carlo simulation study. The proposed model is applied to liver transplant data.

A Semiparametric Bayesian Joint Modelling of Skewed Longitudinal and Competing Risks Failure Time Data: With Application to Chronic Kidney Disease

In clinical and epidemiological studies, when the time-to-event(s) and the longitudinal outcomes are associated, modelling them separately may give biased estimates. A joint modelling approach is required to obtain unbiased results and to evaluate their association. In the joint model, a subject may be exposed to more than one type of failure event (competing risks). Considering the competing event as an independent censoring of the time-to-event process may underestimate the true survival probability and give biased results. Within the joint model, longitudinal outcomes may have nonlinear (irregular) trajectories over time and exhibit skewness with heavy tails. Accordingly, fully parametric mixed-effect models may not be flexible enough to model this type of complex longitudinal data. In addition, assuming a Gaussian distribution for model errors may be too restrictive to adequately represent within-individual variations and may lack robustness against deviation from distributional assumptions. To simultaneously overcome these issues, in this paper, we presented semiparametric joint models for competing risks failure time and skewed-longitudinal data by using a smoothing spline approach and a multivariate skew-t distribution. We also considered different parameterization approaches in the formulation of joint models and used a Bayesian approach to make the statistical inference. We illustrated the proposed methods by analyzing real data on a chronic kidney disease. To evaluate the performance of the methods, we also carried out simulation studies. The results of both the application and simulation studies revealed that the joint modelling approach proposed in this study performed well when the semiparametric, random-effects parameterization, and skew-t distribution specifications were taken into account.

Quantile regression with interval-censored data in questionnaire-based studies

Interval-censored data can arise in questionnaire-based studies when the respondent gives an answer in the form of an interval without having pre-specified ranges. Such data are called self-selected interval data. In this case, the assumption of independent censoring is not fulfilled, and therefore the ordinary methods for interval-censored data are not suitable. This paper explores a quantile regression model for self-selected interval data and suggests an estimator based on estimating equations. The consistency of the estimator is shown. Bootstrap procedures for constructing confidence intervals are considered. A simulation study indicates satisfactory performance of the proposed methods. An application to data concerning price estimates is presented.

Bias due to statistical handling of death and reirradiation in the assessment of duration of response after palliative radiotherapy: a scoping review and analysis of clinical data.

We investigated the influence of handling death and reirradiation on the estimation of duration of response (DOR). First, we performed a scoping review on methods to assess DOR in palliative radiotherapy. Second, we performed three different analyses on a subgroup of patients from a previously published prospective study. The first analysis was a competing risks analysis considering relapse of pain as the event of interest and death and reirradiation as competing events (Analysis A). The second and third analyses were standard survival analyses where the event of interest was a composite outcome of relapse of pain, death, or reirradiation (Analysis B) and relapse of pain (Analysis C), respectively. Death was considered as an event of interest in less than half of the papers, while reirradiation was not considered in any of the studies. Competing risks analysis was not performed in any of the studies. In the analysis of clinical data, competing risks analysis showed that relapse of pain predominated as the cause of the end of response. Median DOR was correctly estimated to be 4.1 months in Analyses A and B, but was overestimated to be 8.1 months in Analysis C. Death and reirradiation should be treated as the events of interest that mark the end of response (as in Analyses A and B) to avoid overestimation of treatment efficacy and an invalid assumption of independent censoring. The definition of end of response remains inconclusive in the assessment of DOR. We recommend competing risks analysis (Analysis A), by which we can estimate cumulative incidence of each event type and evaluate the necessity of reirradiation.

Estimating restricted mean treatment effects with additive-multiplicative hazards models

ABSTRACT The difference in restricted mean survival times between two groups is often of inherent interest in epidemiologic and medical studies. In this paper, we propose a general additive-multiplicative hazards (AMHs) regression model to estimate the restricted mean treatment effects, where the survival time is subject to both dependent and independent censoring. The AMH specifies an additive and multiplicative form on the hazard functions for the survival and censored times associated with covariates, which contains the proportional hazards model and the additive hazards model. By an inverse probability censoring weights scheme, we obtain the estimators of the regression parameters and the restricted mean treatment effects. We establish the large sample properties of the proposed estimators. Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed procedures and the primary biliary cirrhosis patients data are analyzed for illustration.

Weighted U-statistics for likelihood-ratio ordering of bivariate data

Characterisation of marginal distribution and density functions is of interest where data on a pair of random variables (X, Y) are observed. Stochastic orderings between (X, Y) have been studied in statistics and economics. Likelihood-ratio ordering is useful in understanding the behaviour of the random variables. In this article, tests based on U-statistics are proposed to test for equality of marginal density functions against the alternative of likelihood-ratio ordered when (X, Y) are dependent. The tests can be used when the data are either completely observed or subjected to independent univariate right censoring. The asymptotic variances of these tests are complicated and hence, are estimated using jackknife variance estimators. Validity of the jackknife variance estimators in statistical inference based on the proposed tests is demonstrated using simulation studies. The test for uncensored setting has desired size and good power for small sample. The performance of the tests for censored case depends on the sample size, proportion of censoring and the measure of dependence between X and Y. The tests are illustrated on three real data sets chosen in order to bring out various aspects of the tests.

General independent censoring in event-driven trials with staggered entry.

Randomized clinical trials with time-to-event endpoints are frequently stopped after a prespecified number of events has been observed. This practice leads to dependent data and nonrandom censoring, which can in general not be solved by conditioning on the underlying baseline information. In case of staggered study entry, matters are complicated substantially. The present paper demonstrates that the study design at hand entails general independent censoring in the counting process sense, provided that the analysis is based on study time information only. To illustrate that the filtrations must not use abundant information, we simulated data of event-driven trials and evaluated them by means of Cox regression models with covariates for the calendar times. The Breslow curves of the cumulative baseline hazard showed considerable deviations, which implies that the analysis is disturbed by conditioning on the calendar time variables. A second simulation study further revealed that Efron's classical bootstrap, unlike the (martingale-based) wild bootstrap, may lead to biased results in the given setting, as the assumption of random censoring is violated. This is exemplified by an analysis of data on immunotherapy in patients with advanced, previously treated nonsmall cell lungcancer.

Extension of Relative-Risk Power Estimator under Dependent Random Censored Data

In this paper, the considered problem consists in estimation of conditional survival function by right random censoring model in the presence of a covariate. We propose a new estimator of conditional survival function which is extension of relative-risk power estimator of independent censoring and study its large sample properties. We present result of asymptotic normality with the same limiting Gaussian process as for copula-graphic estimator.

Analysis of Survival Data with Competing Risks in ADRD (Alzheimer's Disease and Related Dementias) Research

Abstract Competing risk is an event that precludes the occurrence of the primary event of interest. For example, when studying risk factors associated with dementia, death before the onset of dementia serve as a competing event. A subject who dies is no longer at risk of dementia. This issue play more important role in ADRD research given the elderly population. Conventional methods for survival analysis assume independent censoring and ignore the competing events. However, there are some challenge issues using those conventional methods in the presence of competing risks. First, no one-to-one link between hazard function and cumulative incidence function (CIF), and Kaplan-Meier approach overestimates the cumulative incidence of the event of interest. Second, the effect of covariates on hazard rate cannot be directly linked to the effect of cumulative incidence (the risk). We will discuss two types of analyses in the presence of competing risk: Cause-specific hazard model and Fine-Gray subdistribution hazard model. Cause-specific hazard model directly quantify the cause-specific hazard among subjects who are at risk of developing the event of interest, while Fine-Gray subdistribution hazard model directly model the effects of covariates on the cumulative incidence function. The type of research questions (Association vs. Prediction) may guide the choice of different statistical approaches. We will illustrate those two competing risk analyses using the large national dataset from National Alzheimer’s Coordinating Center (NACC). We will analyze the association between baseline diabetes status and the incidence of dementia, in which death before the onset of dementia is a competing event.