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Association models, like frailty and copula models, are frequently used to analyse clustered survival data and to evaluate within-cluster associations. The assumption of noninformative censoring is commonly applied to these models, though it may not be true in many situations. We consider bivariate competing risk data and focus on association models specified for the bivariate cumulative incidence function (CIF), which is a non-parametrically identifiable quantity. Copula models are proposed which relate the bivariate CIF to its corresponding univariate CIFs, similarly to independently right-censored data, and accommodate frailty models for the bivariate CIF. Two estimating equations are developed to estimate the association parameter, permitting the univariate CIFs to be estimated either parametrically or non-parametrically. Goodness-of-fit tests are presented for formally evaluating the parametric models. Both estimators perform well with moderate sample sizes in simulation studies. The practical use of the methodology is illustrated in an analysis of dementia associations.

Kaplan–Meier survival analysis overestimates cumulative incidence in competing risks (CRs) settings. The extent of overestimation (or its clinical significance) has been questioned, and CRs methods are infrequently used. This meta-analysis compares the Kaplan–Meier method to the cumulative incidence function (CIF), a CRs method. We searched MEDLINE, EMBASE, BIOSIS Previews, Web of Science (1992–2016), and article bibliographies for studies estimating cumulative incidence using the Kaplan–Meier method and CIF. For studies with sufficient data, we calculated pooled risk ratios (RRs) comparing Kaplan–Meier and CIF estimates using DerSimonian and Laird random effects models. We performed stratified meta-analyses by clinical area, rate of CRs (CRs/events of interest), and follow-up time. Of 2,192 identified abstracts, we included 77 studies in the systematic review and meta-analyzed 55. The pooled RR demonstrated the Kaplan–Meier estimate was 1.41 [95% confidence interval (CI): 1.36, 1.47] times higher than the CIF. Overestimation was highest among studies with high rates of CRs [RR = 2.36 (95% CI: 1.79, 3.12)], studies related to hepatology [RR = 2.60 (95% CI: 2.12, 3.19)], and obstetrics and gynecology [RR = 1.84 (95% CI: 1.52, 2.23)]. The Kaplan–Meier method overestimated the cumulative incidence across 10 clinical areas. Using CRs methods will ensure accurate results inform clinical and policy decisions.

Risk estimates from Kaplan–Meier curves are well known to medical researchers, reviewers, and editors. In this study, we determined the proportion of Kaplan–Meier analyses published in prominent medical journals that are potentially biased because of competing events (“competing risk bias”). We randomly selected 100 studies that had at least one Kaplan–Meier analysis and were recently published in prominent medical journals. Susceptibility to competing risk bias was determined by examining the outcome and potential competing events. In susceptible studies, bias was quantified using a previously validated prediction model when the number of outcomes and competing events were given. Forty-six studies (46%) contained Kaplan–Meier analyses susceptible to competing risk bias. Sixteen studies (34.8%) susceptible to competing risk cited the number of outcomes and competing events; in six of these studies (6/16, 37.5%), the outcome risk from the Kaplan–Meier estimate (relative to the true risk) was biased upward by 10% or more. Almost half of Kaplan–Meier analyses published in medical journals are susceptible to competing risk bias and may overestimate event risk. This bias was found to be quantitatively important in a third of such studies.

Background This study examines the effect of prognostic patient and disease characteristics on colorectal cancer (CRC) recurrence after curative resection. We used competing risk analysis with death as a competing risk. This method provides the clinician a perspective into a patient’s actual risk of experiencing a recurrence. Methods A retrospective cohort study of patients diagnosed with CRC who underwent curative resection for CRC from 2003–2007 at the Royal University Hospital in Saskatoon was completed. The outcome of interest was the first CRC recurrence, either local or distant metastasis. Demographic data, tumor characteristics, adjuvant treatment and follow-up data, date of local recurrence or metastasis were recorded from the medical record. Univariate analysis was completed to look at the relationship between each of the prognostic indicators and recurrence. Multivariable modelling (subdistribution regression modelling) was done to identify the main risk factors in determining recurrence. Results Of 148 patients, 38 (25.7%) experienced a recurrence, 16 (10.8%) died without evidence of recurrence, and 94 (63.5%) experienced neither outcome. The median follow-up was 30.5 months (interquartile range 10.6–50). In univariable subdistribution regression, T-stage, N-stage, vascular invasion and positive margins were all predictive of cancer recurrence, with p  ≤ 0.001, with subdistribution hazard ratios for T4 stage at 11.93, T3 stage at 2.46, N2 stage at 10.58, and presence of vascular invasion at 4.27. N-stage remained as the sole predictor in multivariable regression. Cumulative incidence function (CIF) of recurrence at 48 months after surgery was 15%, 27% and 90% for N1/2, N3 and N4 respectively. Conclusion The highest CIF of recurrence was associated with T4 stage, N2 stage, and vascular invasion. Patient’s age, tumour location, type, or histological grade were not found to have a significant effect on the success of CRC surgery in precluding a recurrence.

Under-five child mortality remains a significant public health issue in Bangladesh and other developing countries. Identifying key risk factors within a hierarchical data structure is crucial for improving health system performance. This study employed the Fine-Gray Frailty (FGF) model to Bangladesh Demographic and Health Survey (BDHS) data to assess child mortality from disease-related causes, treating non-disease-related deaths as competing risks in hierarchical framework. The model was also applied by considering non-disease deaths as the main event in a subsequent analysis. Findings reveal that maternal age, child sex, birth order, and regional variation remain significant determinants of mortality after adjusting for the hierarchical data structure. Children of mothers older than 30 years face a significantly higher risk of non-disease deaths compared with those aged 20–30 years. Male children experience higher mortality than females for both disease and non-disease causes. Higher birth order is associated with a lower risk of non-disease mortality. Regionally, Khulna shows a significantly increasing hazard of non-disease deaths than Barisal. This study underscores the value of hierarchical competing risk models for guiding targeted public health interventions.

This study examines nonparametric estimations of a transition probability matrix of a nonhomogeneous Markov process with a finite state space and a partially observed absorbing state. We impose a missing-at-random assumption and propose a computationally efficient nonparametric maximum pseudolikelihood estimator (NPMPLE). The estimator depends on a parametric model that is used to estimate the probability of each absorbing state for the missing observations based, potentially, on auxiliary data. For the latter model, we propose a formal goodness-of-fit test based on a residual process. Using modern empirical process theory, we show that the estimator is uniformly consistent and converges weakly to a tight mean-zero Gaussian random field. We also provide a methodology for constructing simultaneous confidence bands. Simulation studies show that the NPMPLE works well with small sample sizes and that it is robust against some degree of misspecification of the parametric model for the missing absorbing states. The method is illustrated using HIV data from sub-Saharan Africa to estimate the transition probabilities of death and disengagement from HIV care.

Interval sampling is widely used for collection of disease registry data, which typically report incident cases during a certain time period. Such sampling scheme induces doubly truncated data if the failure time can be observed exactly and doubly truncated and interval censored (DTIC) data if the failure time is known only to lie within an interval. In this article, we consider nonparametric estimation of the cumulative incidence functions (CIF) using doubly-truncated and interval-censored competing risks (DTIC-C) data obtained from interval sampling scheme. Using the approach of Shen (Stat Methods Med Res 31:1157-1170, 2022b), we first obtain the nonparametric maximum likelihood estimator (NPMLE) of the distribution function of failure time ignoring failure types. Using the NPMLE, we proposed nonparametric estimators of the CIF with DTIC-C data and establish consistency of the proposed estimators. Simulation studies show that the proposed estimator performs well for finite sample size.

Regression methodology has been well developed for competing risks data with continuous event times, both for the cause-specific hazard and cumulative incidence functions. However, in many applications, including those from the Surveillance, Epidemiology, and End Results (SEER) program of the National Cancer Institute, the event times may be observed discretely. Naive application of continuous time regression methods to such data is not appropriate. We propose maximum likelihood inferences for estimation of model parameters for the discrete time cause-specific hazard functions, develop predictions for the associated cumulative incidence functions, and derive consistent variance estimators for the predicted cumulative incidence functions. The methods are readily implemented using standard software for generalized estimating equations, where models for different causes may be fitted separately. For the SEER data, it may be desirable to model different event types on different time scales and the methods are generalized to accommodate such scenarios, extending earlier work on continuous time data. Simulation studies demonstrate that the methods perform well in realistic set-ups. The methodology is illustrated with stage III colon cancer data from SEER.

Background: An evaluation of the effect of a healthcare intervention (or an exposure) must consider multiple possible outcomes, including the primary outcome of interest and other outcomes such as adverse events or mortality. The determination of the likelihood of benefit from an intervention, in the presence of other competing outcomes, is a competing risks problem. Although statistical methods exist for quantifying the probability of benefit from an intervention while accounting for competing events, these methods have not been widely adopted by clinical researchers. Objectives: (1) To demonstrate the importance of considering competing risks in the evaluation of treatment effectiveness, and (2) to review appropriate statistical methods, and recommend how they might be applied. Research Design and Methods: We reviewed 3 statistical approaches for analyzing the competing risks problem: (a) cause-specific hazard (CSH), (b) cumulative incidence function (CIF), and (c) event-free survival (EFS). We compare these methods using a simulation study and a reanalysis of a randomized clinical trial. Results: Simulation studies evaluating the statistical power to detect the effect of intervention under different scenarios showed that: (1) CSH approach is best for detecting the effect of an intervention if the intervention only affects either the primary outcome or the competing event; (2) EFS approach is best only when the intervention affects both primary and competing events in the same manner; and (3) CIF approach is best when the intervention affects both primary and competing events, but in opposite directions. Using data from a randomized controlled trial, we demonstrated that a comprehensive approach using all 3 approaches provided useful insights on the effect of an intervention on the relative and absolute risks of multiple competing outcomes. Conclusions: CSH is the fundamental measure of outcome in competing risks problems. It is appropriate for evaluating treatment effects in the presence of competing events. Results of CSH analysis for primary and competing outcomes should always be reported even when EFS or CIF approaches are called for. EFS is appropriate for evaluating the composite effect of an intervention, only when combining different endpoints is clinically and biologically meaningful, and the treatment has similar effects on all event types. CIF is useful for evaluating the likelihood of benefit from an intervention over a meaningful period. CIF should be used for absolute risk calculations instead of the widely used complement of the Kaplan-Meier (1 — KM) estimator.

Background Multi-state models are complex stochastic models which focus on pathways defined by the temporal and sequential occurrence of numerous events of interest. In particular, the so-called illness-death models are especially useful for studying probabilities associated to diseases whose occurrence competes with other possible diseases, health conditions or death. They can be seen as a generalization of the competing risks models, which are widely used to estimate disease-incidences among populations with a high risk of death, such as elderly or cancer patients. The main advantage of the aforementioned illness-death models is that they allow the treatment of scenarios with non-terminal competing events that may occur sequentially, which competing risks models fail to do. Methods We propose an illness-death model using Cox proportional hazards models with Weibull baseline hazard functions, and applied the model to a study of recurrent hip fracture. Data came from the PREV2FO cohort and included 34491 patients aged 65 years and older who were discharged alive after a hospitalization due to an osteoporotic hip fracture between 2008-2015. We used a Bayesian approach to approximate the posterior distribution of each parameter of the model, and thus cumulative incidences and transition probabilities. We also compared these results with a competing risks specification. Results Posterior transition probabilities showed higher probabilities of death for men and increasing with age. Women were more likely to refracture as well as less likely to die after it. Free-event time was shown to reduce the probability of death. Estimations from the illness-death and the competing risks models were identical for those common transitions although the illness-death model provided additional information from the transition from refracture to death. Conclusions We illustrated how multi-state models, in particular illness-death models, may be especially useful when dealing with survival scenarios which include multiple events, with competing diseases or when death is an unavoidable event to consider. Illness-death models via transition probabilities provide additional information of transitions from non-terminal health conditions to absorbing states such as death, what implies a deeper understanding of the real-world problem involved compared to competing risks models.