Bias reduction for semi-competing risks frailty model with rare events: application to a chronic kidney disease cohort study in South Korea

In a semi-competing risks model in which a terminal event censors a non-terminal event but not vice versa, the conventional method can predict clinical outcomes by maximizing likelihood estimation. However, this method can produce unreliable or biased estimators when the number of events in the datasets is small. Specifically, parameter estimates may converge to infinity, or their standard errors can be very large. Moreover, terminal and non-terminal event times may be correlated, which can account for the frailty term. Here, we adapt the penalized likelihood with Firth's correction method for gamma frailty models with semi-competing risks data to reduce the bias caused by rare events. The proposed method is evaluated in terms of relative bias, mean squared error, standard error, and standard deviation compared to the conventional methods through simulation studies. The results of the proposed method are stable and robust even when data contain only a few events with the misspecification of the baseline hazard function. We also illustrate a real example with a multi-centre, patient-based cohort study to identify risk factors for chronic kidney disease progression or adverse clinical outcomes. This study will provide a better understanding of semi-competing risk data in which the number of specific diseases or events of interest is rare.

Exploring the impact of stage and tumor site on colorectal cancer survival: Bayesian survival modeling

Colorectal cancer is a prevalent malignancy with global significance. This retrospective study aimed to investigate the influence of stage and tumor site on survival outcomes in 284 colorectal cancer patients diagnosed between 2001 and 2017. Patients were categorized into four groups based on tumor site (colon and rectum) and disease stage (early stage and advanced stage). Demographic characteristics, treatment modalities, and survival outcomes were recorded. Bayesian survival modeling was performed using semi-competing risks illness-death models with an accelerated failure time (AFT) approach, utilizing R 4.1 software. Results demonstrated significantly higher time ratios for disease recurrence (TR = 1.712, 95% CI 1.489-2.197), mortality without recurrence (TR = 1.933, 1.480-2.510), and mortality after recurrence (TR = 1.847, 1.147-2.178) in early-stage colon cancer compared to early-stage rectal cancer. Furthermore, patients with advanced-stage rectal cancer exhibited shorter survival times for disease recurrence than patients with early-stage colon cancer. The interaction effect between the disease site and cancer stage was not significant. These findings, derived from the optimal Bayesian log-normal model for terminal and non-terminal events, highlight the importance of early detection and effective management strategies for colon cancer. Early-stage colon cancer demonstrated improved survival rates for disease recurrence, mortality without recurrence, and mortality after recurrence compared to other stages. Early intervention and comprehensive care are crucial to enhance prognosis and minimize adverse events in colon cancer patients.

Model-based hypothesis tests for the causal mediation of semi-competing risks

Analyzing the causal mediation of semi-competing risks has become important in medical research. Semi-competing risks refers to a scenario wherein an intermediate event may be censored by a primary event but not vice versa. Causal mediation analyses decompose the effect of an exposure on the primary outcome into an indirect (mediation) effect: an effect mediated through a mediator, and a direct effect: an effect not through the mediator. Here we proposed a model-based testing procedure to examine the indirect effect of the exposure on the primary event through the intermediate event. Under the counterfactual outcome framework, we defined a causal mediation effect using counting process. To assess statistical evidence for the mediation effect, we proposed two tests: an intersection-union test (IUT) and a weighted log-rank test (WLR). The test statistic was developed from a semi-parametric estimator of the mediation effect using a Cox proportional hazards model for the primary event and a series of logistic regression models for the intermediate event. We built a connection between the IUT and WLR. Asymptotic properties of the two tests were derived, and the IUT was determined to be a size [Formula: see text] test and statistically more powerful than the WLR. In numerical simulations, both the model-based IUT and WLR can properly adjust for confounding covariates, and the Type I error rates of the proposed methods are well protected, with the IUT being more powerful than the WLR. Our methods demonstrate the strongly significant effects of hepatitis B or C on the risk of liver cancer mediated through liver cirrhosis incidence in a prospective cohort study. The proposed method is also applicable to surrogate endpoint analyses in clinical trials.

A bayesian approach to model the underlying predictors of early recurrence and postoperative death in patients with colorectal Cancer

Objective

This study aimed at utilizing a Bayesian approach semi-competing risks technique to model the underlying predictors of early recurrence and postoperative Death in patients with colorectal cancer (CRC).

Methods

In this prospective cohort study, 284 patients with colorectal cancer, who underwent surgery, referred to Imam Khomeini clinic in Hamadan from 2001 to 2017. The primary outcomes were the probability of recurrence, the probability of Mortality without recurrence, and the probability of Mortality after recurrence. The patients ‘recurrence status was determined from patients’ records. The Bayesian survival modeling was carried out by semi-competing risks illness-death models, with accelerated failure time (AFT) approach, in R 4.1 software. The best model was chosen according to the lowest deviance information criterion (DIC) and highest logarithm of the pseudo marginal likelihood (LPML).

Results

The log-normal model (DIC = 1633, LPML = -811), was the optimal model. The results showed that gender(Time Ratio = 0.764: 95% Confidence Interval = 0.456–0.855), age at diagnosis (0.764: 0.538–0.935 ), T₃ stage (0601: 0.530–0.713), N₂ stage (0.714: 0.577–0.935 ), tumor size (0.709: 0.610–0.929), grade of differentiation at poor (0.856: 0.733–0.988), and moderate (0.648: 0.503–0.955) levels, and the number of chemotherapies (1.583: 1.367–1.863) were significantly related to recurrence. Also, age at diagnosis (0.396: 0.313–0.532), metastasis to other sites (0.566: 0.490–0.835), T₃ stage (0.363: 0.592 − 0.301), T₄ stage (0.434: 0.347–0.545), grade of differentiation at moderate level (0.527: 0.387–0.674), tumor size (0.595: 0.500–0.679), and the number of chemotherapies (1.541: 1.332–2.243) were the significantly predicted the death. Also, age at diagnosis (0.659: 0.559–0.803), and the number of chemotherapies (2.029: 1.792–2.191) were significantly related to mortality after recurrence.

Conclusion

According to specific results obtained from the optimal Bayesian log-normal model for terminal and non-terminal events, appropriate screening strategies and the earlier detection of CRC leads to substantial improvements in the survival of patients.

Statistical models for composite endpoints of death and non-fatal events: a review

The proper analysis of composite endpoints consisting of both death and non-fatal events is an intriguing and sometimes contentious topic. The current practice of analyzing time to the first event often draws criticisms for ignoring the unequal importance between component events and for leaving recurrent-event data unused. Novel methods that address these limitations have recently been proposed. To compare the novel versus traditional approaches, we review three typical models for composite endpoints based on time to the first event, composite event process, and pairwise hierarchical comparisons. The pros and cons of these models are discussed with reference to the relevant regulatory guidelines, such as the recently released ICH-E9(R1) Addendum "Estimands and Sensitivity Analysis in Clinical Trials". We also discuss the impact of censoring when the model assumptions are violated and explore sensitivity analysis strategies. Simulation studies are conducted to assess the performance of the reviewed methods under different settings. As demonstration, we use publicly available R-packages to analyze real data from a major cardiovascular trial.

Fitting a shared frailty illness-death model to left-truncated semi-competing risks data to examine the impact of education level on incident dementia

BACKGROUND

Semi-competing risks arise when interest lies in the time-to-event for some non-terminal event, the observation of which is subject to some terminal event. One approach to assessing the impact of covariates on semi-competing risks data is through the illness-death model with shared frailty, where hazard regression models are used to model the effect of covariates on the endpoints. The shared frailty term, which can be viewed as an individual-specific random effect, acknowledges dependence between the events that is not accounted for by covariates. Although methods exist for fitting such a model to right-censored semi-competing risks data, there is currently a gap in the literature for fitting such models when a flexible baseline hazard specification is desired and the data are left-truncated, for example when time is on the age scale. We provide a modeling framework and openly available code for implementation.

METHODS

We specified the model and the likelihood function that accounts for left-truncated data, and provided an approach to estimation and inference via maximum likelihood. Our model was fully parametric, specifying baseline hazards via Weibull or B-splines. Using simulated data we examined the operating characteristics of the implementation in terms of bias and coverage. We applied our methods to a dataset of 33,117 Kaiser Permanente Northern California members aged 65 or older examining the relationship between educational level (categorized as: high school or less; trade school, some college or college graduate; post-graduate) and incident dementia and death.

RESULTS

A simulation study showed that our implementation provided regression parameter estimates with negligible bias and good coverage. In our data application, we found higher levels of education are associated with a lower risk of incident dementia, after adjusting for sex and race/ethnicity.

CONCLUSIONS

As illustrated by our analysis of Kaiser data, our proposed modeling framework allows the analyst to assess the impact of covariates on semi-competing risks data, such as incident dementia and death, while accounting for dependence between the outcomes when data are left-truncated, as is common in studies of aging and dementia.

Frailty modelling approaches for semi-competing risks data

In the semi-competing risks situation where only a terminal event censors a non-terminal event, observed event times can be correlated. Recently, frailty models with an arbitrary baseline hazard have been studied for the analysis of such semi-competing risks data. However, their maximum likelihood estimator can be substantially biased in the finite samples. In this paper, we propose effective modifications to reduce such bias using the hierarchical likelihood. We also investigate the relationship between marginal and hierarchical likelihood approaches. Simulation results are provided to validate performance of the proposed method. The proposed method is illustrated through analysis of semi-competing risks data from a breast cancer study.

Estimating cross quantile residual ratio with left-truncated semi-competing risks data

A semi-competing risks setting often arises in biomedical studies, involving both a nonterminal event and a terminal event. Cross quantile residual ratio (Yang and Peng in Biometrics 72:770-779, 2016) offers a flexible and robust perspective to study the dependency between the nonterminal and the terminal events which can shed useful scientific insight. In this paper, we propose a new nonparametric estimator of this dependence measure with left truncated semi-competing risks data. The new estimator overcomes the limitation of the existing estimator that is resulted from demanding a strong assumption on the truncation mechanism. We establish the asymptotic properties of the proposed estimator and develop inference procedures accordingly. Simulation studies suggest good finite-sample performance of the proposed method. Our proposal is illustrated via an application to Denmark diabetes registry data.

Accelerated failure time models for semi-competing risks data in the presence of complex censoring

Statistical analyses that investigate risk factors for Alzheimer's disease (AD) are often subject to a number of challenges. Some of these challenges arise due to practical considerations regarding data collection such that the observation of AD events is subject to complex censoring including left-truncation and either interval or right-censoring. Additional challenges arise due to the fact that study participants under investigation are often subject to competing forces, most notably death, that may not be independent of AD. Towards resolving the latter, researchers may choose to embed the study of AD within the "semi-competing risks" framework for which the recent statistical literature has seen a number of advances including for the so-called illness-death model. To the best of our knowledge, however, the semi-competing risks literature has not fully considered analyses in contexts with complex censoring, as in studies of AD. This is particularly the case when interest lies with the accelerated failure time (AFT) model, an alternative to the traditional multiplicative Cox model that places emphasis away from the hazard function. In this article, we outline a new Bayesian framework for estimation/inference of an AFT illness-death model for semi-competing risks data subject to complex censoring. An efficient computational algorithm that gives researchers the flexibility to adopt either a fully parametric or a semi-parametric model specification is developed and implemented. The proposed methods are motivated by and illustrated with an analysis of data from the Adult Changes in Thought study, an on-going community-based prospective study of incident AD in western Washington State.

Semiparametric model for semi-competing risks data with application to breast cancer study

For many forms of cancer, patients will receive the initial regimen of treatments, then experience cancer progression and eventually die of the disease. Understanding the disease process in patients with cancer is essential in clinical, epidemiological and translational research. One challenge in analyzing such data is that death dependently censors cancer progression (e.g., recurrence), whereas progression does not censor death. We deal with the informative censoring by first selecting a suitable copula model through an exploratory diagnostic approach and then developing an inference procedure to simultaneously estimate the marginal survival function of cancer relapse and an association parameter in the copula model. We show that the proposed estimators possess consistency and weak convergence. We use simulation studies to evaluate the finite sample performance of the proposed method, and illustrate it through an application to data from a study of early stage breast cancer.

Joint latent class model for longitudinal data and interval-censored semi-competing events: Application to dementia

Joint models are used in ageing studies to investigate the association between longitudinal markers and a time-to-event, and have been extended to multiple markers and/or competing risks. The competing risk of death must be considered in the elderly because death and dementia have common risk factors. Moreover, in cohort studies, time-to-dementia is interval-censored since dementia is assessed intermittently. So subjects can develop dementia and die between two visits without being diagnosed. To study predementia cognitive decline, we propose a joint latent class model combining a (possibly multivariate) mixed model and an illness-death model handling both interval censoring (by accounting for a possible unobserved transition to dementia) and semi-competing risks. Parameters are estimated by maximum-likelihood handling interval censoring. The correlation between the marker and the times-to-events is captured by latent classes, homogeneous sub-groups with specific risks of death, dementia, and profiles of cognitive decline. We propose Markovian and semi-Markovian versions. Both approaches are compared to a joint latent-class model for competing risks through a simulation study, and applied in a prospective cohort study of cerebral and functional ageing to distinguish different profiles of cognitive decline associated with risks of dementia and death. The comparison highlights that among subjects with dementia, mortality depends more on age than on duration of dementia. This model distinguishes the so-called terminal predeath decline (among healthy subjects) from the predementia decline.

A new flexible dependence measure for semi-competing risks

Semi-competing risks data are often encountered in chronic disease follow-up studies that record both nonterminal events (e.g., disease landmark events) and terminal events (e.g., death). Studying the relationship between the nonterminal event and the terminal event can provide insightful information on disease progression. In this article, we propose a new sensible dependence measure tailored to addressing such an interest. We develop a nonparametric estimator, which is general enough to handle both independent right censoring and left truncation. Our strategy of connecting the new dependence measure with quantile regression enables a natural extension to adjust for covariates with minor additional assumptions imposed. We establish the asymptotic properties of the proposed estimators and develop inferences accordingly. Simulation studies suggest good finite-sample performance of the proposed methods. Our proposals are illustrated via an application to Denmark diabetes registry data.

Quantile Regression Adjusting for Dependent Censoring from Semi-Competing Risks

In this work, we study quantile regression when the response is an event time subject to potentially dependent censoring. We consider the semi-competing risks setting, where time to censoring remains observable after the occurrence of the event of interest. While such a scenario frequently arises in biomedical studies, most of current quantile regression methods for censored data are not applicable because they generally require the censoring time and the event time be independent. By imposing rather mild assumptions on the association structure between the time-to-event response and the censoring time variable, we propose quantile regression procedures, which allow us to garner a comprehensive view of the covariate effects on the event time outcome as well as to examine the informativeness of censoring. An efficient and stable algorithm is provided for implementing the new method. We establish the asymptotic properties of the resulting estimators including uniform consistency and weak convergence. The theoretical development may serve as a useful template for addressing estimating settings that involve stochastic integrals. Extensive simulation studies suggest that the proposed method performs well with moderate sample sizes. We illustrate the practical utility of our proposals through an application to a bone marrow transplant trial.