Modelling dependent censoring in time-to-event data using boosting copula regression

Time-to-event data analyses play a crucial role in various disciplines, including biostatistics (Collett 2003; Klein and Moeschberger 2005; Samoladas et al. 2010; Hassan et al. 2018; De Bin and Stikbakke 2023; Srujana et al. 2024; Stijven et al. 2024). Their focus is on studying the time to the occurrence of an event of interest, such as death, disease progression, treatment response or another specific endpoint. A common challenge is that patients in a study may not experience the event of interest within the study period or may be lost to follow-up before the event occurs due to study dropout. Consequently, censored observations are an inherent and natural characteristic of survival data. Most widely used approaches, such as the Kaplan-Meier estimator (Kaplan and Meier 1958) or the Cox proportional hazards model (Cox 1972), can handle time-to-event data with censored observations, including the most common form of right-censoring. In these approaches, a key assumption is that the survival time T and censoring time C are statistically independent conditional on the covariates. This is illustrated in the left part of Fig. 1, where C provides no information about T given the covariates \({\varvec{X}}\) and vice versa.

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However, this assumption is questionable in many situations. For example, if patients withdraw from a medical trial because their health condition is deteriorating, approaches that assume independent censoring could lead to biased results (Huang and Zhang 2008). This is due to a direct link between the survival and censoring times as illustrated in case a) in the right of Fig. 1. Thus, if we assume that sicker patients drop out of the study due to poor health, censored patients are more likely to be sicker than the non-censored patients and the survival time of the patients may be overestimated. A further dependence can exist if it is caused by unobserved confounding variables, as indicated by case b) in the right of Fig. 1. Here, the unobserved variable U influences survival and censoring times. This could occur if a patient in a trial experiences side effects of a treatment that requires an alternative treatment, causing the patient to drop out of the study.

Several distinct approaches have been developed to account for dependent censoring. These include frailty models, which introduce random effects to capture unobserved heterogeneity, such as that caused by unmeasured covariates, and to induce dependence between survival and censoring times only indirectly through a shared frailty term (e.g., Huang and Wolfe 2002; Schneider et al. 2020). In contrast, copula-based approaches have become increasingly popular for directly modeling the dependence between survival and censoring times  (e.g., Emura and Chen 2016; Deresa and Van Keilegom 2020a, 2020b; Deresa et al. 2022; Midtfjord et al. 2022; Czado and Van Keilegom 2023; Deresa and Van Keilegom 2024b). By specifying a joint distribution through a copula function and marginal models, these methods allow for more flexible and explicit dependence structures, going beyond the implicit modeling of frailty-based approaches. The identifiability of copula models under dependent censoring, however, is challenging because the right-censoring mechanism obviously prevents the simultaneous observation of (T, C) and thus obscures the direct relationship between T and C in the observed data (Tsiatis 1975; Crowder 1991). To ensure model identifiability, some authors assume that the copula is known. The first contribution in this direction was made by Zheng and Klein (1995), who proposed a generalization of the Kaplan-Meier estimator, the so-called copula-graphic estimator. This method was further explored by Rivest and Wells (2001) in the context of Archimedean copulas, assuming a known copula. Since then, numerous approaches have been developed in this vein (see among others, Braekers and Veraverbeke 2005; Huang and Zhang 2008; Chen 2010; Sujica and Van Keilegom 2018; Emura and Chen 2018; Deresa and Van Keilegom 2021). For instance, Chen (2010) extended the copula framework to semiparametric transformation models, using a known copula to characterize the association between survival and censoring times. While this method offers considerable flexibility for modeling marginal distributions, the assumption of a fully known copula function is often unrealistic in practical biostatistical applications. Czado and Van Keilegom (2023) recently proved that a known copula is not necessary for identifying the joint distribution of T and C. They proposed a model based on a parametric copula for the relationship between T and C along with parametric marginal distributions. However, a limitation is that covariates cannot be included into the model. Deresa et al. (2022) addressed this by exploring a copula-based method for bivariate data with left truncation and dependent right censoring, including multiple covariates but only for the margins.

To develop a framework that does not require the copula to be known, while allowing to incorporate covariates for all model parameters (in the spirit of generalized additive models for location, scale and shape, Rigby and Stasinopoulos 2005), we propose a model-based boosting approach for dependent censoring in survival analysis via distributional copula regression. Building on the work of Czado and Van Keilegom (2023) and Deresa et al. (2022), our method utilizes a parametric copula combined with arbitrary parametric marginal distributions. Estimation is carried out via statistical boosting, which facilitates the simultaneous modelling of all distribution parameters as functions of potentially different sets of covariates. Our boosting algorithm is based on the work of Hans et al. (2023), who proposed a general framework for boosting in distributional copula regression.

Our approach comes with three particular merits. First, the dependence between survival and censoring times can be modelled and explained through covariates. This may provide deeper insights and a better understanding of the underlying relationship between survival and censoring times. Second, our approach can handle high-dimensional cases where the number of covariates exceeds the number of observations (\(p > n\)). For such settings, most classical approaches are no longer feasible. Third, the boosting approach has the advantage of including a data-driven variable selection mechanism. This feature is particularly beneficial when dealing with many potential predictors, ensuring that only the most relevant variables are included without compromising predictive power.

Using a recent observational study from oncology that investigates the overall survival of patients with colon cancer we demonstrate the merits of our approach in practical research. The routine data were gathered from a specialized cancer center and include potential predictors such as common clinical variables related to the tumor, patient demographics, and treatment (Seipp et al. 2021). This dataset represents a typical scenario encountered in biostatistical applications, characterized by a relatively high proportion of right-censored observations. Since the censoring mechanism is inherently unknown, it is impossible to determine whether the censoring is dependent or independent. However, in observational studies based on routine data, dependent censoring is more likely to occur due to various factors; for example, both patients with advanced disease and better health are more likely to miss follow-up visits or withdraw from the study (Howe et al. 2010). Nevertheless, the Cox proportional hazards model or a parametric accelerated failure time (AFT) model (Wei 1992; Heller 2024) are often still the methods of choice. We examine how the results of the analysis differ when accounting for associations between survival and censoring times, and how these associations vary with the covariates.

The structure of this article is as follows: Sect. 2 presents distributional copula regression for dependent censoring in time-to-event data and how to perform estimation based on model-based boosting. In Sect. 3, we investigate the performance of the new boosting approach in a simulation study with different scenarios—including conditional independent censoring. We apply our approach to colon cancer data and present the results in Sect. 4 before we finally discuss strengths and limitations of our approach in Sect. 5. Supplementary Materials (SM) contain further simulations and additional results for the application.

We conducted a detailed simulation study to evaluate the performance of CopBoostDepCens with regard to the following questions:

We illustrate the application of our method via an observational study investigating the overall survival of colon cancer patients after surgery based on routine data.

We developed CopBoostDepCens, a model-based boosting approach that relies on parametric marginal distributions and copulas, ensuring model identification. CopBoostDepCens does not assume a known copula and allows to include large numbers of potential predictor variables.

Modelling both survival and censoring times together through a copula in combination with a component-wise boosting approach bridges the gap from a compelling theoretical idea to a practical and versatile modelling option in many disciplines.

We demonstrated through simulations that CopBoostDepCens is able to identify the correct predictors in various data situations, including varying numbers of potential covariates and different censoring rates. While boosting the distributional AFT model (assuming independence) also may yield satisfying results, particularly in situations with a low censoring rate or a weak dependence between survival and censoring times, CopBoostDepCens provides additional insights into the dependence between survival and censoring times, not captured by classical approaches. For instance, in our analysis of the overall survival of colon cancer patients—a scenario commonly encountered by biostatisticians—we identified, among other findings, a negative effect of chemotherapy on the relationship between survival and censoring. This insight could not be obtained using previous models.

Evaluating the predictive performance of our model and comparing it with standard approaches such as the AFT model, is challenging. Many conventional metrics, such as the mean absolute error or concordance index, either exclude censored observations or assume non-informative censoring, which leads to biased results or misleading conclusions (Qi et al. 2023). Even metrics that attempt to account for censoring, such as the Brier score, pose challenges. Traditional implementations estimate the censoring distribution using Kaplan–Meier estimator, which assumes independence between survival and censoring times. In contrast, CopBoostDepCens explicitly models the censoring distribution as function of covariates. This fundamental difference complicates the comparison between models and raises concerns about the appropriateness of using standard evaluation metrics that assume independent censoring.

These limitations underscore the need for developing metrics that adequately capture the model performance under dependent censoring (Foomani et al. 2023). Recent work has started to close this gap: Prince et al. (2025) demonstrates that the Brier score is sensitive to the choice of weighting and provides guidance on inverse probability weighting (IPCW), while Lillelund et al. (2025) demonstrates the bias of traditional metrics and propose a copula-based alternative that uses a known Archimedean copula to estimate \({\hat{G}}(t)\). However, this approach requires a known copula structure and ignores covariates. Our method avoids these constraints: rather than relying on a copula-graphic estimator, our model directly estimates the censoring distribution conditional on covariates and allows the dependency parameter to vary with covariate effects.

An approach related to ours is that of Midtfjord et al. (2022). The authors employ the Clayton copula to account for dependent censoring. However, their model is restricted to the Clayton copula and limited to a fixed dependence parameter. Furthermore, it assumes that censoring is independent of covariates. In contrast, our boosting approach flexibly models all distributional parameters as functions of covariates.

This flexibility and the ability to model each parameter of the margins and the copula based on covariates also comes with the challenge of interpreting the effects of covariates involved in multiple parameters. Further research is warranted to reduce model complexity, for example, by deselecting covariates that have only a minor impact on overall model performance (Strömer et al. 2022, 2025). An inherent limitation of CopBoostDepCens is the difficulty in providing standard errors for the resulting coefficients, which is true for all boosting approaches. Due to early stopping and the resulting shrinkage of effect estimates, there are no closed formulas for standard errors. To address this, permutation tests could be utilized for significance testing and to provide p-values (Mayr et al. 2017; Hepp et al. 2019), though this would further increase computational costs. Another limitation of our boosting approach is that we currently only have empirical evidence from the simulation study and the application on its performance, but no formal theoretical proof regarding the consistency of effect estimates or asymptotics. Together with the missing standard errors, this makes statistical boosting in general most suitable for exploratory data analyses or prediction modelling—not for confirmatory data analyses (Mayr and Hofner 2018; Strömer et al. 2023).

Lastly, our conditional censoring approach is naturally limited to settings with a sufficient number of censored and uncensored individuals. In our simulations, we considered different censoring rates ranging from 20% to 80%. Also it is clear, that for a complex model as ours where we relate all distribution parameters from two marginal distributions as well as a copula parameter to covariates, one needs a reasonable large sample size for reliable estimation and to detect the most informative predictors. In our simulations, we used \(n = 1000\) for most settings, a setting with \(n = 100\) can be found in the Supplementary Material. This limitation is already true for most distributional regression approaches, but for CopBoostDepCens becomes even more pressing, as only parts of the observations can be used for estimating the corresponding marginal distributions.

While our current implementation uses one-parameter copulas, the modular structure of the boosting algorithm would allow the integration of more advanced copula families. These could include, for example, Vine (Czado and Nagler 2022) and mixture copulas (Pan et al. 2025) to better capture asymmetric or time-varying dependencies. However, careful consideration of pair-copula selection, identifiability constraints and sample size requirements is necessary for these extensions, particularly in smaller studies (Barthel et al. 2019). Another potential topic for future research involves extending the approach to accommodate left censored or truncated data (Deresa et al. 2022). Doing so could broaden the applicability of the model to a wider range of survival analysis scenarios. Building on recent advances by Deresa and Van Keilegom (2024a) and Deresa and Van Keilegom (2024b), future work could incorporate semi-parametric margins (e.g., a Cox proportional hazards model for survival times with nonparametric baseline hazards) to relax the parametric assumptions while preserving identifiability under dependent censoring.

Tables and figures referenced in Sects. 3 and 4 are available with this paper online. The code is available at https://github.com/AnnikaStr/CopBoostDepCens.