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Competing-Risk Cure Models: A Five-Axis Systematic Review of Methodological Literature

This paper presents a five-axis systematic review of 26 competing-risk cure models that clarifies their methodological distinctions, compares estimation techniques, and highlights critical gaps in software availability and reproducibility to guide future research and transparent model selection.

Original authors: Nilotpal Sanyal

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nilotpal Sanyal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of how long people live or how long machines last, scientists often face a puzzle: not everyone is at risk of the same ending. Imagine a group of patients being followed after a bone marrow transplant. Some will eventually relapse, some will die from other causes like heart disease, and some will never relapse at all, living out their natural lives free from that specific threat. This second group, the ones who are effectively "cured" of the primary danger, creates a unique statistical challenge. Standard methods for analyzing time-to-event data often assume that everyone is eventually susceptible to the event being studied. When a portion of the population is immune, those standard tools can mislead, painting a picture of risk that never truly disappears. Furthermore, when multiple different endings are possible—such as relapse versus death from another cause—the situation becomes even more complex, because the occurrence of one event prevents the observation of the others.

For decades, researchers have developed various mathematical ways to handle these "cure" fractions and competing risks, but the field has become a patchwork of different approaches. A new systematic review by Nilotpal Sanyal at the University of Texas at El Paso brings order to this scattered landscape. The author examined twenty-six distinct methodological papers, organizing them not by their names, which can be misleading, but by five fundamental questions that define how each model works. The review reveals that while many models share similar titles, they often rely on entirely different assumptions about how cure happens, how risks interact, and how data is processed. The central finding is that there is no single "best" model; rather, the choice of method depends heavily on what the researcher believes about the underlying biology or mechanics of the situation. Crucially, the review also highlights a significant gap in the field: despite the theoretical sophistication of these models, very few are available as ready-to-use software, leaving most researchers to write their own custom code for every new study.

The review begins by clarifying what "cure" actually means in these models. In some approaches, being cured means a person is immune to all possible bad outcomes forever. In others, a person might be cured of the primary disease but still remain vulnerable to other causes of failure, such as an unrelated illness. This distinction is vital because it changes the entire mathematical structure of the prediction. The author then sorts the models based on how they break down the data. Some models treat the population as a simple mix of two groups: those who are cured and those who are not. Others use more intricate structures, such as imagining a hidden number of "risk factors" inside a person, where having zero factors means the person is cured. Still others build the cure directly into the shape of the survival curve itself, allowing the curve to flatten out at a level higher than zero without needing a separate group label.

Another critical axis of the review looks at how these models handle the timing of events and the relationships between them. Do the different risks act independently, or does the presence of one risk change the likelihood of another? Most of the reviewed models assume that the different risks operate independently of each other, but a few attempt to model the complex ways they might be linked. The review also examines how these models deal with missing information, such as when a patient dies but the specific cause is unknown, or when the exact time of an event is only known to have happened within a certain window. The author finds that while some models have sophisticated ways to handle these messy real-world data issues, many others rely on simpler, less flexible assumptions.

Perhaps the most surprising discovery in the review concerns the tools available to use these models. The field is rich with theory but poor in practice. The author found that out of the twenty-six papers reviewed, only a handful provided any public code, and none offered a widely adopted, clearly licensed software package in common languages like R or Python. Most researchers are forced to write their own programs from scratch to fit these complex models. This lack of standardized software makes it difficult to compare results across different studies or to verify that the models are working correctly. The review includes a practical demonstration using public bone marrow transplant data to show how different models can produce different answers for the same group of patients, emphasizing that the results depend entirely on the specific assumptions built into the chosen method.

Ultimately, this work serves as a map for navigating a complicated field. It helps researchers understand that simply picking a model with a familiar name is not enough; they must look under the hood to see how that model defines cure, handles competing risks, and deals with missing data. By organizing the literature along these five clear lines of inquiry, the review provides a framework for making better choices in future research. It also issues a quiet but urgent call for the development of unified, open-source software tools. Until such tools exist, the powerful insights offered by these advanced models will remain locked behind custom code, limiting their ability to improve our understanding of long-term survival in medicine and beyond.

Technical Summary: Competing-Risk Cure Models: A Five-Axis Systematic Review of Methodological Literature

Problem Statement
Competing-risk cure models address time-to-event populations where individuals may not be susceptible to all event types or to a specific event of interest. The methodological literature in this domain has developed through largely separate model families, leading to fragmentation. A primary challenge is that similarly named models (e.g., "mixture cure models") often target different estimands, entail distinct cure mechanisms, assume different latent-risk and censoring structures, and offer varying regression interpretations. Furthermore, the simultaneous presence of competing risks and cure fractions creates fundamental modeling complexities regarding the relationship between cure status, event time, and event type. Existing literature lacks a unified conceptual framework to compare these methods, and reproducibility is uneven, with most implementations relying on custom code and few providing unified, clearly licensed software frameworks.

Methodology
This paper presents a systematic review of 26 methodological contributions, ranging from early mixture formulations to recent semiparametric, Bayesian, defective, latent-count, zero-inflated, clustered, interval-censored, and neural network-based models. The literature is organized along five complementary axes to distinguish assumptions often conflated in the field:

Definition and Scope of Cure: The review distinguishes between global cure (immunity to all competing event types) and cause-specific cure (immunity only to a primary event, with other risks remaining possible). It clarifies how non-curability of competing risks is treated within each definition.
Decomposition of the Joint Distribution and Cure Mechanism: The paper categorizes how the joint distribution $P(T, E, Z | X)$ $P (T, E, Z ∣ X)$ is factorized. Four principal strategies are identified:
- Incidence–Latency (Mixture) Decomposition: The dominant approach, separating a cured group (incidence) from a susceptible group (latency). Variations include baseline-category multinomial logit, binary logistic susceptibility, and zero-inflated mixture/CIF formulations.
- Vertical (Susceptibility) Factorization: A structural approach where susceptibility governs entry into the competing-risk hazard system, conditioning the event type on the time of failure.
- Latent Competing-Causes (Promotion-Time) Construction: Cure is encoded via a zero latent count of failure-initiating causes (e.g., Poisson, negative-binomial, or destructive processes).
- Defective (Improper) Survival Decomposition: Cure is induced by the limiting tail of an improper survival distribution without an explicit cure indicator or mixture weight.
Latency Structure: The review compares parametric (e.g., Weibull, Gompertz, inverse-Gaussian), piecewise-constant, Proportional Hazards (PH), Accelerated Failure Time (AFT), transformation, CIF-based, and nonparametric latency models.
Dependence and Observation-Process Assumptions: This axis separates dependence among latent failure times from failure–censoring dependence, within-cluster association, and masked causes. It highlights that many models specify the observed event distribution without modeling the joint distribution of counterfactual latent failure times.
Estimation Framework: The paper surveys estimation techniques including direct and partial likelihood maximization, Expectation-Maximization (EM) algorithms (including those with neural-network M-steps), estimating equations with Inverse Probability of Censoring Weighting (IPCW), Bayesian computation (MCMC, data augmentation), and copula-graphic estimation.

To illustrate the framework, the author provides a reproducible example using a defective-Gompertz competing-risk cure construction with public bone-marrow-transplant data, demonstrating how fitted tail probabilities must be interpreted according to specific model definitions rather than as a universal metric.

Key Contributions

Taxonomic Framework: The primary contribution is the organization of 26 diverse methodological papers into a five-axis system. This taxonomy clarifies distinctions between models that are often conflated, such as separating explicit cure indicators from defective-distribution tail masses, and distinguishing latent competing-risk dependence from failure–censoring dependence.
Clarification of Estimands: The review explicitly differentiates between global and cause-specific cure, and between mixture formulations, vertical factorizations, and latent-count constructions, showing how these choices affect the interpretation of the cure fraction, Cumulative Incidence Function (CIF), and regression parameters.
Reproducibility Assessment: The paper documents the current state of software availability, noting that while some authors provide code (e.g., in Python, R, or FORTRAN), no widely adopted, clearly licensed R or Python framework unifies the principal constructions. Most implementations rely on custom code, limiting benchmarking and verification.
Illustrative Application: A reproducible defective-Gompertz example is provided to demonstrate the application of the five-axis framework, emphasizing that tail probabilities are model-dependent.

Results

Dominance of Mixture Models: Mixture formulations remain the dominant approach in the literature. However, the review reveals that models with similar names can differ significantly in their underlying mechanisms and assumptions.
Modeling Gaps: Dependence among latent failure times is less frequently modeled explicitly compared to cure or latency structures. Failure–censoring dependence, within-cluster association, and masked causes receive targeted treatment only in smaller subsets of models.
Estimation Diversity: Estimation methods vary widely, from standard likelihood maximization and EM algorithms to advanced techniques like neural-network M-steps, Bayesian data augmentation, and copula-graphic estimation.
Software Limitations: The review finds that reproducibility is limited. Only a small number of papers report public code repositories, and none provide a unified, clearly licensed software package for competing-risk cure models.

Significance
The paper claims that by organizing the literature along these five axes, the proposed taxonomy helps researchers select and report models transparently, compare estimation approaches, and identify specific needs for further theory, software development, benchmarking, and reproducible applications. It aims to resolve confusion arising from fragmented literature where similarly named models may target different estimands. The review does not propose new statistical theory but rather synthesizes existing contributions to provide a common conceptual and mathematical framework for comparison. The author emphasizes that fitted tail probabilities should be interpreted strictly according to the specific model and endpoint definition used, rather than as a basis for all-method comparisons.

Technical Summary: Competing-Risk Cure Models: A Five-Axis Systematic Review of Methodological Literature

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