A Proportional Hazards Model for the Subdistribution of a Competing Risk¶
Why this mattered¶
Before this paper, regression analysis for competing risks was usually framed through cause-specific hazards: useful for etiologic questions, but indirect when the scientific target was the absolute probability of a particular event in the presence of other possible events. Fine and Gray’s central shift was to make the cumulative incidence function itself regression-accessible. By defining a proportional hazards model for the subdistribution hazard, they gave analysts a way to estimate how covariates relate directly to marginal failure probabilities for a specific cause, rather than requiring a two-step reconstruction from multiple cause-specific hazard models.
That made a practical difference in clinical and health-policy settings where the question is not only “what changes the instantaneous rate of one event among those still event-free?” but “what changes the patient’s probability of experiencing this event over time?” In oncology, transplantation, cardiovascular outcomes, and cost-effectiveness analysis, competing events such as death can make ordinary survival interpretations misleading. The paper’s estimation, inference, and prediction machinery made it newly routine to report covariate effects and individualized cumulative incidence estimates on the probability scale that clinicians and decision analysts needed.
Its later influence came from turning competing-risks regression into a standard applied toolkit rather than a specialist reconstruction problem. The “Fine-Gray model” became the default comparator, and often the default method, for studies focused on absolute risk under competing events. Subsequent work on dynamic prediction, risk stratification, validation, causal interpretation, and software implementation repeatedly built around the distinction this paper made operational: cause-specific hazards answer one kind of question, while subdistribution hazards target the cumulative incidence that often drives prognosis and decisions.
Abstract¶
Abstract With explanatory covariates, the standard analysis for competing risks data involves modeling the cause-specific hazard functions via a proportional hazards assumption. Unfortunately, the cause-specific hazard function does not have a direct interpretation in terms of survival probabilities for the particular failure type. In recent years many clinicians have begun using the cumulative incidence function, the marginal failure probabilities for a particular cause, which is intuitively appealing and more easily explained to the nonstatistician. The cumulative incidence is especially relevant in cost-effectiveness analyses in which the survival probabilities are needed to determine treatment utility. Previously, authors have considered methods for combining estimates of the cause-specific hazard functions under the proportional hazards formulation. However, these methods do not allow the analyst to directly assess the effect of a covariate on the marginal probability function. In this article we propose a novel semiparametric proportional hazards model for the subdistribution. Using the partial likelihood principle and weighting techniques, we derive estimation and inference procedures for the finite-dimensional regression parameter under a variety of censoring scenarios. We give a uniformly consistent estimator for the predicted cumulative incidence for an individual with certain covariates; confidence intervals and bands can be obtained analytically or with an easy-to-implement simulation technique. To contrast the two approaches, we analyze a dataset from a breast cancer clinical trial under both models. Key Words: Hazard of subdistributionMartingalePartial likelihoodTransformation model
Related¶
- cite → An Analysis of Transformations — Fine and Gray use Box-Cox-style transformation ideas to model covariate effects flexibly in subdistribution hazards for competing risks.
- cite → Regression Models and Life-Tables — Fine and Gray extend Cox proportional hazards regression to the subdistribution hazard for competing-risk failure data.
- enables → A Randomized Trial of Intensive versus Standard Blood-Pressure Control — The Fine-Gray subdistribution hazards model links them through competing-risk analysis of cardiovascular outcomes in the SPRINT trial.
- cite ← A Randomized Trial of Intensive versus Standard Blood-Pressure Control — The SPRINT trial uses Fine and Gray's subdistribution proportional hazards model to analyze outcomes such as cardiovascular events in the presence of competing risks like noncardiovascular death.
- enables ← An Analysis of Transformations — Box-Cox transformation theory enabled Fine and Gray to model transformed cumulative incidence functions while preserving proportional regression structure for competing risks.
- enables ← Regression Models and Life-Tables — Cox's proportional hazards regression supplied the semiparametric hazard-model template that Fine and Gray adapted to the subdistribution hazard for competing risks.