Competing Risks Analysis

Competing risks analysis is a specialized framework in survival analysis, clinical epidemiology, and demography designed to evaluate time-to-event data when individuals are exposed to two or more mutually exclusive failure types, and the occurrence of any one event precludes (or alters the probability of) the primary event of interest. For example, in an oncology cohort tracking time to local cancer recurrence, death from cardiovascular disease acts as a competing risk because a deceased patient cannot subsequently experience cancer recurrence. Treating competing events as standard independent right-censoring in a Kaplan–Meier estimator violates the core assumption that censored individuals retain the same future event hazard as those remaining in the risk set, producing upward bias in estimated cumulative incidence.

Observable Quantities and Non-Identifiability

In a competing risks setting with K distinct failure causes, each subject experiences an event at failure time T > 0 coupled with a discrete cause indicator D ∈ {1, 2, ..., K} identifying which specific event occurred first. Early actuarial formulations posited a vector of latent failure times (T₁, T₂, ..., T_K), where only T = min(T₁, ..., T_K) is observed. However, as David R. Cox and later Anastasios Tsiatis (1975) proved, the joint distribution of latent failure times is mathematically non-identifiable from observed data without untestable independence assumptions: for any dependent joint distribution of (T₁, ..., T_K), there exists a marginal independent model that generates the same observable distribution of (T, D).

Modern competing risks methodology therefore avoids latent failure times and relies directly on two observable quantities: the cause-specific hazard functions h_k(t) and the cumulative incidence functions CIF_k(t).

Cause-Specific Hazard Function

The cause-specific hazard function h_k(t) measures the instantaneous failure rate from cause k at time t among individuals who remain alive and free of any competing event up to time t:

h_k(t) = lim_{Δt → 0} [P(t ≤ T < t + Δt, D = k | T ≥ t) / Δt]

Because the K causes are mutually exclusive, the overall all-cause hazard rate h(t) of experiencing any first event is the direct sum of the individual cause-specific hazards: h(t) = ∑_k=1^K h_k(t). Consequently, the overall event-free survival probability S(t) = P(T > t) depends on all K cause-specific hazards simultaneously:

S(t) = exp(−∫₀^t ∑_k=1^K h_k(u) du)

Cumulative Incidence Function (Aalen–Johansen Estimator)

The Cumulative Incidence Function (also termed the subdistribution function), CIF_k(t) = F_k(t) = P(T ≤ t, D = k), quantifies the absolute probability that an individual experiences an event of type k on or before time t in the physical presence of all other competing risks:

CIF_k(t) = ∫₀^t S(u⁻) · h_k(u) du

In discrete empirical data with ordered event times t₁ < t₂ < ··· < t_m, where d_ki events of cause k occur among n_i individuals at risk at time t_i, the non-parametric Aalen–Johansen estimator combines the overall Kaplan–Meier estimator of all-cause event-free survival Ŝ(t_i−1) with the empirical cause-specific hazard increment (the jump of the cause-specific Nelson–Aalen estimator):

CIF̂_k(t) = ∑_{ti ≤ t} Ŝ(t_i−1) · (d_ki / n_i)

Unlike naive 1 − KM estimates—whose sum across all causes can exceed 100 percent—the sum of the Aalen–Johansen cumulative incidence estimators across all K causes plus the overall event-free survival probability always satisfies the conservation law: ∑_k=1^K CIF̂_k(t) + Ŝ(t) = 1.

Cause-Specific Proportional Hazards Regression

In a cause-specific Cox proportional hazards model, regression parameters β_k are estimated by fitting a standard Cox model for cause k while treating all individuals who experience a competing event (D ≠ k) as right-censored at the moment their competing event occurs:

h_k(t | X) = h_0k(t) · exp(X^Tβ_k)

While this approach yields valid inference on the instantaneous rate of cause k among those still event-free, the covariate effect on h_k(t) does not translate one-to-one into an effect on the cumulative probability CIF_k(t). If a covariate (such as advanced age or heavy smoking) leaves the cause-specific hazard of a benign condition unchanged (HR₁ = 1.0) but elevates competing mortality (HR₂ = 3.0), fewer older smokers survive long enough to develop the benign condition, causing their cumulative incidence CIF₁(t) to decrease despite the null cause-specific hazard ratio.

Fine–Gray Subdistribution Hazard Model

To establish a direct monotonic link between regression coefficients and the cumulative incidence function CIF_k(t), Jason P. Fine and Robert J. Gray (1999) introduced the subdistribution hazard function λ̃_k(t):

λ̃_k(t) = − [d ln(1 − CIF_k(t)) / dt] = [d CIF_k(t) / dt] / [1 − CIF_k(t)]

In the Fine–Gray model, λ̃_k(t | X) = λ̃_0k(t) exp(X^Tγ_k), individuals who experience a competing event (D ≠ k) are not removed from the risk set; instead, they remain artificially in the risk set indefinitely (weighted by inverse probability of censoring weights when administrative censoring is present) because they can never experience event k. Consequently, a subdistribution hazard ratio exp(γ_k) > 1 guarantees that the cumulative incidence CIF_k(t) is higher across follow-up. Biostatistical guidelines recommend reporting both cause-specific hazard ratios (for biological etiology) and Fine–Gray subdistribution hazard ratios (for clinical prognosis) side by side.

Why does 1 minus the Kaplan–Meier estimate overestimate risk when competing risks exist?

When a competing event such as death is treated as standard right-censoring in a Kaplan–Meier calculation, the algorithm redistributes that patient's future probability of experiencing the primary outcome across the surviving risk set. Because a deceased patient actually has a zero percent chance of experiencing the primary event in the future, treating them as merely lost to follow-up inflates the estimated cumulative probability.

When should researchers use a cause-specific Cox model versus a Fine–Gray model?

Epidemiologists use cause-specific Cox models when investigating biological etiology or treatment mechanisms—answering whether an exposure changes the instantaneous physiological rate of disease among living patients. Researchers use the Fine–Gray subdistribution hazard model when building clinical risk calculators or planning public health capacity—answering which patients have the highest net cumulative probability of experiencing the outcome over a 5- or 10-year window.

How does Gray's test differ from the standard log-rank test?

The standard cause-specific log-rank test compares cause-specific hazard rates h_k(t) across groups by censoring competing events, whereas Robert Gray's (1988) test directly compares the subdistribution hazards and cumulative incidence functions CIF_k(t) across groups. Two groups can have identical cause-specific log-rank test results yet significantly different Gray's test results if their competing mortality rates differ.

1. Fine, J. P., & Gray, R. J. (1999). A proportional hazards model for the subdistribution of a competing risk. Journal of the American Statistical Association, 94(446), 496–509. https://doi.org/10.1080/01621459.1999.10474144
2. Gray, R. J. (1988). A class of K-sample tests for comparing the cumulative incidence of a competing risk. The Annals of Statistics, 16(3), 1141–1154. https://doi.org/10.1214/aos/1176350951
3. Putter, H., Fiocco, M., & Geskus, R. B. (2007). Tutorial in biostatistics: Competing risks and multi-state models. Statistics in Medicine, 26(11), 2389–2430. https://doi.org/10.1002/sim.2712
4. Austin, P. C., Lee, D. S., & Fine, J. P. (2016). Introduction to the analysis of survival data in the presence of competing risks. Circulation, 133(6), 601–609. https://doi.org/10.1161/CIRCULATIONAHA.115.017719