Common Interpretation Errors for Competing-Risk Regression

Two Hazard Ratios, Two Different Interpretations

Competing-risk regression often produces a hazard ratio, but the word “hazard” does not make every hazard ratio a measure of the same quantity. A cause-specific Cox model compares instantaneous event hazards among people who have not yet experienced any event. A Fine-Gray model compares subdistribution hazards, defined through the cumulative incidence function and a different risk-set construction. Neither coefficient is, by itself, a ratio of cumulative risks at a chosen time, and a Fine-Gray subdistribution hazard ratio is not a biological event-rate ratio.

These distinctions matter whenever an analysis is reported in clinical terms. “The cause-specific hazard was twice as high” is not equivalent to “the probability of the event was twice as high.” Likewise, “the subdistribution hazard was 50% higher” does not mean that an individual’s biological rate of experiencing the event was 50% higher. The interpretation depends on the model’s risk set and the event probabilities being described.

Definition: Let \(T\) be the time to the first event, \(J\) its event type, \(k\) the event type of interest, and \(Z\) a covariate. The cause-specific hazard for event \(k\), \(\lambda_k(t\mid Z)\), is the instantaneous rate of event \(k\) at time \(t\), conditional on having experienced no event just before \(t\) and given \(Z\). The cumulative incidence function is \(F_k(t\mid Z)=P(T\le t,J=k\mid Z)\), the probability of experiencing event \(k\) by time \(t\) before any competing event.

A cause-specific Cox model commonly specifies \(\lambda_k(t\mid Z)=\lambda_{k0}(t)\exp(\beta_k Z)\). Under this proportional-hazards model, \(\exp(\beta_k)\) compares the cause-specific hazards for a one-unit covariate difference, holding included covariates fixed. Its risk set at each time consists of individuals who remain free of every event. A competing event removes a person from this risk set, because that person can no longer experience the event of interest as the first event.

The Fine-Gray model instead specifies a proportional relationship for the subdistribution hazard. For a continuous event-time distribution, define that hazard as

\widetilde{\lambda}_k(t\mid Z) = \frac{f_k(t\mid Z)}{1-F_k(t\mid Z)},

where \(f_k(t\mid Z)\) is the event-specific density, the derivative of \(F_k(t\mid Z)\) with respect to time. The Fine-Gray risk-set construction retains people who have already experienced a competing event, with appropriate handling of censoring. Thus its denominator \(1-F_k(t\mid Z)\) is not simply the proportion still free of all events. A subdistribution hazard is tied to the evolution of the cumulative incidence; it is not the ordinary instantaneous event rate among people who remain biologically capable of experiencing a first event.

Why a Cause-Specific Hazard Ratio Is Not a Risk Ratio

The cumulative incidence of event \(k\) depends on both its own cause-specific hazard and the hazards of competing events. With continuous event times and causes \(1,\ldots,K\), define the probability of remaining free of every event just before \(u\) as \(S(u-\mid Z)\). Then

F_k(t\mid Z) = \int_0^t S(u-\mid Z)\lambda_k(u\mid Z)\,du, \qquad S(u\mid Z) = \exp\left\{ -\int_0^u \sum_{j=1}^{K}\lambda_j(v\mid Z)\,dv \right\}.

The integrand combines the target-event hazard with the probability of still being event-free and able to experience that event. Changing a competing-event hazard changes \(S\), and therefore can change \(F_k\), even when \(\lambda_k\) itself does not change. A cause-specific hazard ratio consequently does not determine a cumulative-risk ratio without information about the competing hazards, time, and the covariate patterns being compared.

Justification: Over a short interval of length \(du\), the probability of event \(k\) is approximately \(S(u-\mid Z)\lambda_k(u\mid Z)du\): a person must still be event-free and then experience event \(k\) in that interval. Adding these contributions over time gives the integral for \(F_k\). Since \(S\) depends on the hazards of all event types, the cumulative incidence depends on competing events as well as on the target cause-specific hazard.

Worked Example: Same Cause-Specific Hazard Ratio, Different Risk Ratio

Consider two hypothetical groups followed for 3 years, with constant cause-specific hazards per year. In Group A, the target-event hazard is \(0.08\) and the competing-event hazard is \(0.06\). In Group B, the corresponding hazards are \(0.04\) and \(0.20\). Assume there are no other event types and that the constant-hazard model is suitable for this illustration.

The cause-specific hazard ratio for the target event, A relative to B, is \(0.08/0.04=2.00\). With constant hazards, total event-free survival is \(\exp\{-(\lambda_1+\lambda_2)t\}\), and the target-event cumulative incidence is

F_1(t) = \frac{\lambda_1}{\lambda_1+\lambda_2} \left[1-\exp\{-(\lambda_1+\lambda_2)t\}\right].

For Group A, the total hazard is \(0.08+0.06=0.14\) per year. Substitution gives

F_{1,A}(3) = \frac{0.08}{0.14}\left[1-\exp\{-(0.14)(3)\}\right] = 0.5714(1-e^{-0.42}) \approx 0.1960.

For Group B, the total hazard is \(0.04+0.20=0.24\) per year:

F_{1,B}(3) = \frac{0.04}{0.24}\left[1-\exp\{-(0.24)(3)\}\right] = 0.1667(1-e^{-0.72}) \approx 0.08554.

The 3-year cumulative-risk ratio is therefore \(0.1960/0.08554\approx 2.291\), not 2.00. The difference arises in part because Group B has a larger competing-event hazard, which reduces the probability of remaining event-free long enough to experience the target event. The calculations illustrate why a cause-specific hazard ratio is not itself a cumulative-risk ratio.

Why a Fine-Gray Ratio Is Not a Biological Event-Rate Ratio

The subdistribution hazard is constructed from the event-specific density divided by \(1-F_k(t\mid Z)\). Its denominator includes the probability that the event of interest has not yet occurred, whether or not a competing event has already occurred. The Fine-Gray method represents this feature through a modified risk set that retains people after competing events; censoring must be handled appropriately, often using inverse-probability censoring weights. This risk-set construction is useful for modeling cumulative incidence, but it is not a cohort of people all still at risk in the ordinary biological sense.

Under a proportional subdistribution-hazards model, \(\widetilde{\lambda}_k(t\mid Z)=\widetilde{\lambda}_{k0}(t)\exp(\gamma_k Z)\), the ratio \(\exp(\gamma_k)\) compares subdistribution hazards for a one-unit difference in \(Z\), conditional on modeled covariates. Because the subdistribution hazard is defined through \(F_k\), the coefficient is related to cumulative incidence. But its ratio is still not generally a ratio of cumulative incidences at a fixed time: converting a hazard to a cumulative probability involves its evolution over time, and probabilities do not increase linearly with hazards.

Worked Example: A Subdistribution Hazard Ratio and Five-Year Risk

Suppose, hypothetically, a Fine-Gray model for a particular event has a constant baseline subdistribution hazard of \(0.05\) per year. For a covariate value of 1, suppose the fitted subdistribution hazard ratio is \(1.50\), so the corresponding constant subdistribution hazard is \(0.05(1.50)=0.075\) per year. This constant-hazard setup is a simplifying assumption for the calculation, not a claim that all real subdistribution hazards are constant.

For a constant subdistribution hazard \(\widetilde{\lambda}\), the cumulative incidence satisfies \(dF_k(t)/dt=\widetilde{\lambda}\{1-F_k(t)\}\), with \(F_k(0)=0\). Solving gives \(F_k(t)=1-\exp(-\widetilde{\lambda}t)\). At 5 years, the risks are

F_{k,0}(5)=1-e^{-(0.05)(5)}=1-e^{-0.25}\approx 0.2212,

F_{k,1}(5)=1-e^{-(0.075)(5)}=1-e^{-0.375}\approx 0.3127.

The five-year cumulative-risk ratio is \(0.3127/0.2212\approx 1.414\), not \(1.50\). The model’s 1.50 is a ratio of subdistribution hazards under the stated model, not a 50% increase in five-year risk and not a 50% increase in a biological rate among people still event-free.

Reading Regression Output Precisely

Worked Example: Reporting a Fine-Gray Coefficient

Imagine a hypothetical Fine-Gray analysis of time to a particular complication, with death before the complication as a competing event. The fitted covariate coefficient is \(\hat{\gamma}=0.4055\), with standard error \(0.159\). The estimated subdistribution hazard ratio is \(\exp(0.4055)\approx 1.50\). A large-sample 95% confidence interval on the coefficient scale is \(\hat{\gamma}\pm1.96SE\):

0.4055\pm1.96(0.159) = 0.4055\pm0.3116 = (0.0939,\ 0.7171).

Exponentiating the endpoints gives \((\exp(0.0939),\exp(0.7171))\approx(1.10,2.05)\). A suitable report is: “Conditional on the other modeled covariates, the estimated subdistribution hazard for the complication was 1.50 times as high for a one-unit increase in this covariate; the 95% confidence interval was 1.10 to 2.05.” This describes the fitted Fine-Gray association. It does not say that the complication’s biological event rate was 50% higher, nor does it give a 50% higher cumulative risk.

This interval describes the uncertainty of the estimated model parameter under the model and its sampling assumptions. It is not a 95% probability statement about the fixed parameter. Its interpretation also depends on adequate model specification, including a suitable proportional subdistribution-hazards assumption and appropriate handling of censoring.

Assumptions, Diagnostics, and Alternatives

Interpretation starts with a correctly identified event type and a model appropriate to the question. Cause-specific Cox regression requires appropriate event coding, correct specification of the cause-specific hazard, and a defensible censoring assumption conditional on modeled information. Fine-Gray regression requires correct event coding and appropriate handling of censoring for its modified risk set; inverse-probability censoring weights may be used when needed. For either proportional model, the corresponding hazard ratio is assumed constant over time unless time-varying effects are modeled. Covariate functional forms and relevant interactions also need adequate specification.

Diagnostics should ask whether the model’s assumptions and data handling are credible, rather than whether a hazard ratio can be relabeled as a risk ratio. Check coding of the target and competing events, numbers at risk and event counts over time, censoring patterns, proportionality, and the sensitivity of Fine-Gray results to the censoring-weight approach where applicable. Sparse events, influential observations, extreme weights, or marked time variation can make estimates unstable or the single-number summary misleading. A model can be computationally successful and still have an interpretation that does not match the scientific question.

The cause-specific Cox model is often useful for describing event rates among people still free of all events. Fine-Gray regression directly models a subdistribution hazard linked to the target event’s cumulative incidence. These are different summaries, and the methods need not yield matching-looking coefficients or answer the same question. If the goal is to report absolute probability by a specified time, report a cumulative incidence estimate or model-based cumulative incidence directly, with uncertainty and covariate values made clear. Do not substitute either hazard ratio for that probability.

Common Interpretation Errors

Wrong: “The cause-specific hazard ratio is 2, so the event risk by 3 years is twice as large.” Correction: The ratio compares instantaneous cause-specific hazards among those still event-free. The cumulative-risk ratio also depends on competing events and time.
Wrong: “The Fine-Gray ratio is 1.5, so the biological event rate is 50% higher.” Correction: It compares subdistribution hazards in the Fine-Gray model’s modified risk set. It is not the ordinary event rate among people who remain free of all events.
Wrong: “A proportional hazard ratio of 1.5 means the cumulative risk is 50% higher at every follow-up time.” Correction: Even under a constant proportional subdistribution hazard, the corresponding risk ratio can vary with time.
Wrong: “A statistically precise hazard ratio proves a causal effect.” Correction: A regression coefficient describes an association under its model. Causal interpretation requires additional design and assumptions; it does not follow from the hazard ratio alone.

Key takeaway: Always name the quantity in the model before interpreting its ratio. A cause-specific hazard ratio compares hazards among those still event-free; a Fine-Gray subdistribution hazard ratio compares subdistribution hazards linked to cumulative incidence. Neither is automatically a cumulative-risk ratio, and the Fine-Gray ratio is not a biological event-rate ratio.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A cause-specific hazard ratio for an event is 0.70. What does this ratio compare, and what does it not directly tell you?
In a constant-hazard setting, the target-event hazard is \(0.03\) per year and the competing-event hazard is \(0.07\) per year. Calculate the target-event cumulative incidence by 4 years.
A Fine-Gray model reports a subdistribution hazard ratio of 1.40. Is it correct to report a 40% increase in cumulative risk at every time point? Explain.
Why can the cumulative incidence change when the cause-specific hazard for the target event stays fixed?

Solutions. (1) It compares the instantaneous target-event hazards for the modeled covariate contrast among people still free of every event, conditional on modeled covariates. It does not directly give a cumulative-risk ratio. (2) The total hazard is \(0.03+0.07=0.10\) per year. Thus \(F_1(4)=(0.03/0.10)\{1-e^{-(0.10)(4)}\}=0.3(1-e^{-0.4})\approx0.09890\), or about 9.890% by 4 years. (3) No. The ratio compares subdistribution hazards under the model. The cumulative-risk ratio is a function of follow-up time and is not generally 1.40. (4) The target-event cumulative incidence integrates the target hazard multiplied by the probability of remaining free of all events. A change in a competing-event hazard changes that event-free probability and can therefore change the target-event cumulative incidence.