Two Hazard Ratios, Two Questions
A competing-risks analysis can produce more than one hazard ratio for the same covariate and event cause. That does not necessarily signal a contradiction or a failed analysis. A cause-specific hazard ratio describes the rate of a particular event among people who are still free of every event. A Fine–Gray subdistribution hazard ratio describes a different rate, constructed so that people who have already had a competing event remain represented in the target cause's risk set. Because the risk sets differ, the estimates answer different questions.
The reporting task is therefore not to choose whichever ratio appears more intuitive or more favorable. It is to name each estimand, describe its risk set and outcome, and interpret it on its own terms. Neither ratio is itself a probability of experiencing the event by a chosen time.
In standard notation, write the cause-specific hazard as \(\lambda_k(t\mid X)\) and the subdistribution hazard as \(\tilde{\lambda}_k(t\mid X)\). In a cause-specific Cox model, a coefficient \(\beta_k\) gives a hazard ratio \(\exp(\beta_k)\). In a proportional subdistribution hazards model, a coefficient \(\tilde{\beta}_k\) gives a subdistribution hazard ratio \(\exp(\tilde{\beta}_k)\). The labels matter: a “hazard ratio” without identifying which hazard was modeled is incomplete reporting in a competing-risks analysis.
Why the Risk Sets Lead to Different Interpretations
For the cause-specific hazard, individuals leave the risk set when any first event occurs. Thus, at time \(t\), the comparison is about the instantaneous rate of cause \(k\) among people who have not yet experienced cause \(k\) or any competing event. A covariate association with this hazard concerns the event rate in this event-free group. It is not automatically an effect on the cumulative incidence, because cumulative incidence also depends on the rates of competing causes.
For the subdistribution hazard, an individual who has a competing event is not treated as having experienced cause \(k\), but remains in the constructed risk set. This device makes the subdistribution hazard directly related to the cumulative incidence for cause \(k\). It does not mean that someone with a competing event is still biologically at risk of a first event of cause \(k\). The retained risk-set membership is a feature of the estimand and model, not a literal account of the person's subsequent clinical state.
Here \(S(t)=P(T\geq t)\) is the probability of having had no event by time \(t\), subject to the usual convention at event times. The first identity follows because an event of cause \(k\) at time \(t\) requires both remaining event-free up to \(t\), with probability \(S(t)\), and then experiencing cause \(k\) at its cause-specific rate. The second follows from the hazard definition for the constructed time: its survival function is \(1-F_k(t)\), since the constructed event has occurred by \(t\) exactly when cause \(k\) has occurred by \(t\). Therefore its hazard is its event density divided by its probability of not yet having that constructed event.
These identities show why the two hazard ratios need not agree. The cause-specific hazard is linked to the event-free probability and the rate of cause \(k\); the subdistribution hazard also reflects how the cumulative incidence has accumulated and how competing events shape the event-free probability. Even if a covariate multiplies the cause-specific hazard by a constant, the resulting subdistribution hazard ratio need not be constant over time.
Worked Example: Interpreting Two Reported Coefficients
Suppose a hypothetical adjusted analysis compares a treatment group with a reference group for a target event, with death before the target event as a competing event. Assume that a cause-specific Cox model estimates \(\hat{\beta}_k=0.405\), with standard error \(0.160\), and a proportional subdistribution hazards model estimates \(\hat{\tilde{\beta}}_k=0.182\), with standard error \(0.140\). Both models adjust for the same prespecified covariates.
Cause-specific estimate. Exponentiating the coefficient gives \(\exp(0.405)=1.50\). A Wald 95% confidence interval on the coefficient scale is \(0.405\pm1.96(0.160)=(0.0914,0.719)\). Exponentiating its limits gives an approximate 95% confidence interval of \((1.10,2.05)\) for the cause-specific hazard ratio. In context, conditional on the listed covariates and among individuals still free of any event, the treatment group has an estimated cause-specific rate of the target event 1.50 times the reference-group rate, under the proportional cause-specific hazards model.
Subdistribution estimate. Exponentiating \(0.182\) gives \(\exp(0.182)=1.20\). Its coefficient-scale interval is \(0.182\pm1.96(0.140)=(-0.0924,0.456)\), so the hazard-ratio interval is approximately \((0.912,1.58)\). The treatment group has an estimated subdistribution hazard 1.20 times the reference-group subdistribution hazard, conditional on the listed covariates and under the proportional subdistribution hazards model. This is an association with the constructed hazard linked to the target event's cumulative incidence; it is not a 20% increase in the probability of the target event.
The point estimates differ, as may their uncertainty intervals, because they concern different risk sets and estimands. The first interval excludes 1 while the second includes 1, but that does not establish that the two coefficients differ statistically: they are correlated estimates from the same data, and a direct contrast would require an appropriate variance calculation. Nor does the difference prove that the treatment changes the target event's absolute risk in a particular direction.
Competing Events Can Change the Pattern
A simple constant-hazard calculation illustrates why a cause-specific effect cannot be substituted for a subdistribution effect. Suppose, purely hypothetically, that two groups have constant cause-specific hazards over time: in group A, \(\lambda_1=0.10\) and \(\lambda_2=0.10\) per year; in group B, \(\lambda_1=0.20\) and \(\lambda_2=0.80\) per year. Cause 1 is the target and cause 2 is competing. The cause-specific hazard ratio for cause 1, B versus A, is \(0.20/0.10=2.00\). The hazard for the competing cause is also much higher in group B.
This formula follows by integrating the cause-1 event density \(S(u)\lambda_1\) from \(u=0\) to \(t\), where \(S(u)=\exp[-(\lambda_1+\lambda_2)u]\). At five years, group A's cumulative incidence is
For group B it is
Thus group B has twice the cause-specific hazard of cause 1, yet its five-year cumulative incidence of cause 1 is lower in this hypothetical example because the competing cause occurs so frequently. The cause-specific ratio does not tell us the direction or size of the difference in cumulative incidence by itself. Likewise, a subdistribution hazard ratio is not a five-year risk ratio: it is a hazard-scale summary whose relationship with cumulative incidence depends on its full time pattern.
Worked Example: Checking the Subdistribution Hazard at Five Years
Continue the hypothetical constant-hazard example. For constant cause-specific hazards, the cause-1 event density at time \(t\) is \(f_1(t)=\lambda_1\exp[-(\lambda_1+\lambda_2)t]\). The subdistribution hazard is \(f_1(t)/(1-F_1(t))\).
For group A at five years, \(f_{1,A}(5)=0.10\exp(-1)=0.0368\), while \(1-F_{1,A}(5)=1-0.3161=0.6839\). Therefore \(\tilde{\lambda}_{1,A}(5)=0.0368/0.6839=0.0538\) per year.
For group B, \(f_{1,B}(5)=0.20\exp(-5)=0.00135\), and \(1-F_{1,B}(5)=1-0.1987=0.8013\). Therefore \(\tilde{\lambda}_{1,B}(5)=0.00135/0.8013=0.00168\) per year. The instantaneous subdistribution hazard ratio at five years is approximately \(0.00168/0.0538=0.0312\), not 2.00.
This calculation is not an estimate from a proportional subdistribution hazards model; it is an instantaneous ratio in the specified data-generating example. Its dramatic difference from the cause-specific ratio illustrates that the subdistribution ratio can vary with time. A single proportional subdistribution hazard ratio would be a model-based summary, and its adequacy would need assessment rather than assumption by default.
Assumptions and Checks Before Reporting
Each model requires assumptions that should be stated or checked as appropriate to the analysis. For both approaches, causes must be defined as mutually exclusive types of first event, the time origin and event ascertainment must be clear, and censoring must be non-informative conditional on the modeled covariates for the usual estimation procedures. The covariate coding and adjustment set should be appropriate to the stated question. These conditions do not make an observational association causal; causal interpretation requires additional design and identification assumptions.
A cause-specific Cox model additionally relies on proportional cause-specific hazards for a single constant hazard ratio over follow-up, unless time-varying effects or another model are used. The proportional subdistribution hazards model similarly relies on proportionality of subdistribution hazards for a single constant subdistribution hazard ratio. These are separate proportionality conditions: adequate fit for one does not establish adequate fit for the other. Examine model diagnostics, such as residual-based checks and plots suited to the fitted model, and investigate whether effects vary over time. Also check coding of event type, censoring, and covariates, and assess whether sparse events or unstable fitting make the estimates unreliable.
For Fine–Gray estimation, competing-event individuals contribute to the constructed risk set, while right-censored individuals cannot simply be treated as observed indefinitely. Estimation procedures account for censoring, commonly through weighting; the censoring model and support over follow-up therefore matter. A sparse competing cause, heavy censoring, or very few people remaining under observation can make late-follow-up estimates unstable. Report that limitation rather than presenting a precise-looking ratio without context.
| Reported quantity | Risk-set question | Appropriate interpretation | Do not call it |
|---|---|---|---|
| Cause-specific hazard ratio | Among people still free of every event, how do the instantaneous cause-\(k\) rates compare? | Relative cause-specific event rate, conditional on remaining event-free and the modeled covariates | Risk ratio, cumulative-incidence ratio, or Fine–Gray ratio |
| Subdistribution hazard ratio | How do the cause-\(k\) subdistribution hazards compare, with competing-event cases retained in the constructed risk set? | Relative subdistribution hazard associated with the cumulative incidence of cause \(k\), under the fitted model | Cause-specific hazard ratio or direct percentage-point change in risk |
| Cumulative incidence | What is the probability of cause \(k\) having occurred by a specified time, with competing events present? | Absolute event probability by a stated horizon | Either hazard ratio |
Reporting Both Estimates Clearly
When both estimates are relevant, present them as complementary results rather than competing versions of one quantity. Name the event cause and reference group, provide the model type, point estimate, confidence interval, and (when used) p-value, and state the adjustment set. In prose, use the risk-set interpretation attached to each model. Keep absolute-risk statements for cumulative incidence estimates or other explicitly defined risk predictions.
Worked Example: Writing a Side-by-Side Results Statement
In a hypothetical cohort, treatment is compared with usual care for a target event, with death before that event treated as a competing event. Suppose adjusted cause-specific Cox regression gives a hazard ratio of 1.45 (95% confidence interval 1.10 to 1.91; \(p=0.008\)). An adjusted Fine–Gray model gives a subdistribution hazard ratio of 1.18 (95% confidence interval 0.91 to 1.53; \(p=0.21\)). Both models use the same adjustment covariates, and their proportionality and censoring diagnostics are considered acceptable for the intended summaries.
A suitable report is: “After adjustment for the prespecified covariates, treatment was associated with a higher cause-specific rate of the target event among participants who remained free of any event (cause-specific hazard ratio 1.45, 95% CI 1.10–1.91; \(p=0.008\)). The adjusted Fine–Gray subdistribution hazard ratio was 1.18 (95% CI 0.91–1.53; \(p=0.21\)), providing no clear evidence of a difference in the target event's subdistribution hazard under the fitted model.”
The wording names the distinct quantities and avoids claiming that treatment increased the event probability by 45% or 18%. It also avoids describing the second result as “adjusting away” death: death is a competing event incorporated through the subdistribution estimand, not an ordinary covariate adjustment. Since the two model estimates are dependent, their different p-values alone do not test whether their coefficients differ.
Common Mistakes and Practical Limits
- Reporting an unlabeled “hazard ratio.” State whether it is cause-specific or subdistribution. A reader cannot infer the risk set from the number alone.
- Translating a hazard ratio into a risk percentage. A ratio of 1.45 is not a 45-percentage-point increase, and it is not necessarily a 45% increase in cumulative incidence. Report risks by time horizon separately.
- Calling the subdistribution risk set literal. People with a competing event remain in a constructed risk set for estimation; they are not still event-free or clinically eligible for the first target event.
- Interpreting different p-values as evidence that estimates differ. A formal comparison needs the covariance between the estimates and a defined contrast; separate tests against a null ratio of 1 do not provide that comparison.
- Assuming constant effects without checking. If proportionality is doubtful, qualify a single ratio as an average/model-based summary, consider time-varying effects, or report another estimand suited to the scientific question.
The main limitation is that neither hazard ratio alone communicates absolute event burden. Cause-specific and subdistribution effects can be useful for different questions, but a clinical reader may also need the cumulative incidence at meaningful horizons. Report that quantity with its time point and uncertainty when it is part of the analysis; do not infer it directly from either ratio.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In one sentence, distinguish the risk sets used by cause-specific and subdistribution hazards.
- A cause-specific hazard ratio for a treatment is 0.80. Does this establish that the treatment reduces the target event's cumulative incidence by 20%? Explain.
- Why can an increased competing-event rate cause the target cause's cumulative incidence to fall even when its cause-specific hazard is unchanged?
- A fitted subdistribution hazard ratio is reported as 1.30. Give an appropriate interpretation and name one conclusion that cannot be drawn from it alone.
- Why is it invalid to infer that cause-specific and subdistribution effects differ just because one confidence interval excludes 1 and the other includes 1?
Solutions. (1) Cause-specific hazards compare target-event rates among people still free of every event; subdistribution hazards retain people with competing events in the constructed target-cause risk set. (2) No. It describes a relative cause-specific rate under the fitted model, not a 20% reduction in cumulative incidence; competing-event rates and follow-up time also affect cumulative incidence. (3) More people experience the competing event before they can experience the target event, reducing the probability of remaining event-free and available to experience the target cause. (4) Under the fitted proportional subdistribution hazards model, the group has an estimated subdistribution hazard 1.30 times that of the reference group, conditional on modeled covariates. This alone does not establish a 30% increase in absolute risk. (5) The estimates are dependent, and separate intervals or tests against 1 do not test their difference. A valid comparison requires an appropriate contrast and its variance.