What a Cause-Specific Hazard Ratio Compares
A cause-specific Cox coefficient is easy to misread as a comparison of the probability of experiencing an event. It is not. It compares instantaneous event rates at a particular time among subjects who have not yet experienced any event. That conditioning set matters: a person who has experienced a competing event is no longer event-free and cannot subsequently experience the event of interest as a first event.
Let \(T\) denote the time to the first event, \(J\) its cause, and let \(J=k\) identify the event of interest. A subject is event-free just before time \(t\) if \(T\geq t\). The cause-specific hazard for cause \(k\), conditional on covariates \(X=x\), is the instantaneous rate of cause-\(k\) events among such subjects:
A hazard has units of inverse time, such as events per person-year, and is not itself a probability. The Cox model for cause \(k\) writes this hazard as
where \(\lambda_{k0}(t)\) is an unspecified baseline hazard and \(\beta_k\) is the coefficient vector for cause \(k\). For a covariate contrast from \(x\) to \(x'\), the model implies the cause-specific hazard ratio
The baseline hazard cancels, so this ratio does not depend on \(t\) under the proportional-hazards model. Its interpretation is nevertheless conditional: it compares cause-\(k\) hazards among subjects with the specified covariates who are still event-free at time \(t\). It does not compare the probability of ever experiencing cause \(k\), nor does it automatically describe a causal effect.
Reading the Coefficient on Its Covariate Scale
The size and direction of a coefficient only become meaningful after identifying how its covariate is coded. A positive coefficient gives a hazard ratio above one for an increase in that covariate; a negative coefficient gives a ratio below one. The comparison is always relative to a stated contrast. For a binary treatment coded 1 versus 0, the contrast is treated versus reference. For a continuous covariate, it is per one unit unless another change is specified. Rescaling a covariate changes the coefficient but not the fitted comparisons.
Worked Example: Translating a Continuous-Covariate Coefficient
Suppose a hypothetical cause-specific Cox model for first cardiovascular event has coefficient \(\hat{\beta}_k=0.04\) per one-year increase in age. Interpret the model comparison for people whose ages differ by three years, with other modeled covariates held fixed.
The three-year contrast has estimated log hazard ratio \(3(0.04)=0.12\). Therefore,
Thus, under the fitted model, the older person has an estimated cause-specific hazard about 1.1275 times as large, or about 12.75% higher, than the younger person’s hazard among people still free of every event at the same time. The comparison concerns a three-year age difference, not a three-year increase in the probability of an event. It is conditional on the other included covariates and on the proportional-hazards model being an adequate description.
For a binary covariate, the same rule applies, but the contrast is usually from the reference group to the group coded 1. A hazard ratio below one can be expressed as a percentage lower hazard by calculating \(100(1-HR)\%\). This wording should retain the conditional population and the cause being modeled; omitting either can make the statement sound like an unconditional risk comparison.
Worked Example: Interpreting a Treatment Hazard Ratio and Interval
In a hypothetical observational study, treatment is coded 1 and usual care 0. A cause-specific Cox model for first disease-related death estimates \(\hat{\beta}_k=\log(0.70)=-0.3567\), with standard error \(SE=0.15\). Assume the model is correctly specified for this interpretation and use a normal-approximation 95% Wald interval for the coefficient.
The estimated hazard ratio is \(\exp(-0.3567)=0.7000\). At a given time, among people still free of any event and with the same values of the other modeled covariates, the estimated disease-death cause-specific hazard for treatment is 0.70 times that for usual care. Equivalently, it is estimated to be 30% lower.
The coefficient interval is
Exponentiating both endpoints gives the hazard-ratio interval:
Under the model and the approximation used for this interval, the data are compatible with treatment-to-usual-care cause-specific hazard ratios from about 0.522 to 0.939. In percentage terms, that corresponds to hazards about 47.84% to 6.08% lower, respectively, among subjects still event-free. A 95% confidence interval is a procedure with 95% coverage in repeated samples under its assumptions; it is not a statement that the parameter has a 95% probability of lying in this particular interval. Because the example is observational, the estimate alone also does not establish that treatment caused a reduction.
Why a Hazard Ratio Is Not a Cumulative-Risk Comparison
The cause-\(k\) cumulative incidence by time \(t\) is \(F_k(t\mid x)=P(T\leq t,J=k\mid X=x)\): the probability that cause \(k\) is the first event by time \(t\). It depends not only on the cause-\(k\) hazard but also on the hazards of all competing causes, because those events remove people from the possibility of experiencing cause \(k\) first.
For continuous event times with cause-specific hazards \(\lambda_j(u\mid x)\), the probability of remaining event-free to time \(u\) is \(S(u\mid x)=\exp\{-\int_0^u\sum_j\lambda_j(v\mid x)\,dv\}\). Consequently, the cumulative incidence for cause \(k\) is
This expression shows why a cause-specific hazard ratio alone cannot specify a cumulative-risk ratio. Even if the cause-\(k\) hazard comparison is known, the event-free probability \(S(u\mid x)\) depends on every cause-specific hazard. This distinction does not require assuming independent latent failure times for different causes; it follows from the competing-event process described by the cause-specific hazards.
Worked Example: A Lower Cause-Specific Hazard but Higher Cumulative Incidence
Consider two hypothetical groups with constant cause-specific hazards per year. For the event of interest, group A has \(\lambda_1=0.040\), while group B has \(\lambda_1=0.032\). Their cause-2 competing-event hazards are \(\lambda_2=0.160\) and \(\lambda_2=0.005\), respectively. Assume these constant hazards adequately describe the hypothetical event process over five years.
The cause-1 hazard ratio for B versus A is \(0.032/0.040=0.80\). Thus B has a 20% lower instantaneous cause-1 hazard among subjects still event-free. To calculate cause-1 cumulative incidence under constant hazards, the total hazard is \(\lambda_1+\lambda_2\), so
For group A at five years, the cause-1 share of the total hazard is \(0.040/0.200=0.200\), and the total-hazard exponent is \(-(0.200)(5)=-1\). Therefore,
For group B, the corresponding quantities are \(0.032/0.037=0.8649\) and \(-(0.037)(5)=-0.185\). Thus,
Despite its lower cause-1 hazard, group B has the higher five-year cause-1 cumulative incidence: about 14.61% versus 12.64%. Its competing-event hazard is much lower, so a larger fraction remains available to experience cause 1. This example is a mathematical comparison, not evidence that changing a competing-event rate would cause a particular risk change in a real population.
Assumptions, Diagnostics, and Alternative Questions
A hazard-ratio interpretation is tied to the model and data structure. Event causes must be defined consistently, the event of interest must be coded correctly, and the covariate contrast must match its coding and units. The proportional-hazards form assumes that a covariate contrast has a constant cause-specific hazard ratio over follow-up. If that assumption is materially wrong, one number may conceal a time-varying association; time-specific estimates or a model that allows time-varying effects may be more appropriate. The coefficient is also conditional on the other modeled covariates, not automatically a marginal population comparison.
Before interpreting a reported result, check the event coding, reference category, unit of each covariate, direction of the contrast, and confidence interval. Also ask whether the estimate is precise enough for the substantive claim. Sparse events can produce unstable estimates and wide intervals. The Cox model additionally relies on suitable handling of censoring and on the chosen covariate form being adequate. An association estimated from an observational model is not automatically causal; confounding, selection, and other design issues remain relevant.
The question determines the appropriate summary. A cause-specific Cox model answers a conditional instantaneous-rate question. A cumulative-incidence analysis answers an absolute probability question by a specified time and incorporates the competing causes. Fine-Gray regression is another standard regression approach focused on subdistribution hazards; its coefficient is not interpreted as a cause-specific hazard ratio. These approaches target different quantities, so their numerical estimates need not agree and should not be treated as interchangeable.
| Quantity | Comparison being described | Key interpretation |
|---|---|---|
| Cause-specific hazard ratio | Cause-\(k\) instantaneous hazards among subjects still event-free | Conditional rate comparison at a time |
| Cumulative incidence | Probability of cause \(k\) being the first event by a stated time | Absolute risk that reflects all competing causes |
| Subdistribution hazard ratio | A regression comparison for the subdistribution hazard | A different model quantity, not the cause-specific hazard ratio |
Interpreting a Reported Result in Practice
A complete interpretation names the cause, covariate contrast, event-free conditioning set, and model scale. For example: “For a one-year increase in age, the estimated cause-specific hazard of first cardiovascular event is multiplied by \(\exp(\hat{\beta}_k)\), among individuals still free of every event at the same time and with other modeled covariates held fixed.” Add an interval and its units or reference contrast where available. Avoid replacing “hazard” with “risk,” and avoid causal language unless the study design and assumptions justify it.
Name the cause \(k\), the covariate values being compared, the reference group, and the covariate units.
For a contrast \(x'-x\), calculate \(\exp\{\hat{\beta}_k^\mathsf{T}(x'-x)\}\), rather than interpreting the raw coefficient as a percentage.
Interpret the ratio among subjects still event-free, subject to the model assumptions; use cumulative incidence for a probability-by-time question.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A continuous covariate has cause-specific Cox coefficient \(-0.10\) per unit. What is the hazard ratio for a three-unit increase, and how should it be described?
- A cause-specific hazard ratio is 1.25 for group B versus group A. Is it correct to say that B has a 25% higher cumulative risk by five years? Explain.
- Why can two groups with the same cause-1 hazard have different cause-1 cumulative incidences?
- A model reports a treatment hazard ratio of 0.60. State a correct conditional interpretation and name one reason not to call it a causal effect without further justification.
Solutions. (1) The ratio is \(\exp\{3(-0.10)\}=\exp(-0.30)=0.7408\), so the estimated cause-specific hazard is about 25.92% lower per three-unit contrast among subjects still event-free, holding other modeled covariates fixed. (2) No. It is a 25% higher cause-specific hazard, conditional on being event-free, not a five-year cumulative-risk ratio. The cumulative incidence also depends on competing-event hazards. (3) Their competing-cause hazards, and therefore the probabilities of remaining event-free over time, may differ; cumulative incidence integrates the cause-1 hazard weighted by that event-free probability. (4) Among subjects still free of every event, the treatment group has an estimated cause-specific hazard 0.60 times that of the reference group at a given time, conditional on modeled covariates and the model assumptions. In an observational study, confounding could explain some or all of the association, so the estimate alone does not establish causation.