Why Cause-Specific Hazard Ratios Are Not Risk Effects

Why a Hazard Ratio Is Not a Risk Effect

A cause-specific hazard describes the instantaneous rate of one event type among people who are still free of every event. A cumulative incidence, by contrast, is the probability of having that event by a specified time, before a competing event occurs. These quantities answer different questions. In particular, a treatment effect on the cause-specific hazard does not, by itself, determine the treatment effect on the probability of the event.

The distinction matters because competing events affect how many people remain at risk of the event of interest. Even if treatment doubles the cause-specific hazard for that event, the resulting change in its cumulative incidence depends on what treatment does to the competing-event hazards, as well as on the time horizon. A hazard ratio is therefore not generally a risk ratio, risk difference, or direct summary of the change in event probability.

Definition: Let $A=a$ denote a treatment strategy, $T^a$ the time to the first event under that strategy, and $J^a$ the cause of that event. The cause-specific hazard for cause $k$ at time $t$ is the instantaneous event rate for cause $k$, conditional on having experienced no event before $t$: $$ \lambda_k^a(t)=\lim_{\Delta t\to 0} \frac{P(t\leq T^a<t+\Delta t,\ J^a=k\mid T^a\geq t)}{\Delta t}. $$ The cumulative incidence for cause $k$ by time $t$ is $F_k^a(t)=P(T^a\leq t,\ J^a=k)$. A cause-specific hazard ratio at time $t$ is $\lambda_k^1(t)/\lambda_k^0(t)$, when the denominator is positive. A risk difference at time $t$ is $F_k^1(t)-F_k^0(t)$, and a risk ratio is $F_k^1(t)/F_k^0(t)$, when the denominator is positive.

How Competing Hazards Enter the Risk

Assume first that event times are continuous, the possible first-event causes are mutually exclusive, and all causes are represented in the model. Under treatment strategy $a$, let $S^a(t)=P(T^a\geq t)$ be the probability of remaining free of every event through time $t$. The cause-specific hazards determine this event-free probability and, in turn, the cumulative incidence:

S^a(t)=\exp\left\{-\int_0^t \sum_j \lambda_j^a(u)\,du\right\}, \qquad F_k^a(t)=\int_0^t S^a(u)\lambda_k^a(u)\,du.

The integrand for cause $k$ is the probability of still being event-free at time $u$, multiplied by the instantaneous rate of cause $k$ at that time. Thus, even when $\lambda_k^a(u)$ is known, $F_k^a(t)$ also depends on every competing cause through $S^a(u)$. A larger competing-event hazard can remove people from the event-free population sooner, leaving fewer opportunities for the event of interest to occur.

Key result: A cause-specific hazard ratio compares conditional event rates among the event-free populations under two treatment strategies. A cumulative-incidence contrast compares probabilities of the event by a specified time under those strategies. The hazard ratio alone does not generally determine the risk contrast: the competing hazards and the time horizon also matter.

Worked Calculations Under Constant Hazards

For constant cause-specific hazards, the formula simplifies. With two causes and constant hazards $\lambda_1^a$ and $\lambda_2^a$, the cumulative incidence for cause 1 is

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

This expression makes the two contributions visible: the total hazard determines how quickly events occur, while the share $\lambda_1^a/(\lambda_1^a+\lambda_2^a)$ determines which cause accounts for them.

Worked Example: A Hazard Ratio of Two, but a Different Risk Ratio

Consider a hypothetical randomized study with two possible first events and complete follow-up to 5 years. Suppose the constant hazards per year are $\lambda_1^0=0.10$ and $\lambda_2^0=0.20$ under control, and $\lambda_1^1=0.20$ and $\lambda_2^1=0.30$ under treatment. Assume the exponential competing-risks model is appropriate: hazards are constant within each arm, the two causes describe mutually exclusive first events, and there are no other event types.

The treatment-to-control cause-specific hazard ratio for cause 1 is $0.20/0.10=2.00$. For control, the total hazard is $0.30$ per year, so the 5-year cause-1 risk is

F_1^0(5)=\frac{0.10}{0.30}\left(1-e^{-0.30(5)}\right) =\frac{1}{3}(1-e^{-1.5}) =0.2589566.

For treatment, the total hazard is $0.50$ per year:

F_1^1(5)=\frac{0.20}{0.50}\left(1-e^{-0.50(5)}\right) =0.4(1-e^{-2.5}) =0.3671660.

Calculating contrasts from the unrounded risks gives a risk difference of $0.3671660-0.2589566=0.1082094$, or about 10.8 percentage points, and a risk ratio of $0.3671660/0.2589566=1.418$. Thus, the cause-1 hazard is twice as large under treatment, but the 5-year cause-1 risk is about 1.42 times as large, not twice as large. The hazards and risks here are causal treatment-specific quantities if randomization was implemented as intended, the treatment strategies are well-defined, and follow-up is complete or censoring is handled appropriately.

The Same Hazard Ratio Can Imply Different Risks

The cause-1 hazard ratio alone cannot recover the cause-1 cumulative incidence. The next comparison holds the control hazards and treatment cause-1 hazard fixed, but changes the treatment competing-event hazard. Each scenario has the same cause-1 hazard ratio of 2.00; their treatment risks differ because their competing hazards differ.

Worked Example: Changing the Competing Hazard Changes the Risk Effect

Use the control hazards from the previous example, $\lambda_1^0=0.10$ and $\lambda_2^0=0.20$ per year, and retain the treatment cause-1 hazard $\lambda_1^1=0.20$ per year. The control 5-year cause-1 risk is $0.2589566$, as calculated above. First suppose the treatment competing-event hazard is $\lambda_2^1=0.30$. The treatment cause-1 risk is $0.3671660$, giving a risk difference of $0.1082094$ and a risk ratio of $1.418$.

Now suppose instead that the treatment competing-event hazard is $\lambda_2^1=0.05$ per year. The cause-1 hazard ratio remains $0.20/0.10=2.00$, but the total treatment hazard is $0.25$ per year. Therefore,

F_1^1(5)=\frac{0.20}{0.25}\left(1-e^{-0.25(5)}\right) =0.8(1-e^{-1.25}) =0.5707962.

The 5-year risk difference is $0.5707962-0.2589566=0.3118396$, or about 31.2 percentage points, and the risk ratio is $0.5707962/0.2589566=2.204$. The cause-1 hazard ratio is identical in the two treatment scenarios, while their risk differences are about 10.8 and 31.2 percentage points. The lower competing-event hazard in the second scenario leaves more people event-free and able to experience cause 1.

A hazard ratio need not even agree with the direction of the risk contrast. A treatment can lower the cause-specific hazard of the event of interest but lower the competing-event hazard by more, increasing the cumulative incidence of the event of interest. The following hypothetical example makes this possibility concrete.

Worked Example: A Lower Cause-Specific Hazard but a Higher Risk

Suppose a randomized study has constant hazards per year. Under control, let $\lambda_1^0=0.10$ and $\lambda_2^0=0.40$. Under treatment, let $\lambda_1^1=0.07$ and $\lambda_2^1=0.02$. Consider the 3-year cause-1 risk. The cause-specific hazard ratio is $0.07/0.10=0.700$, indicating a 30% lower cause-1 hazard under treatment.

Under control, the total hazard is $0.50$ per year:

F_1^0(3)=\frac{0.10}{0.50}\left(1-e^{-0.50(3)}\right) =0.2(1-e^{-1.5}) =0.1553740.

Under treatment, the total hazard is $0.09$ per year:

F_1^1(3)=\frac{0.07}{0.09}\left(1-e^{-0.09(3)}\right) =\frac{7}{9}(1-e^{-0.27}) =0.1840382.

The treatment-minus-control risk difference is $0.1840382-0.1553740=0.0286642$, or about 2.87 percentage points, and the risk ratio is $0.1840382/0.1553740=1.184$. Thus, the cause-1 hazard is lower under treatment, yet the cause-1 risk by 3 years is higher. The substantially lower competing-event hazard allows more people to remain at risk for cause 1. Randomization, well-defined treatment strategies, and adequate handling of censoring support a causal interpretation of the arm-specific risks; the numerical hazard assumptions themselves are illustrative, not evidence about any real study.

Why the Hazard Contrast Is Conditional

At time $t$, the cause-specific hazard is conditional on being event-free up to $t$. The event-free people under treatment need not be comparable to the event-free people under control, even in a randomized study. Treatment may affect earlier events, so the composition of each arm’s surviving event-free group can diverge over time. The hazard ratio is consequently a contrast between rates in two treatment-specific risk sets; it is not automatically a comparison of the same people under two counterfactual treatments.

This does not make a cause-specific hazard ratio meaningless. It can describe a treatment-specific conditional rate contrast and may be useful for understanding event timing or modeling cause-specific hazards. The point is narrower: it is not, without additional information and a specified time horizon, a summary of the treatment effect on the probability of the event.

Justification: Over a short interval of length $du$, the probability of a cause-$k$ event is approximately $S^a(u)\lambda_k^a(u)\,du$. Adding these probabilities over time gives the cause-$k$ cumulative incidence: $$ F_k^a(t)=\int_0^t S^a(u)\lambda_k^a(u)\,du. $$ Since event-free survival satisfies $dS^a(u)/du=-S^a(u)\sum_j\lambda_j^a(u)$, solving this differential equation with $S^a(0)=1$ gives $S^a(t)=\exp\{-\int_0^t\sum_j\lambda_j^a(u)\,du\}$. Substitution shows directly that every cause-specific hazard contributes to the cause-$k$ risk through the event-free survival term.

Choosing and Interpreting the Contrast

The appropriate summary depends on the question. If the target is the probability of a particular event by a clinically meaningful time, report treatment-specific cumulative incidences and a risk difference or risk ratio at that time. If the target is a conditional rate among people still event-free, a cause-specific hazard contrast may be relevant. Neither summary replaces the other.

Quantity	Question answered	What it does not say by itself
Cause-specific hazard ratio	How do the instantaneous cause-specific rates compare among people still event-free under each strategy?	The difference or ratio of event probabilities by a specified time
Cumulative incidence $F_k^a(t)$	What is the probability of cause $k$ occurring by time $t$ under strategy $a$, before any competing event?	A conditional instantaneous rate contrast
Risk difference or risk ratio at $t$	How do the treatment-specific probabilities of cause $k$ by time $t$ compare?	A time-independent effect; the contrast can change with $t$
Subdistribution hazard model	How does a model-specific subdistribution hazard vary with covariates?	A risk ratio or risk difference without translating to cumulative incidence

In practice, estimate or model the cause-specific hazards for all relevant causes and combine them to obtain cumulative incidence, or estimate cumulative incidence directly with an appropriate method such as the Aalen–Johansen estimator. Regression coefficients from cause-specific Cox models are conditional hazard contrasts; they are not substitutes for treatment-specific cumulative-incidence estimates. Subdistribution hazard models use a different risk-set construction, but their coefficients are also not automatically risk ratios. The later material on causal interpretation of Fine-Gray coefficients addresses that distinct model.

Assumptions, Diagnostics, and Common Pitfalls

A causal interpretation requires more than calculating a hazard or risk. Treatment strategies must be sufficiently well-defined for consistency; treatment assignment must be randomized or adequately controlled for confounding; each relevant strategy must be possible in the covariate groups being compared (positivity); and censoring must be absent or appropriately handled under defensible assumptions. The constant-hazard formula additionally assumes hazards are constant over the stated interval and that the listed causes exhaust the first-event possibilities. It is a working model, not a general requirement for defining cumulative incidence.

Useful checks include plotting estimated cause-specific hazards over time to assess whether a constant-hazard assumption is plausible, examining whether proportional-hazards assumptions are reasonable when reporting a constant Cox hazard ratio, and checking that event causes and censoring are coded correctly. A nonproportional hazard may make a single hazard ratio an inadequate summary even of the conditional rates. Regardless of model fit, report the time horizon and the cumulative incidence when the substantive question concerns event probability.

Mistake: treating the cause-specific hazard ratio as a risk ratio. A hazard ratio of 2 does not generally mean twice the probability of the event by a chosen time. Calculate cumulative incidences under both strategies, then form the risk contrast.
Mistake: ignoring competing causes. The cause-$k$ risk depends on event-free survival, which depends on all cause-specific hazards. Include competing-event processes in the calculation.
Mistake: assuming a hazard ratio fixes the direction of the risk effect. A lower hazard for the event of interest can coexist with a higher risk if competing-event hazards also change. Compare the risks directly.
Mistake: calling a conditional hazard contrast an effect for the same people at each time. The event-free risk sets can differ by treatment. Describe the hazard ratio as a contrast between treatment-specific conditional rates, not automatically as an individual-level causal effect.
Mistake: rounding intermediate risks before calculating the contrast. Compute the risk difference or ratio using unrounded estimates, then round the final answer. This avoids shifting the reported final digit.

Key takeaway: The cause-specific hazard ratio is a conditional rate contrast; the cumulative incidence is a time-specific event probability. Because competing events shape event-free survival, a hazard ratio alone does not tell you the size—or always even the direction—of the risk effect.

Check Your Understanding

Answer each question, then compare with the solutions below.

In one sentence, distinguish a cause-specific hazard ratio from a cumulative-incidence risk ratio.
Suppose treatment doubles the cause-specific hazard for an event and also doubles the competing-event hazard. Is the treatment risk ratio for the event necessarily 2? Explain.
A treatment has cause-specific hazard ratio 0.8 for an event, but its cumulative incidence for that event is higher at 5 years. Is this mathematically possible? State the key mechanism.
What causal assumptions are needed before treatment-specific cumulative-incidence contrasts can be interpreted causally?

Solutions. (1) The hazard ratio compares instantaneous cause-specific rates among people still event-free under each strategy; the risk ratio compares probabilities of the event by a specified time. (2) No. The risk depends on both cause-specific hazards through event-free survival, so doubling both hazards does not generally double the event probability; the time horizon matters too. (3) Yes. If treatment lowers competing-event hazards sufficiently, more people may remain event-free long enough to experience the event of interest. (4) At minimum, treatment strategies must be well-defined, consistency must hold, treatment must be randomized or confounding adequately controlled, positivity must hold, and censoring must be absent or appropriately handled under defensible assumptions.