Why a Risk Ratio Needs an Event and a Horizon
With competing events, the probability of experiencing a particular event by a given time depends not only on its own event rate, but also on the possibility of experiencing another event first. A ratio of predicted risks therefore needs a clearly specified event, follow-up horizon, and pair of covariate profiles. It is a comparison of absolute event probabilities by that horizon—not a general, time-independent property of the profiles.
For example, a risk ratio for a disease-related event by 2 years compares the predicted cumulative incidences of that event by 2 years. It does not compare the instantaneous event rates among people who remain event-free at each time, and it does not describe the ratio of probabilities among only those who survive event-free to the horizon.
A ratio greater than 1 means the predicted cumulative incidence by \(\tau\) is higher for profile A than for profile B; a ratio below 1 means it is lower. A value of 1.5 means the predicted risk is 1.5 times as large, or 50% higher, not 1.5 percentage points higher. The absolute difference remains useful alongside the ratio because it conveys the difference in probability points.
These are model-based predictions for specified covariate profiles. A contrast between profiles is not automatically a causal effect: a causal interpretation requires appropriate design and causal assumptions in addition to a suitable prediction model. The calculations below concern predicted risks and their ratios.
How Competing Events Enter the Ratio
For an initial event-free state with event types \(1,\ldots,K\), let \(\lambda_j(u\mid x)\) denote the cause-specific hazard for event \(j\) at time \(u\), conditional on being event-free just before \(u\). Under continuous event times, the probability of remaining event-free through time \(u\) is $$ S(u\mid x)=\exp\left\{-\sum_{j=1}^{K}\int_0^u \lambda_j(v\mid x)\,dv\right\}. $$ The cumulative incidence for event \(k\) is the accumulated probability of experiencing that event while still event-free: $$ F_k(t\mid x)=\int_0^t S(u\mid x)\lambda_k(u\mid x)\,du. $$ Thus, competing events enter through \(S\), even when the target of the ratio is only one particular event. The earlier material on Aalen-Johansen absolute risk predictions and model-based absolute risk prediction provides ways to obtain \(F_k\); here the focus is on forming and interpreting ratios from those predictions.
When the risks are differentiable in time, write \(f_k(t\mid x)=S(t\mid x)\lambda_k(t\mid x)\), the rate at which cumulative incidence for event \(k\) accumulates. For profiles A and B with \(F_k(t\mid x_B)>0\), differentiating their ratio gives
The derivative is not generally zero: the ratio is locally constant only when the two profiles’ rates of accumulating target-event risk are proportional to their accumulated risks in the required way. Since \(f_k=S\lambda_k\), those rates depend on both the target-event hazard and the event-free probability, which is affected by all competing hazards. This gives a direct reason that the ratio can change with follow-up time.
Worked Risk-Ratio Calculations
Worked Example: Dividing Two Predicted Risks at a Fixed Horizon
Suppose a fitted model gives hypothetical predicted cumulative incidences of a specified event by 3 years of 0.18 for profile A and 0.12 for profile B. The profiles, event definition, prediction population, and horizon are held fixed.
The ratio is $$ RR_k(3;x_A,x_B)=\frac{F_k(3\mid x_A)}{F_k(3\mid x_B)} =\frac{0.18}{0.12}=1.50. $$
The predicted 3-year risk for profile A is 1.50 times the predicted risk for profile B, or 50% higher. The risk difference is \(0.18-0.12=0.06\), which is 6 percentage points. Both convey useful information: the ratio gives a relative comparison, while the difference gives the absolute separation in predicted probability. These model predictions are hypothetical, not observed study results.
The calculation requires a positive denominator, which holds because \(0.12>0\). No further sampling uncertainty is quantified by this arithmetic; an estimate of the ratio alone does not establish how precisely it is known.
Worked Example: A Ratio That Changes With the Horizon
Consider two hypothetical profiles and a target event \(k\). Suppose predictions from a model with constant cause-specific hazards, measured per year, are \(\lambda_k=0.08\) and competing-event hazard \(\lambda_c=0.12\) for profile A; for profile B they are \(\lambda_k=0.04\) and \(\lambda_c=0.02\). Assume these hazards are constant over the horizons considered, the event types are exhaustive for first events in this illustration, and the model predictions are valid over follow-up.
With two constant hazards, the target-event cumulative incidence is $$ F_k(t)=\frac{\lambda_k}{\lambda_k+\lambda_c} \left\{1-\exp[-(\lambda_k+\lambda_c)t]\right\}. $$ For profile A at 1 year, $$ F_k(1\mid x_A)=\frac{0.08}{0.20}\{1-\exp(-0.20)\} =0.4(0.1813)=0.07251. $$ For profile B at 1 year, $$ F_k(1\mid x_B)=\frac{0.04}{0.06}\{1-\exp(-0.06)\} =0.6667(0.05824)=0.03882. $$ Thus \(RR_k(1;x_A,x_B)=0.07251/0.03882=1.868\), approximately. The predicted 1-year risk is about 1.87 times as large for A as for B.
At 5 years, $$ F_k(5\mid x_A)=0.4\{1-\exp(-1)\}=0.2528,\qquad F_k(5\mid x_B)=0.6667\{1-\exp(-0.30)\}=0.1728, $$ so \(RR_k(5;x_A,x_B)=0.2528/0.1728=1.463\). The ratio has fallen from about 1.87 at 1 year to 1.46 at 5 years. Although the target-event hazard is twice as high for A, A also has a higher competing-event hazard. That competing hazard removes more individuals from the event-free state before they can experience the target event. The resulting cumulative-incidence ratio therefore need not equal the target cause-specific hazard ratio, which is \(0.08/0.04=2\) here.
The constant-hazard assumption makes this calculation possible in closed form; it should not be assumed merely because two risks are reported. In an application, the model and its time structure must be appropriate for the prediction interval.
Worked Example: Ratios Differ Across Event Types
Suppose predictions for two profiles are available at three horizons. All entries below are hypothetical cumulative incidences, not hazards. Each event-specific ratio uses the same horizon in numerator and denominator.
| Event | Horizon | Profile A risk | Profile B risk | Risk ratio A/B |
|---|---|---|---|---|
| Target event 1 | 1 year | 0.04 | 0.02 | 2.00 |
| Target event 1 | 3 years | 0.12 | 0.08 | 1.50 |
| Target event 1 | 5 years | 0.20 | 0.16 | 1.25 |
| Target event 2 | 1 year | 0.03 | 0.05 | 0.60 |
| Target event 2 | 3 years | 0.10 | 0.10 | 1.00 |
| Target event 2 | 5 years | 0.14 | 0.12 | 1.17 |
For event 1 at 3 years, the calculation is \(0.12/0.08=1.50\). For event 2 at 5 years, it is \(0.14/0.12=1.1667\), reported as 1.17. The event 1 ratio declines across these horizons, while the event 2 ratio moves from below 1 to above 1. A risk ratio is therefore specific to both the event and the horizon; it should not be summarized as a universal ordering of the profiles. The displayed event-specific incidences are nondecreasing over time, and their sums at each horizon are at most 1, as required for coherent first-event probabilities.
Assumptions, Diagnostics, and Alternatives
The ratio itself is simple arithmetic, but its interpretation relies on the predictions being meaningful for the intended comparison. At minimum, check that the numerator and denominator use the same event coding, time origin, horizon, and prediction definition; that both profiles are specified clearly; and that predictions are supported by the fitted model and observed follow-up. Model assumptions, censoring handling, and covariate definitions must be appropriate for the underlying predictions. If the horizon lies beyond the follow-up support or requires substantial extrapolation, the quotient can look precise while resting on weak information.
Check the denominator before interpreting a ratio. If it is zero, the ratio is undefined; if it is very small, modest changes in that risk can produce a large change in the ratio. Do not replace an undefined ratio with zero or describe an unstable quotient as a reliable large relative effect. Report the two predicted risks alongside the ratio so readers can see the scale of the probabilities.
A useful diagnostic is to calculate the ratio at several clinically relevant horizons and for each event of interest, while checking that the underlying predicted risks are plausible and coherent. Sharp or unexpected changes may reflect genuine differences in the accumulation of event-specific risk, but can also signal sparse data, weak follow-up support, a poorly fitting model, or sensitivity to model specification. The ratio alone cannot distinguish these explanations. This is a diagnostic prompt, not a substitute for checking the prediction model and its inputs.
The main alternatives answer different questions. A risk difference \(F_k(\tau\mid x_A)-F_k(\tau\mid x_B)\) reports the absolute difference in probability by the horizon. A cause-specific hazard ratio compares instantaneous rates among those still event-free; it does not directly give the ratio of cumulative incidences. A subdistribution-hazard comparison is also a different model-based contrast, not a cumulative-incidence risk ratio. Choose the measure that matches the scientific question, and avoid translating one measure into another without the assumptions and calculations that justify the translation.
A common mistake is to calculate the ratio at one horizon and then call it “the risk ratio” without naming the event or time. The correction is to report, for example, “the ratio of predicted cumulative incidences for event 1 by 3 years, profile A versus profile B,” and to provide the two risks. Another mistake is to interpret a ratio of 1.5 as an increase of 1.5 percentage points. It means 50% higher relative to the denominator; the percentage-point increase is found by subtracting the risks.
Name the event, horizon, profile ordering, and prediction basis.
Confirm they refer to the same event and horizon, are plausible, and that the denominator is positive.
Divide the numerator risk by the denominator risk, then report both risks and interpret the ratio in context.
Check Your Understanding
Answer each question, then compare with the solutions below.
- By 2 years, predicted risks for a specified event are 0.09 for profile A and 0.06 for profile B. What is the risk ratio, and what does it mean?
- Why can a ratio of cumulative incidences change with time even if the cause-specific hazard ratio for the target event is constant?
- At a horizon, the predicted denominator risk is zero. What is the risk ratio, and how should it be reported?
- Predicted risks for one event are 0.10 and 0.20 for profiles A and B. State the risk ratio A/B and the risk difference A minus B.
Solutions. (1) \(0.09/0.06=1.50\): the predicted 2-year risk for A is 1.5 times that for B, or 50% higher. (2) Cumulative incidence accumulates as \(S(t\mid x)\lambda_k(t\mid x)\); the event-free probabilities \(S\) depend on all event hazards and can evolve differently across profiles. (3) The ratio is undefined because division by zero is not defined. Report the two risks and do not assign a finite ratio. (4) The ratio is \(0.10/0.20=0.50\), and the difference is \(0.10-0.20=-0.10\), or 10 percentage points lower for A.