What a Cause-Specific Risk Curve Shows
A cause-specific risk curve answers an absolute-risk question: for a defined prediction target, what is the probability of experiencing a particular event type by each time point? For example, a clinician may want to read the predicted probability of relapse by two years while also seeing the probability of death before relapse. The event types compete because the first event prevents a person from experiencing the other as their first event.
A plotted curve is not a record of the hazard at each time. It shows risk accumulated from the start of follow-up up to each time on the horizontal axis. Its interpretation therefore depends on the event type, time origin, time units, and target population or covariate profile used to make the prediction.
The probability in \(F_j(t)\) is an absolute probability for a particular event type, not the probability of that event in a hypothetical setting where competing events cannot happen. The event-free curve also has a specific meaning: it describes being free of every event included in the prediction, not merely being free of one selected cause.
How the Curves Accumulate
For a continuous-time competing-risks model, let \(\lambda_j(t\mid x)\) denote the cause-specific hazard for event type \(j\): the instantaneous event rate for type \(j\) at time \(t\), among people with profile \(x\) who remain event-free just before \(t\). Let \(\Lambda_j(t\mid x)\) be its cumulative cause-specific hazard. If the modeled event types exhaust the possible first events, the event-free survival and cause-specific cumulative incidences satisfy
The integral explains how to read the curves: at each time, the increase in event type \(j\)'s cumulative incidence depends both on its cause-specific hazard and on the proportion still event-free and therefore able to experience a first event. In a model with jumps in event probability, the corresponding increment is \(dF_j(t\mid x)=S(t-\mid x)d\Lambda_j(t\mid x)\), using event-free survival immediately before the jump.
A short derivation clarifies how the cause-specific curves relate to event-free survival. In continuous time, the total hazard of any first event is \(\sum_j\lambda_j(t\mid x)\). Consequently, the instantaneous decline in event-free survival is \(dS(t\mid x)=-S(t\mid x)\sum_j\lambda_j(t\mid x)\,dt\). The sum of the instantaneous increases in the cause-specific incidences is
Integrating from the start of follow-up, where \(S(0\mid x)=1\) and each \(F_j(0\mid x)=0\), gives \(\sum_jF_j(t\mid x)=1-S(t\mid x)\). Thus, at a given time, the probability of having had any modeled first event is the total accumulated incidence across event types. Small discrepancies in displayed curves or tables can arise from rounding.
Worked Examples
Worked Example: Reading Risk Accumulation From a Table
Suppose a model gives the following hypothetical predictions for a specified patient profile. Event A is relapse, event B is death before relapse, and follow-up time is measured in years.
| Time (years) | Event-free survival \(S(t\mid x)\) | Relapse incidence \(F_A(t\mid x)\) | Death incidence \(F_B(t\mid x)\) |
|---|---|---|---|
| 1 | 0.88 | 0.08 | 0.04 |
| 3 | 0.65 | 0.23 | 0.12 |
| 5 | 0.40 | 0.38 | 0.22 |
At one year, the model predicts an 8% probability of relapse, a 4% probability of death before relapse, and an 88% probability of remaining free of both events. The probabilities add to 1: \(0.08+0.04+0.88=1.00\).
From year 1 to year 3, the relapse incidence increases by \(0.23-0.08=0.15\), or 15 percentage points; the death incidence increases by \(0.12-0.04=0.08\), or 8 percentage points. The combined increase is \(0.15+0.08=0.23\). Over the same interval, event-free survival falls by \(0.88-0.65=0.23\). This agreement reflects the fact that people leaving the event-free state experience one of the modeled first events.
At year 5, the predicted risk of relapse is 38%, not 38% among the people who are still event-free at year 5. The latter group is represented by \(S(5\mid x)=0.40\); the cumulative incidence is measured against the original prediction target.
Worked Example: Constant Cause-Specific Hazards
Consider hypothetical predictions for one profile with constant cause-specific hazards of \(\lambda_A=0.10\) per year for event A and \(\lambda_B=0.05\) per year for event B. Assume these hazards apply throughout the five-year period, the only possible first events are A and B, and the model starts with everyone event-free. Calculate the five-year incidence for each event and event-free survival.
The total hazard is \(0.10+0.05=0.15\) per year, so the event-free survival at five years is
For event A, the cumulative incidence is the integral of event-free survival times the event-A hazard. With constant hazards, this evaluates to
Thus the predicted five-year incidence of event A is about 35.2%. For event B,
The event-B incidence is about 17.6%. Together, the event incidences sum to \(0.3517556+0.1758778\approx0.5276334\), which equals \(1-S(5)=1-0.4723666\), apart from rounding. These are cumulative probabilities from the start of follow-up. They are not the hazards multiplied by five: multiplying \(0.10\) per year by five years gives a cumulative hazard of 0.50, not a 50% event-A probability.
Worked Example: Comparing Cause Mix at a Fixed Time
Suppose hypothetical predictions for two specified profiles give the following values at year 4. Event A and event B are the only modeled first events.
| Profile | \(F_A(4\mid x)\) | \(F_B(4\mid x)\) | \(S(4\mid x)\) |
|---|---|---|---|
| Profile P | 0.32 | 0.18 | 0.50 |
| Profile Q | 0.20 | 0.30 | 0.50 |
For profile P, the predicted probability of any first event by year 4 is \(0.32+0.18=0.50\), while for profile Q it is \(0.20+0.30=0.50\). Both profiles therefore have a 50% predicted probability of remaining event-free at that time. The cause-specific risks nevertheless differ: profile P has a 12-percentage-point higher event-A incidence, while profile Q has a 12-percentage-point higher event-B incidence.
This comparison shows why event-free survival alone does not describe the composition of risk. It also does not establish that changing a person's profile would cause the incidence to change: these are model-based predictions for specified profiles, and causal interpretation requires additional design and assumptions.
Reading Curves Carefully
When reading a plotted prediction, first identify what the prediction conditions on. It may refer to a particular covariate profile or to a target population summarized by a standardization procedure. A curve for one profile should not be described as an average risk for a different population. Next, check the time origin, the units on the horizontal axis, the cause labels, and whether the vertical axis is displayed as a proportion or a percentage.
A cumulative-incidence curve should stay level or rise as time advances: once a first event of type \(j\) has occurred by a time, that event remains counted in \(F_j\) at later times. Event-free survival should stay level or fall. These are basic consequences of the definitions and useful visual checks when interpreting a graph. They do not, on their own, establish that the fitted model is well calibrated or that its predictions are reliable.
Keep three quantities distinct:
- Cumulative incidence: the probability of a specified first-event type by time \(t\), measured from the start of follow-up.
- Event-free survival: the probability of having experienced none of the modeled first events by time \(t\).
- Cause-specific hazard: the instantaneous rate of a particular event among those still event-free at that time.
Cause-specific hazard models combine hazards with the event-free probability to produce cumulative incidence. A Fine-Gray model instead targets the subdistribution hazard associated with cumulative incidence. Either modeling strategy may yield cause-specific risk predictions, but the hazard parameter itself should not be read as an absolute probability. In particular, \(1-\exp\{-\Lambda_j(t)\}\) based on only one cause-specific cumulative hazard generally does not equal the competing-risks cumulative incidence \(F_j(t)\), because it omits the effect of competing events on who remains event-free.
Common Mistakes and Limitations
- Calling a curve a hazard. A cumulative-incidence curve is an accumulated probability; its height is not an instantaneous rate. Read its increase over an interval as additional risk accumulated over that interval.
- Using the wrong denominator. \(F_j(t)\) is a probability among the original prediction target, not a probability among only those event-free at \(t\). The latter group is described by conditioning on remaining event-free.
- Ignoring competing events. A person who experiences a competing first event is no longer at risk of experiencing another first event. A cause-specific risk curve must reflect that removal from the event-free group.
- Confusing censoring with an event type. Censoring means the event status is not observed after a person's last follow-up; it is not itself a competing clinical event in the risk interpretation.
- Over-interpreting a prediction. Curves depend on the fitted model, its inputs, and the stated target. A predicted probability is not a guarantee for an individual and does not by itself demonstrate a causal effect.
Predictions also rely on the model being appropriate for the time period and covariate profiles where they are used. A curve may be mathematically interpretable while still being poorly supported by the observed data or inaccurate in a new population. Read the stated horizon and target alongside the plotted values, and avoid extending a prediction beyond the range it was intended to describe.
Check Your Understanding
Answer each question using cumulative incidence as a probability from the original prediction target.
- At year 2, a hypothetical prediction gives \(F_A(2)=0.14\), \(F_B(2)=0.06\), and \(S(2)=0.80\). Interpret all three values in context.
- Between years 2 and 4, \(F_A\) rises from 0.14 to 0.23 and \(F_B\) rises from 0.06 to 0.11. By how much did the predicted probability of any first event increase over this interval?
- With constant cause-specific hazards \(\lambda_A=0.08\) and \(\lambda_B=0.12\) per year, calculate the two-year event-free survival and cumulative incidence for each event type.
- Why is a cause-specific hazard of 0.10 per year not the same statement as a 10% one-year cumulative incidence?
- A model prediction is 25% for event A by year 3. Does this mean 25% of people who remain event-free at year 3 will experience event A? Explain.
Solutions. (1) By year 2, the predicted probabilities are 14% for event A, 6% for event B, and 80% of remaining free of either modeled event; the values sum to 1. (2) The increase is \((0.23-0.14)+(0.11-0.06)=0.09+0.05=0.14\), or 14 percentage points. (3) The total hazard is \(0.20\) per year, so \(S(2)=\exp(-0.20\times2)=\exp(-0.40)\approx0.6703\). The total event probability is \(1-0.6703=0.3297\). Therefore \(F_A(2)=\frac{0.08}{0.20}(0.3297)\approx0.1319\), and \(F_B(2)=\frac{0.12}{0.20}(0.3297)\approx0.1978\). (4) A hazard is an instantaneous rate among those still event-free, whereas cumulative incidence accumulates probability over time and accounts for competing events. (5) No. The 25% is the predicted probability, from the original target, of having experienced event A by year 3; those still event-free at year 3 are a different, smaller group.