Why Interpret Both Event Types?
In a competing-risks setting, a person can experience the event of interest or a competing event, but not both as the first event. A treatment may change the risk of either event, and these changes jointly determine how often the primary event is observed by a given time. A lower primary-event risk is therefore clinically important, but by itself it does not show that treatment prevented the primary event: an increase in competing events could instead leave fewer people alive and event-free long enough to experience it.
The relevant interpretation is a joint one. At a prespecified time horizon, compare the treatment-specific risks of the primary event, the competing event or events, and remaining event-free. Whether a shift between these outcomes is beneficial depends on the clinical consequences of each outcome, not just on which risk is lower.
The causal risk difference for event type \(k\) is \(\Delta_k(t)=F_k^1(t)-F_k^0(t)\). A negative value means a lower risk of that event under treatment by time \(t\); it is an absolute difference in probability, often reported in percentage points. The difference in event-free risk is \(\Delta_S(t)=S^1(t)-S^0(t)\). Because the outcomes at time \(t\) partition the population, these contrasts obey \(\Delta_1(t)+\Delta_2(t)+\Delta_S(t)=0\). A risk ratio can supplement a difference, but does not replace the joint absolute-risk interpretation.
What the Joint Contrasts Can Say
A treatment-specific cumulative incidence is a population risk under a specified treatment regime. Its causal interpretation requires that the regimes are well defined and that the risks are identified for the target population. In particular, the usual causal requirements include consistency, exchangeability for treatment assignment, positivity, and appropriate handling of censoring. The earlier material on exchangeability, positivity, consistency, censoring, and standardization sets out these conditions. If they are not credible, the contrasts may describe observed or adjusted associations rather than causal effects.
Even when causal identification is justified, the risk contrasts answer a limited but useful question: how does assignment to one treatment regime rather than another change the probability of each first-event outcome by time \(t\)? They do not, on their own, identify a biological mechanism. For example, a lower primary-event risk could arise alongside a higher risk of a harmful competing event. Conversely, a lower competing-event risk may leave more people at risk of the primary event, increasing its observed cumulative incidence even if treatment does not increase the primary event’s conditional rate among those still event-free.
The following accounting identity explains why the event risks must be read together. Divide follow-up into intervals \(1,\ldots,K\). Under regime \(a\), let \(q_{k,j}^a\) be the conditional probability of event type \(k\) in interval \(j\), given no event before that interval. With two event types, the probability of remaining event-free through interval \(j\), denoted \(S_j^a\), updates by removing both event types:
Each interval’s contribution to primary-event risk is its primary-event probability among those still event-free, multiplied by the probability of reaching that interval event-free. Thus, changing the competing-event probabilities can change the number who remain available to experience the primary event. The primary-event risk can change even when the conditional primary-event probabilities \(q_{1,j}^a\) do not.
Worked Example: Lower Primary Risk With More Competing Events
Suppose hypothetical standardized risks by 2 years are 0.20 for the primary event, 0.10 for the competing event, and 0.70 for remaining event-free under control. Under treatment, they are 0.14, 0.22, and 0.64. Each set sums to 1, as required.
The primary-event risk difference is \(0.14-0.20=-0.06\), or 6 fewer primary events per 100 people by 2 years. The competing-event risk difference is \(0.22-0.10=0.12\), or 12 more competing events per 100 people. The event-free risk difference is \(0.64-0.70=-0.06\), or 6 fewer people event-free per 100. The identity is satisfied: \(-0.06+0.12-0.06=0\).
For an illustrative cohort of 1,000 people in each regime, these risks correspond to 200 primary events, 100 competing events, and 700 event-free people under control, versus 140, 220, and 640 under treatment. The total risk of either event is \(0.20+0.10=0.30\) under control and \(0.14+0.22=0.36\) under treatment. Thus, fewer primary events coexist with more competing events and more events overall. This is a shift in the observed distribution of first events; the primary-risk reduction alone does not establish that treatment prevented the primary event.
Worked Example: A Favorable Joint Risk Pattern
Consider different hypothetical 2-year risks: under control, \(F_1^0=0.20\), \(F_2^0=0.10\), and \(S^0=0.70\); under treatment, \(F_1^1=0.12\), \(F_2^1=0.05\), and \(S^1=0.83\). Both sets again sum to 1.
The primary-event difference is \(0.12-0.20=-0.08\), an 8-percentage-point reduction. The competing-event difference is \(0.05-0.10=-0.05\), a 5-percentage-point reduction. Event-free risk increases by \(0.83-0.70=0.13\), or 13 percentage points. The total event risk falls from \(0.20+0.10=0.30\) to \(0.12+0.05=0.17\), a 13-percentage-point reduction.
This pattern is more reassuring than a lower primary risk accompanied by a higher competing risk: both types of event are less common by 2 years, and more people remain event-free. If both component outcomes are clinically undesirable, this is a favorable joint risk pattern. It still does not, by itself, prove a specific biological mechanism for the primary-event reduction.
Worked Example: Lower Competing Risk Can Raise Primary Risk
To isolate the effect of changing who remains at risk, consider a hypothetical two-interval model. In each interval, the listed probabilities are conditional on being event-free at its start. The primary-event probability is 0.10 in interval 1 and 0.20 in interval 2 under both regimes. The competing-event probability is 0.30 in interval 1 and 0.10 in interval 2 under control; under treatment it is 0.10 in each interval. The probabilities of no event in each interval are the remainders.
Under control, the probability of reaching interval 2 event-free is \(1-0.10-0.30=0.60\). Primary-event risk across both intervals is \(0.10+0.60(0.20)=0.22\). Competing-event risk is \(0.30+0.60(0.10)=0.36\), leaving \(1-0.22-0.36=0.42\) event-free.
Under treatment, the probability of reaching interval 2 event-free is \(1-0.10-0.10=0.80\). Primary-event risk is \(0.10+0.80(0.20)=0.26\). Competing-event risk is \(0.10+0.80(0.10)=0.18\), leaving \(1-0.26-0.18=0.56\) event-free. Although the conditional primary-event probabilities are unchanged in both intervals, the treatment primary-event risk is 0.04 higher because more people remain event-free to experience the primary event in interval 2. In this constructed example, reducing the competing risk is favorable if that event is harmful, even though primary-event cumulative incidence rises. A higher primary-event risk is not automatically evidence that treatment worsened the primary event’s underlying process.
Comparisons, Diagnostics, and Limits
A useful report places the two event-specific risks side by side with the event-free risk at the same time horizon. The comparison should state the target population, treatment regimes, time horizon, and the absolute risks as well as their differences. If a relative measure is also reported, define its numerator and denominator and keep the absolute differences visible; a relative measure alone can conceal the magnitude and direction of the other event’s change.
| Observed joint pattern | Careful interpretation | What it does not establish |
|---|---|---|
| Primary risk decreases; competing risk increases | The observed first-event distribution shifts toward the competing event; assess both events’ clinical consequences. | That treatment prevents the primary event or improves overall health. |
| Both event risks decrease; event-free risk increases | A favorable joint risk pattern if both events are undesirable. | A particular causal mechanism for either risk change. |
| Competing risk decreases; primary risk increases | More people may remain at risk long enough to experience the primary event. | That treatment increases the primary event’s conditional rate. |
Several checks help detect an incomplete or misleading interpretation. Confirm that risks for mutually exclusive first events and event-free status sum to 1, allowing for rounding. Check that both regimes use the same time horizon and target population. Examine whether estimates are supported by treatment overlap and adequate follow-up, and whether censoring was handled under defensible assumptions. Large changes in the competing-event risk, sparse data, or wide uncertainty around either risk difference should temper conclusions. For uncertainty about a joint pattern, separate confidence intervals do not by themselves provide a joint confidence statement; the covariance between estimated risks matters.
Cumulative incidence is not the same quantity as a cause-specific hazard, which describes an instantaneous event rate among people still event-free. Nor is the observed cumulative incidence under treatment the same as a hypothetical risk under elimination of the competing event. These are different estimands, answering different questions. A competing-event-free hypothetical analysis may help address a separate question, but it should not be substituted for the treatment-policy risk without stating the change in estimand and its assumptions.
The central limitation is interpretive: marginal risks describe what happens to the population’s distribution of first events under each regime, but do not reveal which individuals would have experienced which events under both regimes. In particular, they cannot alone separate direct prevention of the primary event from changes in competing events or in the people who remain at risk. Keep the conclusion at the level supported by the estimand, and explain the clinical importance of each component outcome.
Check Your Understanding
For each question, use the risk contrasts and event-free accounting to give a conclusion in context.
- At 1 year, control risks are 0.15 for the primary event, 0.05 for the competing event, and 0.80 event-free. Treatment risks are 0.10, 0.12, and 0.78. Calculate all three risk differences and describe the pattern.
- Why can a lower competing-event risk cause the primary-event cumulative incidence to increase even if conditional primary-event probabilities do not change?
- Under one regime, risks are 0.25 for the primary event and 0.15 for the competing event. What is event-free risk at that horizon, and what assumption about event types is needed?
- True or false: if treatment lowers the primary-event cumulative incidence, it has been shown to prevent the primary event’s underlying process. Explain.
Solutions. 1. The primary difference is \(0.10-0.15=-0.05\), the competing-event difference is \(0.12-0.05=0.07\), and the event-free difference is \(0.78-0.80=-0.02\). The changes sum to zero. Treatment is associated with 5 percentage points fewer primary events, 7 points more competing events, and 2 points fewer people event-free; the primary-risk reduction alone is not evidence of overall benefit or primary-event prevention. 2. A lower competing-event probability can leave more people event-free to enter later intervals. Their primary-event probabilities then contribute to cumulative incidence, so the total primary risk can rise even when conditional primary-event probabilities are unchanged. 3. Event-free risk is \(1-0.25-0.15=0.60\), or 60%, assuming the two listed event types are exhaustive and mutually exclusive first events by that horizon. 4. False. It shows a lower population risk of experiencing the primary event by the stated time under treatment, if causal identification assumptions hold. The change may also reflect competing-event shifts and does not alone identify a prevention mechanism.