Reporting Causal Contrasts With Competing Events

Why Precise Reporting Matters

A causal risk contrast is interpretable only when the intervention, outcome, and time horizon are clear. With competing events, that requirement is especially important: a person who experiences one first event can no longer experience a different first event. A lower risk of the event of interest may therefore accompany a higher risk of the competing event, without implying that the treatment has a uniformly beneficial effect.

A useful report states the causal question before presenting an estimate. It names the population, treatment strategies, event types, follow-up horizon, estimand, identification assumptions, and absolute risks. The introductory material on choosing causal estimands distinguishes the questions that different estimands answer; here the emphasis is on making the selected question auditable and its results interpretable.

Definition: Let \(A\) denote a baseline treatment strategy, with \(A=1\) and \(A=0\) denoting the strategies being compared. Let \(T^a\) be the potential time to first event under strategy \(a\), and let \(J^a\) be its event type. For event type \(k\), the cumulative incidence by time \(\tau\) under strategy \(a\) is \(F_k^a(\tau)=P(T^a\leq \tau,\ J^a=k)\). A risk difference is \(\Delta_k(\tau)=F_k^1(\tau)-F_k^0(\tau)\). A negative difference means lower risk of event type \(k\) under strategy 1 by \(\tau\).

The definition concerns the first event of each type by \(\tau\), not an unrestricted count of all later events. The strategy must also be specified: for example, assignment to treatment at baseline with subsequent care managed as usual is not the same intervention as sustained adherence to a specified regimen. In a treatment-policy estimand, outcomes are considered under the assigned strategy regardless of later treatment changes; competing events remain part of the natural course under that strategy.

What a Complete Report Specifies

A compact report can still be precise if it supplies the following information. The target population might be all eligible patients at treatment initiation, or a clearly defined subgroup. The strategies should identify what is initiated, when, and what happens after deviations or discontinuation. Define each event using the operational event criteria, state which events compete with the event of interest, and explain how simultaneous or uncertain event classifications are handled when relevant. Give a clinically meaningful horizon and its units, such as 365 days.

Then name the estimand and its scale. For example, report each strategy-specific cumulative incidence and the risk difference in percentage points or events per 1,000 people. A risk ratio may supplement the absolute contrast, but it does not replace it: the same relative effect can correspond to quite different absolute differences at different baseline risks. Give uncertainty intervals when available, and say how they were obtained. A confidence interval describes the uncertainty of the procedure under its assumptions; it is not a probability statement about whether the fixed parameter lies inside this particular interval.

The causal interpretation also depends on identification. In an observational baseline-treatment analysis, state that consistency, conditional exchangeability given measured covariates, and positivity are assumed. Consistency links a person’s observed outcome under the received strategy to that strategy’s potential outcome; exchangeability requires no unmeasured confounding conditional on the covariates; positivity requires a positive probability of each strategy across covariate patterns in the target population. For longitudinal strategies, the corresponding requirements apply over treatment and covariate histories. Also state the censoring assumptions used to identify risks through \(\tau\), and how loss to follow-up was handled. Randomization can support exchangeability at baseline, but it does not by itself solve informative loss to follow-up or unclear outcome definitions.

Name the target.
Specify the population, treatment strategies, event types, and time horizon in clinical terms.

Name the estimand.
State whether the target is a treatment-policy contrast or a contrast under another explicitly defined intervention, and give the risk scale.

State identification conditions.
List the assumptions needed for the observed data to identify the target and describe how censoring and missing event information were addressed.

Present risks before contrasts.
Report the strategy-specific cumulative incidences, then the absolute contrast, its uncertainty, and a contextual interpretation.

Worked Reports

Worked Example: A Treatment-Policy Contrast at One Year

Suppose a hypothetical randomized study compares a new treatment (\(A=1\)) with control (\(A=0\)) among eligible adults. The first events are recurrence and death before recurrence. The estimated one-year cumulative incidences are 0.14 for recurrence and 0.18 for death under treatment, and 0.20 for recurrence and 0.10 for death under control. Assume treatment strategies are well defined, follow-up and event classification are adequate, censoring before one year is independent of the future event process under the analysis, and the risk estimates are valid for the randomized target population.

For recurrence, the risk difference is

\Delta_{\text{recurrence}}(1\text{ year})=0.14-0.20=-0.06.

Thus, treatment is associated with an estimated 6-percentage-point lower one-year recurrence risk, or 60 fewer first recurrences per 1,000 people, under the treatment-policy strategies. The recurrence risk ratio is \(0.14/0.20=0.70\), a 30% lower relative risk. These are different scales describing the same event-specific contrast.

The competing-event risks must be reported too: the death risk difference is \(0.18-0.10=0.08\), or 80 more first deaths per 1,000 under treatment. Since the first events are mutually exclusive, the risk of either first event is 0.30 under control and 0.32 under treatment: \(0.20+0.10=0.30\), while \(0.14+0.18=0.32\). The corresponding event-free probabilities through one year are \(1-0.30=0.70\) and \(1-0.32=0.68\). This arithmetic is a useful consistency check; it does not make the combined outcome a substitute for the event-specific results.

A concise report would say: “In this hypothetical randomized population, the estimated one-year cumulative incidence of first recurrence was 14% under assignment to treatment and 20% under assignment to control, a risk difference of −6 percentage points (−60 per 1,000). The corresponding first-death risks were 18% and 10%. These are treatment-policy contrasts over one year.” Without a confidence interval, the report should not imply that the observed difference is statistically distinguishable from zero.

Worked Example: Separating a Policy Risk From a Hypothetical Risk

Consider a hypothetical observational analysis of recurrence with death as the competing event. Under treatment-policy strategies, suppose the estimated one-year recurrence risks are 0.20 under control and 0.14 under treatment. The treatment-policy risk difference is \(0.14-0.20=-0.06\), or 6 fewer recurrences per 100 people by one year under treatment. These risks include the fact that some people die before recurrence.

Now suppose investigators also define a distinct hypothetical intervention that prevents death through one year while otherwise specifying how the intervention operates. A model produces hypothetical recurrence risks of 0.26 under control and 0.19 under treatment. The contrast for this separate estimand is

$$ 0.19-0.26=-0.07, $$

or 7 fewer recurrences per 100 people by one year under treatment in that hypothetical setting. The two risk differences differ by 1 percentage point: \(-0.07-(-0.06)=-0.01\). This calculation does not show that either contrast is preferable. The hypothetical contrast answers a different question and requires a well-defined death-prevention intervention and additional assumptions to identify its consequences. It must not be described as the observed-world treatment-policy effect or as simply “removing” deaths from the data.

A responsible report would label the second result explicitly as hypothetical, describe the intervention and assumptions, and present it separately from the treatment-policy risk. If no plausible intervention or sufficiently clear counterfactual meaning can be specified, the hypothetical contrast should not be reported as though it were an ordinary observed risk.

Worked Example: Reporting a Dynamic Treatment Strategy

Suppose a hypothetical cohort study compares two strategies for patients with a newly diagnosed condition: initiate treatment within seven days unless a stated contraindication occurs, versus do not initiate it during the first 30 days. Death before hospitalization is the competing first event, and hospitalization is the event of interest. At 180 days, the estimated hospitalization risks under the two strategies are 0.11 and 0.15, respectively.

The absolute contrast is \(0.11-0.15=-0.04\), which is 4 percentage points or 40 fewer first hospitalizations per 1,000 people under the initiation strategy. Suppose a hypothetical analysis gives a 95% confidence interval for the risk difference of \((-0.075,-0.005)\). In context, the interval ranges from 75 fewer to 5 fewer hospitalizations per 1,000 under that strategy, under the analysis assumptions. Its exclusion of zero indicates compatibility with a nonzero contrast at the corresponding two-sided 5% level, conditional on the method and assumptions; it does not establish that the assumptions hold.

Because the strategies depend on treatment and contraindication histories, the report must state that measured histories were sufficient to control time-varying confounding, that each strategy remained possible at relevant histories, and that consistency and appropriate censoring assumptions were used. A report that gives only “risk was reduced by 4%” is inadequate: it does not make clear whether 4% means a relative reduction or an absolute difference, and it omits the event, horizon, and strategies.

Justification and Checks

For mutually exclusive first-event types \(1,\ldots,K\), the event of experiencing any first event by \(\tau\) is the disjoint union of the events \(\{T^a\leq\tau,J^a=k\}\). Additivity of probability therefore gives

P(T^a\leq\tau)=\sum_{k=1}^{K}P(T^a\leq\tau,J^a=k) =\sum_{k=1}^{K}F_k^a(\tau).

If the categories exhaust the possible first events and there is no event by \(\tau\) otherwise, the probability of remaining free of a first event is \(1-\sum_{k=1}^{K}F_k^a(\tau)\). This identity justifies the arithmetic check in the first example. It also explains why reporting only one event type can hide important changes in the competing-event risk. The identity applies to probabilities for the same strategy, population, and horizon; estimates from different populations or time horizons should not be combined this way.

Reporting choice	What it answers	What to state
Cause-specific cumulative incidence and risk difference	How the probability of each first-event type by a chosen horizon differs between strategies	Each risk, the absolute difference, event definitions, strategies, and horizon
Risk ratio	How large one strategy-specific risk is relative to the other	Both risks as well as the ratio; do not present the ratio as an absolute change
Cause-specific hazard contrast	How instantaneous event rates among those still event-free compare over time	That it is a hazard-scale contrast, not a risk difference or direct probability contrast
Hypothetical competing-event intervention	What risk would be under a separately specified intervention affecting the competing event	The intervention, its target population and horizon, and the extra identification assumptions

A cumulative incidence risk is often the most direct scale for communicating absolute prognosis and causal risk differences. A hazard ratio can be useful for a different question, but it does not state how many events occur by a fixed time and can be difficult to interpret when hazards vary over time. Likewise, a cause-specific hazard analysis that treats competing events as leaving the risk set does not, by itself, report the probability of the event of interest in the presence of competing events. The chosen summary should match the estimand, not merely the model that was fitted.

Diagnostics, Limitations, and a Common Pitfall

Before interpreting a reported contrast, check that the stated risks are on the same scale and refer to the same population and time horizon. Verify that event-specific cumulative incidences are nonnegative and, for mutually exclusive first-event categories, sum to no more than one. Check that the reported absolute difference equals treatment risk minus comparator risk and that its sign agrees with the wording. For observational analyses, review covariate balance or model diagnostics relevant to confounding control, inspect whether treatment probabilities approach zero in important histories, and assess whether loss to follow-up could depend on future event risk. These checks can reveal problems; they cannot prove unmeasured exchangeability or positivity.

A common mistake is to censor people when they experience a competing event and then describe the resulting event-of-interest quantity as the actual-world probability of that event by the horizon. That reasoning ignores that a competing event prevents the first event of interest from occurring later. The correction is to report the cumulative incidence for the event of interest under the specified strategy, retaining the competing event as part of the event process. If the scientific question instead imagines preventing the competing event, define that intervention and report the corresponding hypothetical estimand separately.

Even a carefully reported estimate is limited by its assumptions, measurement quality, and target population. An absolute contrast at 180 days does not determine the contrast at one year; changing the horizon changes the estimand. A result in the study population may not transport to patients with different treatment options or baseline risks. Finally, an estimated causal contrast is not evidence that treatment caused an outcome unless the design and identification assumptions support that interpretation.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

Under control, one-year risks of event A and competing event B are 0.12 and 0.08. Under treatment, they are 0.09 and 0.15. What are the risks of either first event and the risk difference for event A?
Why must a treatment-policy report name what happens to participants who discontinue treatment?
An analysis reports a risk ratio of 0.80 but gives neither strategy-specific risks nor a horizon. What important information is missing?
What additional question does a hypothetical intervention that prevents a competing event answer compared with a treatment-policy estimand?

Solutions. (1) The risks of either first event are \(0.12+0.08=0.20\) under control and \(0.09+0.15=0.24\) under treatment. The event-A difference is \(0.09-0.12=-0.03\), or 3 fewer event-A outcomes per 100 by one year. (2) Discontinuation rules clarify the treatment strategy; a treatment-policy estimand generally follows outcomes after discontinuation as part of the assigned strategy’s course. (3) The absolute risks, time horizon, target population, event definition, and strategies are needed to interpret the relative contrast in context. (4) It asks about outcomes under a distinct intervention on the competing event, not the risks under the natural competing-event process associated with each treatment strategy.