Any-Event Effects Versus Cause-Specific Effects

Why the Choice of Outcome Matters

When several mutually exclusive event types can occur, “Did an event happen?” and “Did this particular event happen?” are different questions. A treatment may reduce the risk of one event type while increasing the risk of another. The total frequency of any event can therefore show little change, even when the distribution of causes changes substantially. It can also increase while the risk of a clinically important event decreases.

The comparison in this tutorial is between causal effects on risks measured by the same fixed time \(t\). The effects concern the event that would occur under each treatment, not merely the event observed among people who happened to remain event-free. The earlier material on treatment-specific cumulative incidence and total effects on cause-specific risk provides the relevant causal-outcome context.

The cause-specific risk is the probability of having event type \(k\) first by time \(t\). A competing event before type \(k\) prevents type \(k\) from being the first event in this definition. The any-event risk instead counts every event type in the specified set. Both are risks in the full population under a treatment intervention; neither is a risk calculated only among those who remain event-free.

The Exact Relationship Between the Contrasts

Suppose the event types are mutually exclusive and exhaustive for the first event being studied. Then each person who experiences an event by \(t\) has exactly one event type. The events \(\{T^a\le t,J^a=k\}\), over \(k=1,\ldots,K\), are disjoint, and their union is \(\{T^a\le t\}\). Additivity of probability gives:

Subtract the control risk from the treatment risk on both sides. Because the sum is finite, subtraction distributes across the terms:

This identity is the key to interpreting the contrasts. A decrease in one cause-specific risk can be offset by an increase in another. The identity applies to risk differences at the same horizon and for the same event set. It does not say that the effect on any event is the effect on every individual cause.

Risk ratios have a different relationship. If \(F_{\mathrm{any}}^0(t)>0\), then:

provided each component ratio in the expression is defined. The weights are the shares of the control any-event risk attributable to each cause, and sum to one. Thus, when all control cause-specific risks are positive, the any-event risk ratio is a baseline-risk-weighted average of the cause-specific risk ratios. It is not their unweighted average. If a control cause-specific risk is zero, its component ratio is undefined; the overall any-event ratio can still be defined, but this weighted representation must be handled without that ratio.

Worked Comparisons Across Event Types

Assumptions, Estimation, and Checks

The identities above are mathematical consequences of the event definitions. Interpreting the risks and their differences as causal effects requires additional conditions. Treatment must correspond to well-defined interventions, and consistency requires that a person's observed outcome under the treatment actually received agrees with the corresponding potential outcome. There should be no interference between individuals, so one person's treatment does not change another person's potential outcome. In a randomized study, random assignment supports exchangeability of treatment groups. In an observational study, causal identification generally requires conditional exchangeability given measured covariates and adequate positivity: each treatment must be possible within the covariate patterns used for the comparison.

Censoring also matters when risks are estimated from follow-up data. The approach used to estimate the treatment-specific risks must appropriately account for censoring; a common sufficient condition for standard approaches is independent censoring, possibly conditional on measured covariates. Inadequate follow-up, informative loss to follow-up, incorrect cause coding, or failure to adjust for confounding can compromise estimation or causal interpretation. Randomization does not repair outcome misclassification or informative censoring automatically.

Before interpreting a comparison, check that both treatment groups use the same time horizon, event definitions, and target population. Confirm that first-event categories are mutually exclusive, that the categories counted in any event match those used in the cause-specific risks, and that the reported risks add to the any-event risk. If they do not, there may be an omitted event category, inconsistent denominators or horizons, or a coding error. Sampling uncertainty also matters: differences between estimated risks are not necessarily evidence of a nonzero causal effect. Any-event and cause-specific estimates are dependent because the component outcomes are related; uncertainty for a sum or contrast should account for their joint sampling variability.

These risk contrasts answer questions about probabilities by a stated time. Cause-specific hazards and regression coefficients, including those from cause-specific Cox or Fine-Gray models, are different summaries and are not interchangeable with these risk differences or risk ratios. The earlier material on competing-event rates and cause-specific Cox coefficients distinguishes regression summaries; for the present comparison, the outcome is a risk under each treatment at the chosen horizon.

Common Mistakes and Limitations

• Equating a target-event reduction with an any-event reduction. A lower risk for one cause does not guarantee a lower total risk. Add the risks for all included event types before making an any-event claim.
• Calling an unchanged any-event risk “no treatment effect.” The aggregate risk may be unchanged while the cause distribution shifts, as in the first worked example. State which outcome the effect refers to.
• Interpreting a cause-specific risk as a risk among survivors. The defined risk is a population probability by time \(t\), with competing first events preventing the target event from occurring first. It is not a comparison restricted to people who remain event-free.
• Averaging cause-specific risk ratios without weights. The any-event risk ratio uses the control cause-specific risks as weights, not equal weights in general.
• Changing the time horizon or event set between comparisons. A sum identity cannot reconcile risks defined over different periods or different event categories.

The choice between any-event and cause-specific contrasts depends on the scientific question. Any-event risk summarizes the probability of experiencing at least one event in the specified set. Cause-specific risks retain information about which event occurred first. Neither alone describes the full clinical consequences of a treatment: event severity, timing, recurrent events, and patient preferences may also matter. Report the horizon, event definitions, risks under both treatments, and the contrasts that answer the stated question.