Why Cause-Specific Risks Must Sum To Failure Risk

Why an Accounting Identity Matters

A competing-risks analysis may report a separate absolute risk for each type of first event. If a patient can experience one of several mutually exclusive first events, those risks are not unrelated summaries: together, they account for the chance of experiencing any first event by the chosen horizon. This is an accounting identity, not a claim that the causes act independently or that their risks can be calculated in isolation.

The identity is useful both for interpretation and for checking results. If the reported cause-specific risks do not add to the reported risk of any event, the discrepancy may reflect rounding, an omitted cause, different target populations or horizons, or estimates that were not constructed coherently. It is not automatically evidence that the underlying probability rule has failed.

Here, \(T\leq t\) includes an event occurring exactly at the horizon. An individual with no event by \(t\) contributes to none of the cause-specific incidences at that horizon. If \(S(t)=P(T>t)\) denotes the probability of remaining free of the first event through \(t\), then the same identity can be written as \(F_{\mathrm{any}}(t)=1-S(t)\). Together, the cause-specific incidences partition the probability of failure; the event-free probability is the remaining part.

The Identity Follows From a Partition

The key condition is that each individual who has a first event by \(t\) has exactly one cause in the list. Define \(A_k(t)=\{T\leq t,\ J=k\}\), the event that the first event occurs by \(t\) and is of cause \(k\). The sets \(A_1(t),\ldots,A_K(t)\) are disjoint: an individual cannot have two different causes for the same first event. If the cause list is exhaustive, their union is exactly the event \(\{T\leq t\}\).

Additivity of probability for a finite collection of disjoint events now gives the result. Equivalently, the indicator that any first event has occurred by \(t\) is the sum of the indicators for the cause-specific events. Taking expectations yields the same equation.

This short argument does not require independent causes, constant event rates, or a particular hazard model. It requires a well-defined first event, a common time origin and horizon, and a cause classification that is mutually exclusive and exhaustive. The risks are probabilities for the same target population and the same event definition. If one cause can be assigned to an individual only after observing a later event, the first-event classification must still follow the stated rule.

Assumptions and What the Identity Does Not Say

At the population level, the identity is a probability rule. Its assumptions are primarily about how the outcome is defined and classified, rather than about a statistical model:

• One first event is being counted. Later events are not additional first events. For a question about recurrent events or multiple transitions, this partition of first-event causes is not, by itself, the relevant accounting rule.
• Causes are mutually exclusive for the first event. An event with genuinely simultaneous or ambiguous causes needs a prespecified classification rule, such as adjudication or an explicit joint category.
• The cause list is exhaustive. “Other,” unknown, or unclassified causes must either be included as categories or explicitly left out of the reported sum. A sum over only some causes is not the risk of any event.
• All terms refer to the same estimand. Use a common population, time origin, horizon, and definition of failure. A five-year risk in one subgroup cannot be added to a five-year risk in another subgroup to obtain a population risk.

The identity itself does not require independent censoring because it is a statement about the true population probabilities. Censoring matters when those probabilities are estimated from observed data. Estimation requires appropriate methods and assumptions about how censoring relates to event time, possibly conditional on measured variables. Treating a competing event as ordinary censoring when estimating a cause-specific cumulative incidence can target the wrong quantity; the relevant risk counts the competing event as preventing a later first event of the other cause. These estimation issues are distinct from the partition identity.

For a coherent set of estimated cumulative incidences constructed from the same competing-risks analysis, the estimated all-cause failure probability should agree with their sum, apart from numerical rounding. Separately fitted or otherwise incompatible estimates need not add exactly. Such a mismatch signals that the estimates may not form one common probability partition; it does not change the population identity.

Checking Reported Risks

When a reported sum does not match the stated all-cause risk, check the definitions before attributing the difference to a calculation error. First verify that every relevant first-event type is represented, including unknown or “other” events. Then check that all risks use the same horizon, time origin, target population, and denominator or estimation target. Confirm that a cause-specific risk is a cumulative incidence—not a cause-specific hazard or a net risk—and check whether estimates were rounded before being added.

A small discrepancy may be explained by rounding: for example, unrounded risks of 0.124 and 0.076 sum to 0.200, whereas independently rounded values might display as 0.12 and 0.08. A material discrepancy is more likely to reflect different definitions, omitted categories, different populations, censoring handled inconsistently, or separate modeling procedures that do not produce a coherent set of competing-risk probabilities.

The main alternatives answer different questions. A cause-specific hazard describes the instantaneous rate of a particular cause among people who are still event-free; it is not the probability of that cause by a horizon, so adding cause-specific hazards does not produce the all-cause cumulative risk. A net risk describes a hypothetical setting in which competing events are removed; it is not one of the observed competing-cause cumulative incidences in the partition above. The introductory material on cumulative incidence distinguishes these absolute-risk targets. The accounting identity applies to the actual first-event causes in the stated population, not to hypothetical risks under a different event process.

A final limitation is interpretive: summing risks establishes a probability accounting relationship, not a causal explanation. It does not imply that changing one cause’s hazard will leave the other causes’ cumulative incidences unchanged. Each cause-specific incidence is part of a set of competing outcomes, and a complete interpretation must preserve the common target population and event process.