Can cumulative incidence decrease over time?

For a fixed cohort and event definition, the true function cannot decrease because past occurrences remain counted.

Is it always one minus survival?

For any first event collectively, yes under the corresponding survival definition. For one type among competing events, generally no.

Can a lower cause-specific hazard coexist with a higher event probability?

Comparisons can be affected by competing hazards and their timing, so hazard and cumulative-incidence comparisons need separate interpretation.

AI Glossary / Statistics and probability / Intermediate

Cumulative incidence

Definition

Cumulative incidence is the probability that a specified event has occurred by a particular time. It concerns an event defined to occur once, or a first occurrence. In competing-risks analysis, the cumulative incidence function for event type k is F_k(t) = P(T <= t, J = k), where T is the first-event time and J identifies its type.

What it means

The measure answers a practical question: “By this deadline, what proportion of the starting population will have experienced this event?” A time horizon is essential. A one-month probability and a five-year probability describe different quantities even if the event definition is identical.

When several mutually exclusive first events are possible, the probability of one event depends on the others. A component permanently retired before failure can no longer experience its first failure while in service. Treating retirement merely as missing follow-up changes the question and can distort the probability relevant to the actual operating environment.

How it works

In a closed cohort with complete follow-up to a common horizon, estimate cumulative incidence by dividing the number experiencing the specified event by the number initially at risk. This simple fraction requires clear eligibility and event definitions.

For right-censored follow-up without competing events, one minus the Kaplan-Meier survival estimate can estimate cumulative incidence under suitable censoring assumptions. With competing events, use a method that accounts for them, such as the Aalen-Johansen estimator.

Conceptually, accumulate the event-specific hazard while weighting by the probability of remaining free of every competing event just beforehand. Thus cumulative incidence is not simply the accumulated hazard. Loss to follow-up still requires appropriate assumptions; recognizing competing events does not solve informative missingness or misclassified outcomes.

Three practical examples

A maintenance team estimates the probability of a first equipment failure within a year, explicitly accounting for permanent retirement before failure.
A subscription analyst models whether an account first closes through customer cancellation or another terminal closure route during a fixed period.
An AI-supported workflow forecasts the probability that a pending case reaches a specified terminal outcome before alternative outcomes make that event impossible under the case definition.

Think of it this way

Think of people starting together and leaving through different one-way doors. The cumulative incidence for one door is the share that has used it by a deadline. People who already left through another door cannot later leave through the first one. Ignoring those other exits creates a different and potentially misleading picture of the original group's destinations.

Worked example

Consider an original complete-follow-up example with 100 newly installed components. At month one, 30 are permanently retired without failure. At month two, 20 of the remaining 70 experience their first failure. The other 50 remain operating through the three-month horizon.

The cumulative incidence of failure by three months is 20/100 = 0.20. Retirement has cumulative incidence 30/100 = 0.30, and the probability of neither event is 50/100 = 0.50. These three proportions sum to one.

The failure contribution can also be written as 0.70 times (20/70) = 0.20: 70% are still event-free immediately before the failures, and 20 of those 70 fail.

If retirement is instead treated as ordinary censoring in a failure-only Kaplan-Meier calculation, one minus estimated survival becomes 20/70, approximately 0.28571. That is not the observed probability of failure in the original cohort with retirement operating as a competing event. The example demonstrates the estimand difference; it does not identify what would happen if retirement were somehow prevented.

Illustrative case study

In this fictional case, Foundry Systems trains an AI model to estimate first-failure risk for rented devices. Its initial evaluation removes devices returned permanently before failure and calculates a failure probability from the remaining devices.

Reviewers identify that permanent return ends the rental-service failure process being studied. They define failure and return as separate first-event outcomes, retain both in evaluation, and report each cumulative incidence at fixed horizons. They separately document genuinely missing device histories.

The revised report supports planning under the actual rental policy. It does not claim to estimate a causal effect of preventing returns. This invented case illustrates why the event definition and competing outcomes must match the operational question.

Common misconceptions and mistakes

Cumulative incidence is a probability, not an incidence rate per unit of exposure time. A rate such as failures per thousand operating hours cannot be read directly as a percentage without additional modeling.

It also differs from cumulative hazard, which is not bounded by one. Hazard ratios compare hazard functions, not directly cumulative probabilities at a chosen horizon.

Do not count recurrent events as though each were a different member of the starting population. A first-failure cumulative incidence and an expected number of failures answer different questions. Competing-event definitions must also be substantively credible: a temporary interruption need not permanently prevent the target event.

Business use

Businesses can use cumulative incidence for warranty planning, service capacity, and forecasts of terminal customer or case outcomes. Specify the starting population, time origin, event, competing events, and horizon before choosing a method.

Compare predictions with observed event probabilities using evaluation methods that accommodate censoring. Report uncertainty and the amount of follow-up supporting each horizon. A curve extending far beyond the available observations is heavily model-dependent. If a business policy changes the frequency of competing events, historical probabilities may no longer transfer to the new setting.

Check your understanding

Check your understanding: Why is the example's failure probability 20% rather than about 28.6%?

Answer: It refers to all 100 starting components while retaining retirement as a competing outcome.

What does the remaining 50% represent?

Answer: Components experiencing neither first event by the horizon.

Would counting every later repair visit produce the same measure?

Answer: No; recurrent-event counts require a different outcome definition and analysis.

Related terms

Survival analysis
Kaplan-Meier estimator
Censoring
competing risks
cumulative incidence function
Aalen-Johansen estimator
cause-specific hazard
cumulative hazard

Learn next

Compare a cause-specific hazard with its cumulative incidence curve. Then explore how changing a competing-event process affects the target event's probability, even if its own cause-specific hazard stays fixed.

Sources

Sources: Frank Harrell, Glossary of Statistical Terms, https://hbiostat.org/glossary/, cumulative incidence definition. R survival package authors, competing risks and multistate-model vignette, https://cran.r-project.org/web/packages/survival/vignettes/compete.pdf, for Aalen-Johansen and cumulative-incidence relationships. The component example and fictional business case are original teaching material.

Previous: Cumulative forecasting Next: CUR matrix approximation