Competing Risks and Cumulative Incidence

Time to default is not observed in isolation when prepayment or maturity can happen first. Once either occurs, later default under the original contract is impossible.

These events compete because their ordering is mutually exclusive for the first-event question. The event taxonomy must follow the economic process.

Combining all exits estimates time to any exit. Separating them asks which exit occurs first, and a hypothetical world without competitors asks a stronger, less observable question.

Kaplan–Meier treats censored units as representative of those remaining at risk. A competing event instead reveals that the target event can no longer occur first.

Censoring prepaid loans while estimating default effectively imagines they retained the same future default opportunity as continuing loans.

One minus that Kaplan–Meier curve therefore estimates a net quantity under removal of competitors and generally exceeds real-world default cumulative incidence.

For cause k, the cause-specific hazard is the instantaneous rate of cause k among units with no event yet.

A Cox model can censor other causes to estimate covariate associations with this hazard. Here censoring is a risk-set device, not an estimate of actual cause probability.

Cause-specific hazard ratios are useful for mechanism and etiologic questions, but they do not map directly to cumulative incidence without all causes' hazards.

The cumulative incidence function for cause k is the probability that cause k occurs first by time t.

It combines the current cause-specific hazard with the probability of surviving every cause until each time. Competing hazards therefore shape the target probability.

The cause-specific cumulative incidences sum to the probability of any event, leaving event-free survival as the remainder.

Fine–Gray regression links covariates to a subdistribution hazard constructed to model cumulative incidence directly.

Its risk set retains people with a prior competing event through weighting conventions. The resulting subdistribution hazard ratio is not an ordinary current-risk rate ratio.

Use it when covariate association with cumulative incidence is the target. Do not translate its coefficient into a direct probability multiplier.

A borrower feature may increase both prepayment and default hazards. Faster prepayment can remove high-risk loans before default and lower observed default incidence.

Thus a covariate can raise the cause-specific default hazard yet have a weak or opposite association with default cumulative incidence.

This is not a contradiction. The two estimands answer different questions about current event flow and realized first-event probability.

Credit studies may distinguish default, prepayment, maturity, restructuring, and loan sale. Fund studies may separate liquidation, merger, and database disappearance.

Event misclassification moves observations between causes and changes every risk set. Ambiguous exit codes should not silently become censoring.

Independent loss to follow-up is still required for identification. Calendar cutoff differs from an unclassified disappearance correlated with distress.

Define the first-event question, event hierarchy, horizon, time origin, delayed entry, censoring, and how ties or ambiguous outcomes were resolved.

Plot event-free survival and each cumulative incidence so probabilities reconcile. Report absolute incidence at useful horizons with uncertainty.

Use cause-specific models for event dynamics and Fine–Gray only when its cumulative-incidence estimand matches the decision, clearly labeling every ratio.