A machine retired from service cannot subsequently experience its first failure while operating in that fleet. Retirement is therefore different from losing its telemetry connection. The first event changes what can happen; the second changes what researchers can observe. Time-to-event analyses need to preserve that distinction.
TL;DR: A competing event prevents the event of interest as defined by the study. Estimate the probability of each event in the presence of the others when that is the practical question. Cause-specific hazards describe rates among units still event-free; cumulative incidence describes probabilities from the starting population. Neither a cause-specific nor a subdistribution hazard ratio is automatically a risk ratio or a causal effect.
Open the fleet ledger before choosing a model
This article follows an invented fleet of one hundred industrial pumps from installation. The primary endpoint is the first mechanical failure before permanent retirement. Retirement ends service permanently, while a temporary loss of telemetry does not. The example concerns reliability research and uses simplified event times to make the accounting visible.
At the end of the observation period, twenty pumps have failed, thirty have been retired without a prior failure, and fifty remain operating without either event. Assume initially that every pump has complete follow-up to that point. The observed failure proportion is twenty percent. Retirement accounts for thirty percent, and the probability of remaining in the initial event-free state is fifty percent.
Those three quantities sum to one because the states are mutually exclusive and collectively exhaustive under the study's first-event definition. A pump cannot appear in both the first-failure and first-retirement counts. A later inspection of a retired pump may discover a defect, but that is not automatically the same endpoint as failure while operating before retirement.
The endpoint definition must make this distinction operational. Does a planned replacement count as retirement? Does a pump moved to another site remain in the fleet? Is a temporary shutdown a competing event, an intermediate state, or simply a period without exposure? The answers depend on the scientific question, not on which coding choice makes the analysis easiest.
Competing-risks methods separate event-specific rates from event-specific probabilities. Andersen and colleagues explain why the simple relationship between one hazard and one risk no longer holds when several event types can occur first. This is the conceptual starting point for the ledger. Competing risks: possibilities and pitfalls
The analyst records one time origin, a clear event code, and the time of the first relevant transition for every pump. The raw file also preserves later information rather than deleting it. Later transitions may be useful for a different multistate analysis, but they should not silently alter the meaning of the current first-event endpoint.
Exhibit A: why censoring retirements changes the answer
To expose the arithmetic, suppose all thirty retirements occur at month three and all twenty failures occur at month six. The remaining fifty pumps operate through month twelve. There is no loss to follow-up in this toy dataset. Ties within each event time are intentional and make the calculation easy to inspect.
At month three, the probability of remaining free of either event falls from one to 0.70. At month six, twenty of the seventy still-operating pumps fail. The conditional failure proportion at that time is twenty divided by seventy. Multiplying it by the probability of reaching that time event-free gives 0.70 times twenty divided by seventy, or 0.20.
That multiplication is the essential accounting behind cumulative incidence: an event-specific increment is weighted by the probability of still being eligible to experience a first event. The retirement probability remains 0.30, and event-free probability becomes 0.50. The calculation agrees with the direct proportions because follow-up is complete.
Now run a different calculation that treats the thirty retirements as ordinary censoring and applies one minus Kaplan–Meier for failure alone. At month six, it uses twenty failures among seventy remaining pumps, producing approximately 28.6 percent. That number is larger than the observed twenty percent failure probability in the original fleet.
| Calculation at twelve months | Result | Question represented |
|---|---|---|
| Failure cumulative incidence | 20 percent | Probability of failure before retirement |
| Retirement cumulative incidence | 30 percent | Probability of retirement before failure |
| Remaining event-free | 50 percent | Probability of neither first event |
| One minus failure-only Kaplan–Meier with retirement censored | About 28.6 percent | A different quantity, not the observed failure probability |
The fourth row should not be casually labeled the failure risk if retirement did not exist. Giving it that causal interpretation requires a defined hypothetical intervention and additional assumptions. Simply changing an event code to censored does not show what the retired pumps would have done under continued operation.
This distinction matters even when the numerical difference is small. The size of the distortion depends on event timing and frequency, but the question has changed as soon as competing events are treated as if they only removed observation. A graph can look smooth and statistically sophisticated while estimating the wrong target.
Exhibit B: censoring is about incomplete observation
Introduce a second version of the fleet in which ten operating pumps lose telemetry before the study ends. Their later failures or retirements are unknown. These units have not necessarily ceased to be capable of either event. Their records are censored at the last time their status is known under the defined observation process.
Estimating event probabilities from the remaining follow-up now requires assumptions about censoring. A common requirement is that censoring be independent of the future event process, possibly conditional on measured variables used in the analysis. If telemetry fails more often in pumps already showing severe vibration, treating the missing follow-up as uninformative may be implausible.
A maintenance log could help recover some outcomes. If a pump's telemetry stops but its retirement is recorded independently, the analyst should use the available event information rather than censoring it automatically. Data linkage can improve the study more directly than fitting a more elaborate model to an unnecessarily incomplete record.
Administrative study end is another form of censoring. A pump still operating when data collection closes has an unknown future event time, but its known history remains informative. The report should distinguish this planned endpoint from early loss to follow-up because the mechanisms and potential biases differ.
The competing-risk label is therefore not an intrinsic property of an event in every study. Permanent retirement competes with failure while in service, but it might be an intermediate transition in a study of total lifecycle costs. Transfer to another owner could be a competing event for one operational endpoint and a follow-up problem for another. The estimand determines the classification.
A useful data review asks, for each exit code, whether the event of interest became impossible under the stated definition or merely unobserved. The answer should be documented before outcomes are summarized. This prevents analysts from using censoring as a miscellaneous category for every inconvenient end to a record.
A cause-specific hazard follows the pumps still in service
The cause-specific failure hazard describes the instantaneous failure rate among pumps that have not yet failed or retired. Its risk set contains the pumps still eligible for a first event. When a pump retires, it leaves that risk set because it can no longer fail first while operating in the fleet.
A cause-specific model can be useful for understanding associations with the transition from operating to failed. But its hazard ratio compares rates within evolving event-free groups. Those groups can change composition over time, especially if a characteristic also affects retirement. The ratio should not be read as a fixed comparison of cumulative failure probabilities.
Suppose two maintenance policies have the same cause-specific failure hazard but different retirement hazards. The policy that retires pumps earlier leaves fewer pumps exposed to failure over a given horizon. Its cumulative incidence of failure can therefore be lower even though the failure rate among pumps still operating is unchanged.
A simple constant-hazard example makes this precise. Assume a failure hazard of 0.10 per year under both policies. Under policy A, retirement hazard is 0.10 per year; under policy B, it is 0.30. Under this deliberately simplified model, five-year failure cumulative incidence equals the failure hazard divided by the total hazard, multiplied by one minus the exponential of minus total hazard times five.
For policy A, that is 0.5 multiplied by one minus the exponential of minus one, approximately 31.6 percent. For policy B, it is 0.25 multiplied by one minus the exponential of minus two, approximately 21.6 percent. The cause-specific failure hazards are identical, while the failure probabilities differ by about ten percentage points.
These are model calculations, not empirical claims about maintenance. They show that cumulative incidence depends on the competing event process as well as the event-specific hazard. Reporting only the failure hazard leaves out information needed to estimate how many pumps from the original fleet will fail by a specified date.
The converse also matters. A lower cause-specific failure hazard does not guarantee a lower cumulative failure probability if the policy also keeps many more pumps in service. A reliability program could reduce the rate of failure among operating units while increasing the time during which units remain exposed. The policy's practical value requires the full outcome picture.
A cumulative-incidence curve follows the starting population
The cumulative-incidence function answers a direct question: what is the probability that a specified event has occurred first by time t in the presence of the other events? For the fleet manager, it can describe how many installed pumps are expected to fail before retirement over a chosen horizon.
The Aalen–Johansen estimator provides the corresponding nonparametric state-probability accounting under suitable observation assumptions. In a basic competing-risks setup, its event-state estimates are cumulative incidence estimates. Official documentation for the survival package illustrates this relationship within a broader multistate framework. Competing risks and multistate models
The analysis should show the retirement curve alongside the failure curve when retirement materially affects interpretation. A lower failure curve can be operationally attractive, but it may have been purchased through accelerated replacement. The accompanying retirement probability and costs help distinguish improved durability from reduced exposure through withdrawal.
Time horizons should match decisions. A one-year failure probability may guide warranty reserves, while a five-year probability informs replacement planning. A single hazard ratio cannot replace both quantities. Curves can also cross, making an unqualified statement that one policy is better incomplete without a horizon and an outcome definition.
Uncertainty belongs around estimated probabilities and contrasts at the relevant horizons. The report should state how uncertainty was calculated and avoid presenting the end of a sparsely supported curve with the same confidence as earlier follow-up. A few long-observed pumps cannot necessarily support a precise prediction for the entire installed population far into the future.
Numbers at risk help readers understand that support, but their definition should be clear. For cause-specific event-free follow-up, a pump leaves the at-risk group after either first event. A subdistribution model may use a different mathematical risk set. Reusing one unlabeled risk table across different model interpretations can create confusion even when each individual calculation is correct.
Fine–Gray changes the model, not the need to define the question
Fine and Gray proposed a regression model for the subdistribution hazard associated with cumulative incidence. This offers a way to relate covariates to the event probability structure while accounting for competing events through a particular mathematical construction. Its risk set differs from the ordinary set of units still event-free. Fine–Gray original paper
The resulting subdistribution hazard ratio is not the ratio of failure probabilities at a chosen time. It is also not the instantaneous failure-rate ratio among currently operating pumps. Translating it into a practical statement requires the model's cumulative-incidence predictions, a horizon, and the relevant covariate values or target population.
The proportional subdistribution hazards assumption is a modeling assumption, just as proportional cause-specific hazards is an assumption in a different model. Neither should be accepted solely because the software converges. The study should assess whether the model adequately represents the observed time pattern and use alternatives when the intended claim requires more flexibility.
Choosing between cause-specific and subdistribution models is not a contest in which one method universally corrects the other. They parameterize different aspects of the event process. A study may use cause-specific models to build cumulative-incidence predictions by combining the relevant hazards, or use another justified model directly. The question and assumptions should drive that choice.
For the fleet report, the primary deliverable remains an estimated failure probability under specified conditions, accompanied by retirement probability. Regression coefficients can support that deliverable, but they should not displace it. A manager deciding how many replacement units to stock needs interpretable quantities, not only a model label and a statistically significant ratio.
The causal question needs another layer of design
An observational comparison between maintenance policies may be confounded. Older pumps, harsher operating environments, and different utilization levels can influence both policy selection and outcomes. Competing-risks software does not automatically address those differences. A causal claim requires a design and assumptions that connect observed comparisons to the policy intervention of interest.
Even random assignment does not make every hazard contrast a simple causal effect. Conditioning on remaining event-free can select different groups under the two policies as follow-up progresses. A cumulative-incidence contrast under randomized assignment can address a total policy effect under the relevant follow-up assumptions, but a claim about an effect that excludes retirement pathways is a different estimand.
Young and colleagues develop a causal framework showing why decisions about competing events and censoring correspond to different causal questions and identifying conditions. Their work cautions against treating an analytical convention as if it uniquely determined the scientific target. Causal framework for competing events
In the pump example, one question asks what happens if the fleet adopts the complete maintenance policy, including its retirement decisions. Another asks what the policy's failure effect would be if retirement behavior were somehow held fixed. The second question needs a meaningful intervention and stronger assumptions; it cannot be answered simply by removing retired pumps from the file.
Research on separable effects considers settings in which treatment components can be distinguished in a way that supports more specific causal questions. That framework is useful because it forces investigators to articulate what would actually be changed. It is not a generic permission to decompose any observed association into independent mechanisms. Separable effects research
For a practical study, the protocol should name the total policy effect if that is what stakeholders need and resist adding unsupported mechanistic claims afterward. If the decomposition matters for designing a better policy, it may require additional data or a different experiment. Honest limits on interpretation are more valuable than a causal story inferred from whichever coefficient looks favorable.
The ledger can grow into a multistate model
The first-event analysis intentionally stops distinguishing later histories. A pump that fails, is repaired, and returns to service remains counted as having experienced its first failure. This is appropriate for the chosen endpoint, but it cannot answer questions about repeated breakdowns, total downtime, or the probability of being operational at a later date.
Those questions may require a multistate or recurrent-event framework. The states could include operating, failed awaiting repair, operating after repair, and permanently retired. Transitions would represent actual changes between states, with a clear time scale and observation process. The model should reflect the engineering process rather than treating every event as an absorbing failure.
A composite endpoint such as first failure or retirement is another option, but it changes the meaning of success. It measures time to either event and may be useful for certain planning questions. It does not distinguish an unwanted failure from an intentional retirement. Components should remain visible when they have different consequences.
The data architecture should therefore preserve event type, timing, and sequence even if the first report uses a simpler endpoint. Collapsing every exit into a binary failure flag can make later questions impossible to answer accurately. A careful event ledger is reusable because it preserves distinctions, not because it commits every analysis to one model.
Ambiguous event order cannot be repaired by alphabetical codes
A maintenance database may record only the day of failure and retirement. If both appear on the same date, the analyst needs a rule for determining whether failure prompted retirement or retirement occurred before any qualifying failure. Sorting event labels alphabetically would create a reproducible result with no scientific justification. The resolution rule should use the available operational evidence and be specified consistently.
Some records will remain ambiguous. The report can describe their frequency and compare plausible classifications when the distinction could affect the result. A sensitivity analysis that assigns uncertain cases to different event types is more informative than pretending that an arbitrary tie-breaking convention established the true sequence. This is especially important when the intervention itself changes how promptly events are documented.
Event misclassification can alter the apparent mechanism of a policy. If failures followed by immediate replacement are routinely recorded only as retirement, the failure curve may look artificially favorable while the retirement curve absorbs the missing events. The total first-event count could remain correct, so checking only the sum of all exits would not reveal the problem.
An adjudication sample can compare database codes with service reports, sensor histories, and replacement authorizations. Reviewers should use a consistent endpoint definition, and where feasible should not let knowledge of policy assignment influence classification. The goal is to understand how reliably the observed categories represent the events the study claims to measure.
Late entry changes who is eligible to appear
Suppose the database begins observing pumps only when they enter a monitoring contract, sometimes years after installation. Pumps that failed or retired before joining are absent. An analysis that treats the observed fleet as though every pump had been followed from installation would select survivors without acknowledging that selection.
The study must distinguish the time origin from the time observation begins. Appropriate delayed-entry methods can account for entry under their assumptions, but those assumptions deserve scrutiny when enrollment depends on a pump's condition. Alternatively, the research question may be reframed around outcomes after monitoring enrollment, with the starting population described accordingly.
The distinction affects comparisons between organizations. A fleet observed from installation and another observed only after several successful years do not offer interchangeable evidence about early reliability. Matching their calendar follow-up lengths does not align their lifecycle exposure. A clear eligibility and entry schedule prevents a misleading comparison before any competing-risks model is fitted.
Close the report with probabilities that reconcile
The final fleet report states the starting population, time origin, first-event definitions, competing events, censoring mechanisms, and follow-up horizon. It presents the cumulative incidence of each relevant event and the remaining event-free probability. In the complete toy example, those are twenty, thirty, and fifty percent, which reconcile to the original one hundred pumps.
It separately reports any cause-specific or subdistribution model results with their exact interpretation and assumptions. It explains missing follow-up and avoids calling censored pumps permanently event-free. Where a policy reduces failures by retiring more equipment, the report shows that mechanism in the outcome accounting instead of hiding it behind a favorable headline.
The essential discipline is to keep two questions separate: what can still happen to a unit, and what can researchers still observe about it? Once those questions are explicit, the choice of estimator becomes easier to justify. Without them, even a technically correct survival model can answer a question that nobody intended to ask.