Competing exits: estimate the event that actually interests you
If a company can leave a study through either acquisition or failure, estimating failure probability requires accounting for both exits. Acquisition removes a company from the future failure risk set in the defined study, but it is not merely an unknown future failure time. Treating it as ordinary censoring and reporting one minus a failure-only survival curve answers a different question.
The deliverable is a probability table for mutually exclusive states that sums to one: still under the initial state, acquired, and failed. Define the observational unit and its terminal events carefully. A legal entity continuing after an asset sale may require a different model from a company whose acquisition permanently ends the study’s standalone-company endpoint.
Specify absorbing states before counting exits
Assume a hypothetical cohort of ten standalone companies, followed completely through year two. Every company begins active at time zero. Under this study definition, the first qualifying acquisition or failure is an absorbing endpoint: once either occurs, the company cannot later enter the other endpoint in this analysis.
This is a definition of the research process, not a claim that an acquired business can never encounter future distress. If post-acquisition failure is the intended outcome, acquisition is an intermediate state and the analysis needs additional transitions. Do not change the state diagram after seeing which endpoint gives a preferable result.
Therneau, Crowson and Atkinson’s multi-state vignette, sections 1 and 2.1, distinguishes competing terminal states and describes the Aalen–Johansen probability update. Its section on cause-specific hazards also explains why separate hazard estimation is different from computing probabilities in states. The company arithmetic below is an original illustration.
Reconcile a complete ten-company history
At year one, two companies are acquired and none fail. At year two, two of the eight remaining companies fail. The other six remain active at the reporting horizon. There is no loss to follow-up or other censoring before year two.
| Time | At risk just before time | Acquisitions | Failures | Remaining active after time |
|---|---|---|---|---|
| 1 | 10 | 2 | 0 | 8 |
| 2 | 8 | 0 | 2 | 6 |
Because follow-up is complete, the year-two state probabilities are simply acquired 2/10=0.20, failed 2/10=0.20 and active 6/10=0.60. They add to one. Use this elementary tabulation as a check on any more general estimator applied to the same complete data.
The failure fraction is not 2/8 when the target is the probability of failure by year two for the original cohort. Two eighths is the conditional proportion failing at the second event time among companies still active immediately beforehand. The cumulative probability must also include the probability of reaching that risk set.
Reproduce the Aalen–Johansen update
Let S be the probability of remaining in the initial active state, F_A the cumulative acquisition probability and F_F the cumulative failure probability. Initially (S,F_A,F_F)=(1,0,0). At an event time with risk set Y and cause-specific event counts d_A and d_F, use the state immediately before the time:
increment F_A = S_before × d_A/Y;
increment F_F = S_before × d_F/Y;
S_after = S_before × [1−(d_A+d_F)/Y].
At year one, the acquisition increment is 1×2/10=0.20 and S becomes 0.80. At year two, the failure increment is 0.80×2/8=0.20 and S becomes 0.60. The acquisition probability remains 0.20. Every step conserves total probability.
The important factor is S for avoiding all competing endpoints up to that time, not a survival curve that considers only failures. The cumulative incidence of failure combines its cause-specific occurrence with the chance of still being eligible to fail in the actual observed process.
Diagnose the failure-only Kaplan–Meier result
If the two acquisitions are coded as ordinary censoring in a failure-only analysis, the curve has no failure drop at year one. At year two it drops by 2/8, giving failure-only survival 0.75 and 1−KM=0.25.
That 0.25 disagrees with the complete-data failure proportion 0.20. It should not be relabeled “the actual chance of failure by year two.” It is a net failure-style quantity produced by treating competing exits as censoring. Interpreting it as the probability in a world where acquisitions were eliminated requires additional causal assumptions; merely recoding the event indicator does not identify such an intervention.
This does not mean that coding other exits as censored is always incorrect for every task. A cause-specific hazard model uses the active risk set and can estimate the failure transition rate while removing companies after another endpoint. The mistake is converting that single-cause hazard analysis into an actual cumulative failure probability without considering the other transition hazards.
Two populations can have similar failure hazards among active companies but different failure cumulative incidence because acquisitions remove companies at different rates. A cause-specific hazard comparison and a probability comparison therefore need distinct target labels and interpretations.
Handle tied events and unknown causes explicitly
For a separate boundary example, suppose two acquisitions and two failures all occur at the same recorded time among the original ten. Both cause increments use the same pre-time risk set 10 and S_before1. The result is F_A0.20, F_F0.20 and S0.60.
Do not first reduce S for acquisition and then apply the failure increment using the old denominator 10. That inconsistent update would give failure 0.16. Either correctly resolve a known actual ordering with its corresponding risk sets, or treat the recorded tied events jointly under the stated method. Arbitrary within-day ordering is not new evidence.
An unknown exit cause is another mechanism. If the exact terminal exit is verified but its type is unresolved, it is not simply “no event.” Preserve an unknown absorbing category or another explicitly justified missing-cause analysis. Dropping it can make known-cause probabilities appear better determined than they are. A later source resolving the cause should reassign that probability with a documented evidence change.
Separate observation loss from competing endpoints
With genuine right censoring, an active company’s future endpoint is unknown because observation ended. A suitable independent-censoring condition is needed for an estimated probability curve to represent the target population. If firms disappear from coverage precisely because distress is escalating, using an Aalen–Johansen function does not repair that observation bias automatically.
The complete ten-company example avoids this issue only to make the state accounting transparent. A real study must retain losses, censoring dates and reasons. More complex pathways such as recovery, repeated incidents or post-acquisition failure need their own state diagram and cannot be compressed into the original three states without changing the question.
| Review step | Evidence to preserve | Response to a conflict |
|---|---|---|
| Define states | company unit, endpoints and absorbing rule | revise the target before fitting |
| Establish chronology | effective event dates and time resolution | keep unknown ordering explicit |
| Build risk sets | active companies just before each time | remove prior terminal exits consistently |
| Update probabilities | all cause counts and S_before | check each step sums to one |
| Handle missingness | censoring versus unknown exit cause | avoid converting either into a known cause |
| Interpret output | cumulative incidence or cause-specific hazard | do not interchange the targets |
If an acquisition notice and a failure notice describe the same restructuring, investigate the prespecified classification rule and legal timeline. Counting both as separate terminal first events can exceed the cohort size or create impossible state histories. Keep the source records even when one classification is ultimately selected.
Update the analysis when event dates, causes, entity boundaries or observation-loss information change. Keep a versioned state table and the separate risk-set counts. The central check is straightforward: the probability assigned to actual failure must share the cohort’s total probability with all other states that can occur first.
Show the effect of unresolved cause information
Extend the complete-data illustration with a later year-three exit among the six active companies, verified as terminal but with its cause unresolved. The unknown-state increment is 0.6×1/6=0.1, leaving active probability 0.5, acquisition 0.2, known failure 0.2 and unknown exit 0.1. The four categories still sum to one.
If the unknown cause can only be acquisition or failure under the prespecified definition, the actual failure probability lies between 0.2 and 0.3 until the cause is resolved. This is a logical cause-allocation range, not a sampling confidence interval. Calling the exit ordinary censoring would hide already known terminal information; automatically assigning failure would overstate what the records establish. When a later legal document resolves the cause, move the 0.1 to its supported category and preserve the source that changed the allocation.
Check whether an acquisition source records an announcement or a completed transaction. An announced deal that later fails may not satisfy the prespecified absorbing endpoint. Save the effective-status evidence before moving a company out of the active risk set.
FAQ
Can acquisition always be treated as ordinary censoring?
Not when acquisition is a competing terminal endpoint and the target is actual failure cumulative incidence. Its transition must enter the probability calculation.
Is a cause-specific hazard useless for studying failure?
No. It answers a different rate question. To obtain actual cumulative probabilities, the competing transition processes must also be considered.