Observed First Events and Hidden Failure Times
In competing-risks data, each individual can experience at most one first event. If that event occurs, it prevents the other causes from being observed as first events for that individual. This makes a basic distinction important: the data record which cause happened first and when, but they do not record when another cause would have happened if the first event had not occurred.
A latent failure-time formulation makes the distinction explicit. For two causes, let \(T_1\) and \(T_2\) denote the times to each cause in a hypothetical setting where the other cause does not preclude it. The observed time and cause are \(T=\min(T_1,T_2)\) and \(J=\arg\min(T_1,T_2)\), assuming no tie. With more than two causes, the observed time is the minimum of all latent times, and \(J\) names the cause attaining that minimum. The latent time for the losing cause is not part of the observed record.
For a continuous event time, the cause-specific hazard for cause \(k\) is the instantaneous rate of a cause-\(k\) first event among individuals still event-free just before time \(t\):
This hazard concerns observable first events and the event-free risk set. It does not require independence between latent failure times. But the fact that it can be defined and estimated from first-event data does not mean those data reveal the full distribution of each \(T_k\), or their dependence.
Why the Latent Times Are Not Identified
The observation rule maps a pair \((T_1,T_2)\) to \((T,J)\). If cause 1 happens first at time \(t\), then the record tells us \(T_1=t\) and \(T_2>t\). It does not distinguish \(T_2=t+0.1\) from \(T_2=100\). Both pairs produce the same observed first event. The same reasoning applies when cause 2 happens first.
This is a problem of identifiability. A latent quantity is identified by the observed data only if different possible values of that quantity cannot yield the same distribution of observed records. Here, many latent-time distributions can yield exactly the same distribution of \((T,J)\). Consequently, observing more individuals from the same competing-risks process can estimate the first-event distribution more precisely, but cannot by itself resolve which of those latent distributions generated it.
Worked Example: Two Latent Scenarios, the Same First Events
Consider a hypothetical population of two equally weighted individuals. In Scenario A, their latent time pairs \((T_1,T_2)\), in years, are \((1,2)\) and \((2,1)\). In Scenario B, the pairs are \((1,100)\) and \((100,1)\). Assume exact event times, no censoring, and no ties.
In both scenarios, the first individual has a cause-1 event at year 1, because \(T_1<T_2\). The second has a cause-2 event at year 1, because \(T_2<T_1\). Thus each scenario produces exactly the same observed records: one cause-1 first event at year 1 and one cause-2 first event at year 1.
Yet the latent time to cause 1 has values 1 and 2 in Scenario A, giving a mean of \((1+2)/2=1.5\) years. In Scenario B it has values 1 and 100, giving a mean of \((1+100)/2=50.5\) years. The same calculation applies to cause 2. The observed data cannot distinguish these sharply different latent means.
This is not sampling error: the example gives identical observed records, even before considering uncertainty from a finite sample. It demonstrates structural nonidentifiability of the latent quantities from first-event records alone.
Cause-Specific Hazards Describe the Observed Process
Cause-specific hazards summarize the rates of observed first events among those who remain event-free. They are useful for describing the event process, and their definition does not depend on treating the latent times as independent. But they do not supply the missing times for individuals who have already experienced a different cause.
Worked Example: Identical Cause-Specific Interval Probabilities
Suppose a hypothetical cohort of four individuals is followed without censoring. In the first year, one has a cause-1 event and one a cause-2 event. The remaining two are event-free at year 1; in the second year, one has a cause-1 event and one a cause-2 event. Use year-long intervals, and assume events are recorded at the end of the relevant interval.
For illustration, the discrete cause-specific interval probability is the probability of a cause-\(k\) event in an interval, conditional on being event-free at its start. In the first year, all four are at risk, so the observed probabilities are \(1/4=0.25\) for cause 1 and \(1/4=0.25\) for cause 2. In the second year, two are at risk, so each cause has probability \(1/2=0.50\) in that interval. These are conditional event probabilities for the specified intervals, not continuous-time hazard rates.
Now assign latent times to the same records in two ways. For each cause-1 event at time \(t\), set \(T_1=t\); for each cause-2 event, set \(T_2=t\). In one assignment set the unobserved competing time to \(t+1\). In another, set it to \(t+100\). In each assignment the competing time is later than the observed event, so the first event, risk sets, and interval probabilities are unchanged.
Therefore both assignments yield the same cause-specific interval probabilities: 0.25 for each cause in year 1 and 0.50 for each cause in year 2. But the latent times after the observed first events differ substantially. The probabilities describe how the observed risk set experiences first events; they do not identify the hidden latent times.
In continuous time, the same logic applies to \(\lambda_k(t)\). It is determined by the rates of observed cause-\(k\) events among those still event-free. A cause-specific hazard is not the hazard of \(T_k\) in the entire original population regardless of other events: after a competing first event, that individual is no longer in the observed event-free risk set. The hazard also does not imply a distribution for the time that would follow if the competing event were somehow removed.
A Direct Check of the Information in the Record
The observation rule itself gives a short justification for the limits of what can be learned. Suppose a cause-1 event is observed at time \(t\). Under the two-cause, no-tie setup, the record implies
Every value of \(T_2\) strictly greater than \(t\) is consistent with that same record. The mapping from latent times to observed data therefore discards information about the exact losing time. Repeating this observation across individuals can reveal patterns in first-event times and causes, but cannot recover information that the observation rule never records. This reasoning applies symmetrically to an observed cause-2 event.
For an individual, consider the times assigned to each possible cause, such as \((T_1,T_2)\).
Record the minimum time and the cause attaining it. If cause 1 wins, this fixes \(T_1\) and places only a lower bound on \(T_2\).
Change \(T_2\) to another value above \(T_1\). The observed time and cause stay the same, so the exact losing time is not identified.
If latent times can tie, the observation rule also needs a stated tie-handling convention. In many continuous-time models ties have probability zero. Neither convention changes the core point: observing only the first event does not reveal the time of a cause that did not occur first.
What Extra Assumptions Can and Cannot Do
A model may impose a distribution for the latent times, a particular dependence structure, or independence between \(T_1\) and \(T_2\). Such assumptions can yield predictions about unobserved times, but those predictions are not identified by the competing-risks data alone. They depend on the model restrictions. An independence assumption, for example, is not a consequence of observing first events and is not needed to define an observed cause-specific hazard.
Worked Example: A Hypothetical Time After a Competing Event
Suppose two hypothetical individuals each have a cause-1 first event: one at year 1 and one at year 2. In a latent-time scenario, assign their unobserved cause-2 times as years 2 and 4. Their mean latent time to cause 2 is \((2+4)/2=3\) years. In another scenario, keep the observed cause-1 times at years 1 and 2, but assign latent cause-2 times of years 20 and 40. Their mean latent time to cause 2 is \((20+40)/2=30\) years.
Both scenarios produce exactly the same observed cause-1 events: their cause-2 times exceed the corresponding cause-1 times. Thus the observed records do not determine the mean time to cause 2 for these individuals. If the question is instead what would happen under an intervention preventing cause 1, interpreting either latent assignment as the intervention outcome requires further assumptions about what that intervention means and how it affects the process.
This distinction matters when translating a descriptive analysis into a scientific claim. Cause-specific hazards answer questions about the observed first-event process among those still event-free. They do not, without additional causal assumptions, answer what a person's time to another cause would have been after a competing event was prevented. Other summaries, such as the probability of observing a particular cause by a given time, also describe observable outcomes; they do not reveal the unobserved losing time.
Diagnostics, Alternatives, and Limitations
There is no diagnostic plot or goodness-of-fit test that can verify the true distribution of latent times solely from first-event records. A model for latent times may fit the observed event times and causes well while making incorrect or untestable claims about the unobserved portions. Useful checks therefore focus on the observable implications and the assumptions:
- Check that the model reproduces the observed first-event times and causes, not just its latent-time assumptions.
- Compare conclusions across plausible latent dependence or distribution assumptions when the scientific question depends on hidden times.
- Label latent-time conclusions as assumption-dependent, and distinguish them from estimates of observed first-event quantities.
- Check whether censoring, ties, or an ill-defined hypothetical intervention introduces further complications; these do not resolve the basic nonidentifiability.
The main alternative is to keep the estimand tied to what was observed. Cause-specific hazards describe instantaneous first-event rates among the event-free. Cumulative incidence describes the probability of an observed cause-specific event by time \(t\). These answer different descriptive questions, but neither identifies the latent time to a cause that did not occur first. A latent failure-time model can be useful if its assumptions are scientifically defensible, but its hidden-time results should be presented as conditional on those assumptions rather than as direct empirical findings.
A common mistake is to reason, “The cause-1 hazard is known, so the time to cause 1 is known even for people who had cause 2 first.” The correction is that the hazard describes observed cause-1 events in the event-free risk set. For a person with a cause-2 first event, the observed record supplies no cause-1 event time; a claim about it must come from additional assumptions or a clearly specified intervention model.
Check Your Understanding
Try each question, then compare your reasoning with the solutions.
- If an individual has a cause-1 event at year 3, what does the record tell you about \(T_1\) and \(T_2\), assuming no ties?
- Two latent-time models produce the same distribution of observed first-event times and causes but different distributions of \(T_2\). Is \(T_2\)'s distribution identified by those observed data?
- Does estimating a cause-specific hazard require independence between latent failure times? Does it identify the time to a competing cause after that cause has not occurred first?
- A proposed model reports a mean latent time to cause 2 after a cause-1 event. What should be checked before interpreting that mean as an empirical result?
Solutions. (1) The record gives \(T_1=3\) years and \(T_2>3\) years; it does not give the exact value of \(T_2\). (2) No. Distinct latent distributions yielding the same observed distribution demonstrate nonidentifiability. (3) No independence assumption is needed to define the cause-specific hazard from observed first events. The hazard does not identify the hidden time to the competing cause. (4) Check the latent-time distribution and dependence assumptions, and, if the claim is about preventing cause 1, the assumptions that connect the latent time to that intervention. The mean should be described as model-dependent unless those assumptions are justified.