Handling Simultaneous Or Tied Event Types

Why the Event-Type Definition Matters

A competing-risks analysis requires a rule for assigning an event type. The rule is straightforward when one first event has one clearly recorded cause. It needs more care when one person has more than one cause recorded at the same time. The assigned type determines which event count enters each cause-specific hazard, and therefore what each hazard describes.

First distinguish two situations. A tied recorded time across people means two or more people have event times recorded as equal, perhaps because times were rounded to the same day. Each person may still have one unambiguous cause. Multiple causes recorded for one person means the record does not identify a single cause for that person’s event, or indicates that two causes occurred together. A tie across people does not, by itself, create a joint event type.

Definition: In the usual first-event competing-risks setup, \(T\) is the time to the first event and \(J\) is its event type, with \(J\) taking one mutually exclusive value for each defined cause. If two causes are recorded together for one first event, the analyst must specify how that record maps to \(J\): for example, a separate joint category, a documented priority rule, or a different model for joint transitions.

For cause \(k\), the cause-specific hazard is the instantaneous rate of a type-\(k\) event among people who have not yet had any event. In continuous time, its definition is

\lambda_k(t)=\lim_{\Delta t\to 0} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t)}{\Delta t}.

This definition presumes that the event categories are well specified. For data recorded in discrete intervals, the corresponding quantity is the conditional probability of a type-\(k\) event in the interval, given no event before the interval. It is a probability per interval, not automatically a continuous-time rate. The coding rule determines which observed events count as type \(k\) in either case.

Options for Coding a Jointly Recorded Event

If two causes are recorded for the same first event, there is no universally correct way to turn that record into a single-cause label. The appropriate choice depends on the scientific question and what the data establish. Common options include:

Use a distinct joint category. Define a type such as “A and B together.” The type-\(A\) hazard then concerns A-only events, not events recorded as A and B together.
Apply a prespecified priority rule. Assign the record to one cause according to a clinical or operational rule. The resulting hazard describes events classified to that cause under that rule.
Use a model that represents joint or multiple transitions. This may be appropriate when the scientific target is a transition involving multiple conditions rather than a single first-event cause.

A priority rule should not be presented as if it revealed which cause “really” occurred first unless the available information supports that ordering. Likewise, splitting one joint record fractionally between causes is not the ordinary observed cause-specific hazard: it replaces the observed category with a weighted assignment. Such weighting requires its own justification and changes the interpretation.

Worked Example: A Joint Category Versus a Priority Rule

Hypothetical data. At the start of a one-day interval, 100 people have not yet had an event. During that interval, 4 have a type-A event only, 3 have a type-B event only, and 2 have A and B recorded together as their first event. Assume these categories cover all events in the interval and are recorded consistently.

With a distinct joint category, the three conditional event probabilities are:

\widehat p_A=\frac{4}{100}=0.04,\qquad \widehat p_B=\frac{3}{100}=0.03,\qquad \widehat p_{AB}=\frac{2}{100}=0.02.

These are probabilities during this one-day interval among people event-free at its start. Under this coding, \(0.04\) describes A-only events; it does not include the two joint events.

Now apply a rule assigning every A-and-B record to A. Then the type-A count is \(4+2=6\), and the type-B count remains 3:

\widehat p_A=\frac{6}{100}=0.06,\qquad \widehat p_B=\frac{3}{100}=0.03.

If instead the rule assigns each joint record to B, the estimates are \(4/100=0.04\) for A and \((3+2)/100=0.05\) for B. In all three codings, the probability of any event is \(9/100=0.09\). What changes is how events are divided among types and thus what the type-specific hazards mean. The one-day probabilities are not continuous-time rates.

Why the All-Event Hazard Can Stay the Same

A useful check is whether recoding changes the total number of people with any first event. If every record remains classified as an event, moving a joint event between causes can change cause-specific hazards without changing the all-event hazard. The reason is a partition of the event set: each event belongs to exactly one category under a chosen coding.

\widehat p_{\mathrm{any}} =\frac{\text{number with any event in the interval}}{\text{number event-free at its start}} =\sum_k \widehat p_k.

In the example, the distinct-category probabilities sum to \(0.04+0.03+0.02=0.09\); under the A-priority rule, the cause-specific probabilities sum to \(0.06+0.03=0.09\). This identity is a consequence of the categories forming a complete, non-overlapping partition. It does not mean the type-specific interpretations are interchangeable.

The same distinction matters for the subdistribution hazard, whose risk-set construction is tied to the cumulative incidence of a specified event type. In discrete intervals, if \(F_k\) is the probability of a type-\(k\) event by an interval boundary, the interval subdistribution hazard can be written as the increment in \(F_k\) divided by the probability that type \(k\) has not occurred previously. Changing which events count as type \(k\) can therefore change both the increment and the denominator.

Worked Example: Coding Changes a Later Subdistribution Hazard

Hypothetical data. In a cohort of 100 people, the first interval contains 5 A-only events, 4 B-only events, and 1 jointly recorded A-and-B event. In the second interval, 3 additional A-only events occur. Assume no other events and no loss of records in these intervals. Compare the second-interval subdistribution hazard for A under two coding rules.

With a separate joint category, the first-interval A cumulative incidence is \(F_A=5/100=0.05\). The second-interval increment is \(3/100=0.03\), using the original cohort denominator for the cumulative incidence. Thus:

\widehat h_A^* =\frac{0.03}{1-0.05} =\frac{0.03}{0.95} =0.03158.

Under an A-priority rule, the joint event is also counted as A, so the first-interval A cumulative incidence is \(6/100=0.06\). The second-interval A increment remains \(3/100=0.03\), and the subdistribution hazard becomes:

\widehat h_A^* =\frac{0.03}{1-0.06} =\frac{0.03}{0.94} =0.03191.

Both values are conditional event probabilities on the subdistribution-hazard scale for the second interval, given that A has not occurred previously; they are not probabilities that A is the eventual event type. The difference arises because the coding rule changes whether the joint event counts as a prior A event. The numerical comparison is meaningful only alongside the stated event definition.

Tied Recorded Times Are Not Necessarily Joint Events

Suppose two people have events recorded on the same calendar day, one labelled A and the other B. If each person has one clear cause, there are two events with tied recorded times, not one person with a joint event. The cause-specific event counts can be tabulated by type at that recorded time. The tie may still matter for methods that require event ordering within a time interval, but it does not justify merging the two people’s event types.

Conversely, a single person’s record may list two causes on the same date because only dates, not times, were captured. That record does not establish whether the causes were truly simultaneous or whether one preceded the other within the day. If the scientific definition requires the first cause, the available time resolution may not identify it. A prespecified interval-based definition or a joint category may be more faithful to the data than inventing an ordering.

Worked Example: Two People With the Same Recorded Event Time

Hypothetical data. At the start of a recorded day, 50 people are event-free. On that day, two different people have type-A events and one other person has a type-B event. The timestamps are recorded only to the day, but each person has one unambiguous cause. Assume these are the only events that day.

The observed one-day type-specific event probabilities are \(2/50=0.04\) for A and \(1/50=0.02\) for B. The probability of any event is \(3/50=0.06\). The shared recorded date does not create an A-and-B event for any individual. If the three people’s event times had been recorded more finely, their within-day ordering might be known; the date-only data do not provide that detail, but no within-person cause ambiguity is present here.

If, instead, one of those people had both A and B listed on the same date, the data would require an explicit rule for that person’s event type. The fact that other people also had events that day would not resolve the individual’s ambiguity.

Assumptions, Diagnostics, and Limits

Before estimating or interpreting a cause-specific hazard with tied or joint records, check the data and the event definition:

One record per first event. Confirm whether multiple labels refer to one person’s single first event or to events in different people. Do not treat a tied calendar date across people as a joint event.
Time resolution. Establish whether equal times are exact or arise from rounding, date-only recording, or grouped follow-up. Do not infer exact simultaneity from a coarse timestamp.
Mutual exclusivity. Verify that the chosen categories assign each first event to exactly one category. Overlapping categories make the sum of type-specific event probabilities count some events more than once.
Rule consistency. Apply the same joint-event or priority rule throughout the data and across groups being compared. If the rule changes, a hazard comparison may reflect coding as well as event patterns.
Scientific target. State whether the target is A-only events, A including joint events, or a joint A-and-B event. The labels alone are not enough to determine that target.

A common mistake is to divide a joint record equally between two causes and then call the resulting numbers observed cause-specific hazards. The arithmetic may preserve the total event count, but neither assigned fraction is an observed event of its stated type. Unless a justified allocation model is being used, report the joint category separately or use a clearly described classification rule.

A priority rule can make data analyzable, but it does not recover unknown ordering. A joint category preserves the recorded co-occurrence but may be sparse, making its hazard imprecise. If times are coarsened, treating a whole interval as one instant can also conceal within-interval order. No coding choice can restore information that was not recorded; conclusions should be limited to the event definition the data support.

Key takeaway: A hazard is defined for a particular event type and risk set. When causes are simultaneous or recorded together, define that type before counting events. A joint category, a priority rule, and a weighted allocation produce different estimands and must not be described as equivalent.

Check Your Understanding

Answer each question, then compare with the solutions below.

At a time point, two different people have events recorded on the same day, one A and one B. Does this alone define a joint event for either person?
In a one-interval cohort of 80 event-free people, 3 have A-only events, 2 have B-only events, and 1 has both causes recorded for one first event. What is the one-interval probability of any event, and what are the A and B probabilities if the joint record is assigned to A?
Why does a joint-event category change the interpretation of the type-A hazard compared with assigning joint events to A?
What can be concluded if a person has A and B recorded on the same date but the recording system captures no time of day?

Solutions. (1) No. The events belong to different people; equal recorded times across people do not by themselves create a within-person joint event. (2) Any event: \((3+2+1)/80=0.075\). With the joint record assigned to A, the A probability is \((3+1)/80=0.05\), and the B probability is \(2/80=0.025\). Their sum is \(0.075\). (3) Under the joint-category definition, type A includes A-only events; under A-priority coding, it also includes records with both causes. Those hazards refer to different event sets. (4) The date does not establish whether the causes were exactly simultaneous or which came first. The analysis needs a stated joint-event or classification rule consistent with the available resolution.