Event Histories as Counting Processes

From an Event Record to a Process

A time-to-event record is often stored as two numbers: a follow-up time and an indicator of whether the event was observed. A counting-process representation expresses the same information as a function of time. Before the event occurs, the function is zero; once an observed event occurs, it jumps to one. A censored observation does not produce an event jump. This time-indexed representation makes the event history explicit, while preserving the distinction between an event and the end of observation.

The setup here is one subject, one terminal event of interest, and observation beginning at a fixed time origin, denoted 0. Let \(T\) be the subject’s event time, which may be unobserved, and let \(C\) be the time at which observation would end because of censoring. The recorded follow-up ends at the earlier of these times. This representation describes the data; by itself, it does not assert that censoring is independent of the event time.

With the convention that an event takes priority if event and censoring times tie, \(\delta=1\) when \(T=C\). Other tie conventions are possible, but one must be specified if ties can occur. Under this convention, \(N(t)\) records whether an event has been observed by time \(t\), and \(N^C(t)\) records whether censoring has ended follow-up by time \(t\). Each process stays at zero until its relevant observed endpoint, then stays at one. In this single-terminal-event setup, neither process can jump more than once.

Right-continuity means that the value at a time \(t\) agrees with the values immediately to its right: \(N(t)=\lim_{s\downarrow t}N(s)\). Thus, if the event occurs at time \(u\), it is counted at \(u\), not only after \(u\). The left-hand value \(N(u-)\) describes the value immediately before \(u\). The jump at \(u\), written \(\Delta N(u)=N(u)-N(u-)\), is 1 for an observed event and 0 otherwise. The notation \(\Delta N(u)\) denotes a process jump; it is distinct from the event indicator \(\delta\).

How Event and Censoring Times Appear in Jumps

The formula gives the path directly. If an event is observed at \(Y=u\), then \(\delta=1\), so \(N(t)=0\) for \(t<u\) and \(N(t)=1\) for \(t\geq u\). The event process has a jump of size 1 at \(u\). The censoring process remains zero. If follow-up instead ends in censoring at \(Y=c\), then \(\delta=0\): \(N(t)=0\) throughout observed follow-up, including at \(c\), while \(N^C(t)\) jumps from 0 to 1 at \(c\).

A useful identity follows from these definitions. Exactly one observed endpoint occurs by time \(t\) if follow-up has ended by \(t\), and neither process records an endpoint before then. Therefore,

This identity is a check on the encoding: for one subject, the two processes cannot both jump under the stated tie convention. It does not tell us whether the endpoint was an event or censoring without examining the separate processes.

Why the Paths Are Right-Continuous

The path property can be checked directly rather than assumed. For an observed event at \(u=Y\), the process is \(N(t)=I(t\geq u)\). For any \(t<u\), there is a sufficiently small right-hand neighborhood of \(t\) that remains before \(u\), so the process is zero there. At \(t=u\), its value and all values immediately to the right are one. For \(t>u\), its value and nearby right-hand values are also one. Hence \(N(t)=\lim_{s\downarrow t}N(s)\) at every \(t\). The same argument applies to \(N^C(t)=I(t\geq c)\) for an observed censoring time \(c\); an identically zero process is right-continuous as well.

The path is a step function because the event record changes only at its observed endpoint. Its jump size is obtained by subtracting the value just before the endpoint from the value at the endpoint. For an observed event at \(u\), \(N(u-)=0\) and \(N(u)=1\), giving \(\Delta N(u)=1\). For censoring at \(c\), \(N(c-)=N(c)=0\), giving \(\Delta N(c)=0\), while \(N^C(c-)=0\) and \(N^C(c)=1\). These statements encode the endpoint type without treating censoring as a failure.

Scope, Checks, and Common Errors

The process representation is useful because it preserves the timing and type of an observed endpoint. A pair \((Y,\delta)\) is a compact record; \(N(t)\) and \(N^C(t)\) make explicit when the event or censoring endpoint occurs. This form is convenient when later methods refer to event counts over time. It does not, on its own, supply a probability model, an estimate, or a claim about the relationship between event and censoring times.

The definition assumes exact endpoint times and observation from the chosen time origin. With interval censoring, the event is known only to lie within an interval, so a single exact event-time jump cannot be identified from the record. With delayed entry, the history is not observed from time 0 and that feature must be represented. If event and censoring times can tie, the convention must be stated; changing it can change which process jumps at the tied time.

A frequent error is to put a jump in \(N(t)\) at every end of follow-up. That treats censoring as if the event had occurred. The correction is to use the event indicator: \(N(t)=I(Y\leq t,\delta=1)\). A second error is to infer an event time equal to the censoring time for a censored subject. Censoring says only that no event was observed by the censoring time; the event time remains unknown. Finally, writing a process that jumps only for times strictly greater than the event time would fail the specified right-continuous convention: the event is included at its time \(u\), through \(I(t\geq u)\).