Event Times, Censoring Times, And Risk Sets

Why Event Times and Risk Sets Matter

In incomplete event-time data, a record may show that an event occurred at a known time, or it may show only that the subject was observed event-free up to a certain time. Those records provide different information. Before using the data to describe event experience, an actuary must identify which subjects were still under observation and had not yet experienced the event at each event time. That group is the risk set.

The key distinction is between the time an event happens and the time observation ends. A subject whose observation ends without a recorded event is not known to have had an event at that time. The subject may still belong to the risk set just before a later event, provided observation continued at least that long.

Definition: For subject $i$, let $T_i$ be the time from a defined starting point to the event of interest, and let $C_i$ be the time from that starting point to the end of observation if the event has not already been observed. The observed time is $Y_i=\min(T_i,C_i)$. The event indicator $\Delta_i$ equals 1 if the event is observed at $Y_i$, and 0 if observation ends at $Y_i$ before the event is observed. Thus, $(Y_i,\Delta_i)$ records the observed time and whether it is an event or a right-censoring time.

These definitions distinguish the underlying times $T_i$ and $C_i$ from the information actually recorded. For a censored subject, the event time is not known from that record; the observation establishes only that the subject remained event-free up to the censoring time. In the examples below, event and censoring times do not coincide, so no tie convention is needed.

Definition: At an observed event time $t$, the risk set immediately before $t$, written $R(t)$, consists of subjects still under observation and event-free just before $t$. With right-censored records and no simultaneous event/censoring ambiguity, it is $$ R(t)=\{i:Y_i\geq t\}. $$ A subject with an event at $t$ is included: that subject was at risk immediately before the event occurred.

How to Read the Risk-Set Condition

The inequality $Y_i\geq t$ is a compact way to check two requirements. First, subject $i$ cannot have had an observed event before $t$. Second, the subject’s observation must have continued through the time just before $t$. If $Y_i<t$, the record ended earlier, either because an event occurred or because the subject was censored; either way, the subject is not in $R(t)$.

At the event time itself, an event subject is included in the risk set just before the event, then contributes an event at $t$. By contrast, a subject censored earlier is not in the risk set at a later event time. The risk set is therefore not simply “everyone whose record contains an event,” nor is it “everyone with a record in the data.” It is the set still eligible to experience the event at the time being considered.

A basic justification follows directly from the observation process. If $Y_i\geq t$, the observed record has not ended before $t$, so the subject remains under observation up to just before $t$. If $Y_i<t$, observation ended before $t$, so the subject cannot be counted as under observation just before $t$. Therefore the records satisfying $Y_i\geq t$ are exactly those that meet the risk-set definition in this setting. This is a membership rule, not by itself a claim about how censoring relates to event times.

i\in R(t)\quad\Longleftrightarrow\quad Y_i\geq t

The event indicator answers a related but different question. A subject is an event at $t$ only when $Y_i=t$ and $\Delta_i=1$. A subject can be in $R(t)$ without having an event at $t$: for example, their observed time may be later than $t$. The risk set is the pool at risk just before $t$; the event records identify which members experience the event at $t$.

Worked Example: Classifying Records at an Event Time

A hypothetical study records the following elapsed times, in months, from enrollment. Each pair gives observed time $Y_i$ and event indicator $\Delta_i$. Identify the risk set immediately before the event at month 6, and state how many subjects in it have an event at month 6.

A: $(2,1)$; B: $(4,0)$; C: $(6,1)$; D: $(7,0)$; E: $(9,1)$. These are record data only; no probability distribution is assumed.

Method. Compare each observed time with $t=6$. Include a subject if $Y_i\geq6$, then use $\Delta_i$ to identify which included subjects have an event at 6.

A has $Y_A=2<6$, so A is not included. B has $Y_B=4<6$, so B is not included either; B was censored before month 6. C has $Y_C=6$, so C is included, and $\Delta_C=1$ shows that C has the event at month 6. D has $Y_D=7\geq6$, so D is included, but D has no event at month 6 because its observed time is later. E has $Y_E=9\geq6$, so E is also included and has no event at month 6.

Therefore $R(6)=\{C,D,E\}$, and its size is $3$. Exactly one member, C, has an event at month 6.

Check. The two records ending before month 6 are excluded; the three records ending at or after month 6 are included. The event subject at month 6 is part of the risk set immediately before that event.

Common mistake avoided: Counting only subjects with $\Delta_i=1$ would give C and E, omitting D. But D is still under observation and event-free just before month 6, so D belongs in the risk set even though D is later censored.

Worked Examples

Worked Example: A Censored Record Before the Event Time

In a hypothetical follow-up study, the observed times are measured in weeks. The records are P: $(1,1)$, Q: $(3,0)$, R: $(5,1)$, S: $(8,0)$, and U: $(10,1)$. Find the risk set immediately before the event at week 5, its size, and the number of events at week 5.

Method. Apply $Y_i\geq5$ to each record. Do not treat a censoring time as an event time.

P has $Y_P=1<5$, so P is excluded. Q has $Y_Q=3<5$, so Q is also excluded. Although Q has a recorded time, its indicator is 0: observation ended at week 3 without an observed event. R has $Y_R=5$, so R is included; $\Delta_R=1$ identifies its event at week 5. S has $Y_S=8\geq5$, so S is included. U has $Y_U=10\geq5$, so U is included.

Thus $R(5)=\{R,S,U\}$, with size $3$. Among those three members, R is the one with an event at week 5, so the number of events at that time is $1$.

Check. The included observed times are $5,8,10$, all of which satisfy $Y_i\geq5$. The earlier censoring time, week 3, does not provide evidence of an event at week 3.

Common mistake avoided: Treating Q’s censoring time as an event would wrongly label Q as an event at week 3. Its record says only that Q was observed without a recorded event until week 3.

Worked Example: Separating Risk-Set Membership from Event Status

A hypothetical device study reports observed times in days: device 1, $(3,1)$; device 2, $(6,0)$; device 3, $(8,1)$; device 4, $(11,0)$; and device 5, $(14,1)$. For the event at day 8, determine the risk set immediately before day 8, identify which records in that set have an event at day 8, and calculate the proportion of the risk set experiencing an event then.

Method. First form the membership set using $Y_i\geq8$. Then identify events at day 8 using both $Y_i=8$ and $\Delta_i=1$.

Device 1 has $Y_1=3<8$, so it is excluded. Device 2 has $Y_2=6<8$, so it is also excluded; its time is a censoring time, but that does not keep it under observation until day 8. Device 3 has $Y_3=8$, so it is included, and $\Delta_3=1$ means it has the event at day 8. Device 4 has $Y_4=11\geq8$, so it is included but has no event at day 8. Device 5 has $Y_5=14\geq8$, so it is also included but has no event at day 8.

The risk set is $R(8)=\{3,4,5\}$, so its size is $3$. The number of events at day 8 is $1$. The event count divided by the risk-set size is $$ \frac{1}{3}=0.3333 $$ to four decimal places.

This proportion is the fraction of the risk set with an event at that time in these records. The calculation does not turn the censored observation for device 2 into an event, and it does not imply that every member of the risk set experienced an event.

Check. The risk-set members have observed times $8,11,14$, all at least 8. Of these, only device 3 has both observed time 8 and event indicator 1.

Common mistake avoided: Using all five devices as the denominator ignores that devices 1 and 2 were no longer under observation just before day 8. The denominator for this risk-set proportion is $3$, not $5$.

A Reliable Membership Procedure

For a particular event time, use the same sequence every time. This prevents confusion between “still being followed” and “having an event.” It also makes clear that risk-set membership is determined by comparing observed times with the specified event time, while event status is determined from the event indicator.

Specify the time.
Identify the event time $t$ and the time unit used in the records.

Compare observed times.
For each subject, include the record in $R(t)$ exactly when $Y_i\geq t$.

Identify events at $t$.
Among the risk-set members, a record with $Y_i=t$ and $\Delta_i=1$ represents an event at $t$.

Keep censoring distinct.
A record with $\Delta_i=0$ ends in censoring, not an observed event. If its observed time is before $t$, it is not in $R(t)$.

Common Exam Traps

Excluding the event subject. The risk set is defined immediately before the event. A subject whose event occurs at $t$ was at risk just before $t$, so include that subject in $R(t)$.
Counting only event records. Subjects with $Y_i>t$ can belong to $R(t)$, even when their eventual record is censored or their event occurs later.
Calling censoring an event. When $\Delta_i=0$, the observed time marks the end of observation without a recorded event. It does not establish that an event happened then.
Using $Y_i>t$ instead of $Y_i\geq t$. That strict inequality would wrongly omit a subject whose event is exactly at $t$. The risk set is assessed just before the event, so that subject is included.
Ignoring the time unit. A comparison is meaningful only when $Y_i$ and $t$ use the same unit. Convert, for example, months to years before comparing if the records and requested event time use different units.

These membership rules presume the stated right-censored record structure. If an event and a censoring time coincide, whether the event is considered observed depends on the recording convention. The examples here avoid simultaneous event and censoring times; tied-time handling requires an explicit convention.

Key takeaway: For right-censored records without simultaneous event/censoring ambiguity, a subject belongs to the risk set immediately before event time $t$ exactly when $Y_i\geq t$. Use $\Delta_i$ separately to determine whether that subject has an observed event at $t$.

Check Your Understanding

For each problem, identify risk-set membership from the observed times and event indicators. Treat all times as being in the same unit, and assume there are no tied event and censoring times.

A hypothetical follow-up has records A: $(2,1)$, B: $(3,0)$, C: $(5,1)$, and D: $(7,0)$. Find the risk set immediately before the event at time 5 and the number of events at time 5. (Answer: $R(5)=\{C,D\}$; 1 event.)
Records are 1: $(1,0)$, 2: $(4,1)$, 3: $(8,0)$, 4: $(9,1)$, and 5: $(12,0)$. Find the risk set immediately before time 9 and state which member has an event at 9. (Answer: $R(9)=\{4,5\}$; subject 4 has the event.)
A study records P: $(3,1)$, Q: $(6,0)$, R: $(10,1)$, S: $(13,0)$, and T: $(15,1)$. For the event at time 10, give the risk-set size and the number of members who are censored after time 10. (Answer: risk-set size $3$, with 1 member censored after time 10, subject S.)
At time 7, the records are A: $(5,0)$, B: $(7,1)$, C: $(8,0)$, D: $(11,1)$, and E: $(12,0)$. Find the risk set, its size, and the event count at time 7. (Answer: $R(7)=\{B,C,D,E\}$; size $4$; event count $1$.)

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