Introduction
A risk set is a count of the individuals still under observation and eligible to experience the event at a particular time. To update it correctly, keep track of when each record leaves the set. An individual whose event is observed leaves after that event; an individual whose record is right-censored leaves after the censoring time, even though the event itself was not observed.
The distinction between just before and just after a time matters. A record censored at time \(t\) is still in the risk set immediately before \(t\), because it was observed up to that time. It is not in the risk set immediately after \(t\). The same before-and-after convention applies to an observed event.
This convention makes the timing precise: an individual with \(T_i=t\) is included just before \(t\) and excluded just after \(t\). For the examples below, observed times across individuals are distinct. If several records have the same observed time, the count still decreases by the number removed at that time, but the ordering of tied events and censoring requires additional care.
How a Single Record Changes the Risk Set
Suppose there are \(n\) individuals at the start of observation. At any time before a record's observed time, that record contributes one person to the risk set. Once its observed time is reached, it is removed: the event indicator tells us whether the removal followed an observed event or censoring.
Here \(Y(t)\) is the risk-set size immediately before \(t\), \(Y(t+)\) is the size immediately after \(t\), and \(I(\cdot)\) equals 1 when its condition is true and 0 otherwise. The notation \(t+\) means just after \(t\), not a separate observation at some fixed later time.
If there is one observed time at \(t\), exactly one individual is removed there, whether that time is an event or a censoring time. If the distinct observed times are \(t_1,t_2,\ldots\) in increasing order, and \(r_j\) records are removed at \(t_j\), the update is:
Under the distinct-time condition used here, \(r_j=1\) at each observed time. The next risk-set size before the next observed time is the preceding after-time size, provided no one enters observation in between. In these examples, observation starts together and nobody enters later.
Why Censoring Removes a Record Only After Its Censoring Time
A right-censored observation tells us that the individual was observed up to a known time without a recorded event by that time. It does not tell us what the event time would have been afterward. Risk-set membership follows the observation actually available: the record contributes through the censoring time, and cannot contribute after observation has ended.
The result follows directly from the definition. For a censored individual, \(C_i<U_i\), so \(T_i=C_i\). If \(t\le C_i\), then \(T_i\ge t\), and the individual is counted in \(Y(t)\). If \(t>C_i\), then \(T_i<t\), so that individual is not counted. At \(t=C_i\), the record is in the risk set immediately before the censoring time, but not in the set immediately after it.
The same indicator logic applies to an event record: it is in the risk set immediately before its event time and leaves immediately afterward. What differs is the information recorded at removal. A censored record supplies no observed event at that time; an event record does. This tutorial concerns membership and counting, not whether a censoring mechanism supports a particular statistical estimate.
Chronological Risk-Set Accounting
A reliable procedure is to begin with the initial number observed, sort the observed times from earliest to latest, and subtract each record once at its observed time. Keep the count just before each time separate from the count just after it. The event indicator labels why a record was removed; it does not change the fact that the record leaves the risk set after its observed time.
Count all individuals who have begun observation and have not yet had an observed time.
For each record, use \(T_i\) and \(\Delta_i\) to identify its time and whether it is an event or censoring observation.
The record at that time is still included immediately before its observed time.
Subtract one for each record whose observed time has just occurred, whether event or censoring.
Worked Example: Updating After Alternating Events and Censoring
A hypothetical portfolio begins with six observed claim-reporting durations. The ordered records are: an event at time 2, censoring at time 3, an event at time 5, censoring at time 7, an event at time 8, and censoring at time 10. Find the risk-set size immediately before and after each observed time.
Method: chronological risk-set accounting. Start with \(Y=6\). Each record remains counted up to its observed time and is subtracted once afterward.
At time 2, all six are present immediately before the event, so \(Y(2)=6\). After the event, the count is \(6-1=5\).
At time 3, the five remaining individuals are present immediately before censoring, so \(Y(3)=5\). After censoring, the count is \(5-1=4\).
At time 5, \(Y(5)=4\), and after the event the count is \(4-1=3\). At time 7, \(Y(7)=3\), and after censoring it is \(3-1=2\). At time 8, \(Y(8)=2\), and after the event it is \(2-1=1\). Finally, \(Y(10)=1\), and after censoring it is \(1-1=0\).
Check: Six records were removed in total, so the final count is \(6-6=0\). The common mistake to avoid is subtracting a censored record before its censoring time; it belongs in the count immediately before that time.
Worked Example: Applying the Event-Indicator Convention
For four hypothetical individuals, the event and censoring times \((U_i,C_i)\) are \((4,6)\), \((7,3)\), \((5,5)\), and \((9,8)\). Use the stated convention that \(\Delta_i=1\) when \(U_i\le C_i\). Then find the risk-set size immediately before time 5 and immediately after time 5.
Method: determine each observed record before counting. For the first individual, \(T_1=\min(4,6)=4\), and \(\Delta_1=1\). For the second, \(T_2=\min(7,3)=3\), and \(\Delta_2=0\). For the third, \(T_3=\min(5,5)=5\), and \(\Delta_3=1\) because the equality is assigned to the event. For the fourth, \(T_4=\min(9,8)=8\), and \(\Delta_4=0\).
The ordered observed records are therefore censoring at 3, event at 4, event at 5, and censoring at 8. Initially there are four individuals. The censoring at 3 leaves \(4-1=3\); the event at 4 leaves \(3-1=2\). Both individuals remaining just before time 5 are in \(Y(5)\), so \(Y(5)=2\). The event at 5 then removes one, giving \(Y(5+)=2-1=1\).
Check: The remaining individual is the one with \(T_4=8\), so the after-time count of 1 is consistent. The common mistake to avoid is leaving the third record’s status undefined because \(U_3=C_3\); the tie convention makes it an event.
Worked Example: Finding Membership at an Intermediate Time
A hypothetical study starts with seven individuals. Their distinct observed times and types, in order, are: event at 1, censoring at 2, event at 3, censoring at 4, event at 6, censoring at 7, and event at 9. Find the risk-set sizes immediately before time 6, immediately after time 6, and immediately before time 9.
Method: count removals before each requested time. At time 1, the risk set starts at 7 and becomes \(7-1=6\). At time 2, it becomes \(6-1=5\). At time 3, it becomes \(5-1=4\). At time 4, it becomes \(4-1=3\). No other observed time occurs between 4 and 6, so \(Y(6)=3\).
The event at time 6 removes one individual, giving \(Y(6+)=3-1=2\). Censoring at time 7 then changes the count to \(2-1=1\). No observed time occurs between 7 and 9, so \(Y(9)=1\).
Check: Before time 6, the records with observed times 6, 7, and 9 remain, giving three individuals. After the event at 6, two remain; after censoring at 7, one remains. The common mistake to avoid is using the after-time count at 6 as \(Y(6)\); the event at 6 is still in the risk set immediately before 6.
Common Mistakes and a Practical Check
Risk-set errors usually come from mixing up a record’s observed time with the time after which it is no longer at risk. A short audit after calculating can catch most of them.
- Removing a record too early. A record observed at time \(t\) is included in the risk set immediately before \(t\); subtract it after \(t\).
- Treating censoring as if it were an event. Both kinds of records leave the risk set after their observed time, but only an event record has an observed event at that time.
- Keeping a censored record in later risk sets. Once its censoring time has passed, the record is no longer observed and cannot be counted as still at risk.
- Leaving equality undefined. State how \(U_i=C_i\) is classified. Here, equality is assigned to an observed event through \(\Delta_i=1\) when \(U_i\le C_i\).
- Subtracting a record more than once. Each individual has one observed time and is removed once. Check that the total number removed never exceeds the starting count.
As a final check, compare a calculated risk set with the records themselves: just before \(t\), count the records whose observed times are at least \(t\); just after \(t\), count those whose observed times are greater than \(t\). These definitions give the same answer as chronological subtraction and make the time boundary explicit.
Check Your Understanding
For each problem, distinguish the count immediately before a time from the count immediately after it.
- A hypothetical group has five individuals. One has an event at time 2 and another is censored at time 4; no other observed time occurs by time 4. Find \(Y(2)\), \(Y(2+)\), and \(Y(4)\). (Answer: \(Y(2)=5,\;Y(2+)=4,\;Y(4)=4\).)
- Seven individuals have distinct observed records in this order: censoring at 1, event at 3, censoring at 5, event at 6. Find \(Y(5)\), \(Y(5+)\), and \(Y(6)\). (Answer: \(Y(5)=5,\;Y(5+)=4,\;Y(6)=4\).)
- Four individuals have \((U_i,C_i)\) pairs \((2,9)\), \((8,4)\), \((6,6)\), and \((7,10)\). With equality assigned to the event, find \(Y(6)\) and \(Y(6+)\). (Answer: \(Y(6)=2,\;Y(6+)=1\).)
- Eight individuals have distinct observed times: censoring at 1, event at 2, censoring at 4, event at 7, and censoring at 9; the other three have observed times later than 10. Find \(Y(7)\), \(Y(7+)\), and \(Y(10)\). (Answer: \(Y(7)=5,\;Y(7+)=4,\;Y(10)=3\).)
- A study starts with nine individuals. Four are removed at distinct times before time 6: events at 1 and 4, and censoring at 2 and 5. One of the five remaining individuals is censored at time 6; no other record occurs between times 5 and 8. Find \(Y(6)\), \(Y(6+)\), and \(Y(8)\), given that an event occurs at time 8. (Answer: \(Y(6)=5,\;Y(6+)=4,\;Y(8)=4\).)