Why Record an Observed Time and an Event Indicator?
In an incomplete loss or survival record, the event of interest may not have occurred by the time observation ends. For example, a hypothetical insurer might follow a policy from its start until a claim is reported, or until the study closes. If the study closes first, the claim time is unknown: the record establishes only that the policy remained without a reported claim up to the closing time. A single number cannot distinguish that case from one in which the claim actually occurred at that time.
The standard representation keeps two pieces of information: the time at which observation ends and an indicator of whether the event was observed. Together they distinguish an exact event time from a right-censored observation. The pair is a compact record of what the data establish, not a replacement for the unobserved event time.
The time origin and units must be consistent. If \(T\) is measured in days since policy inception, \(C\) must also be measured in days since policy inception. The pair then reports an observed duration in days, not a calendar date. A record with \(\Delta=1\) is an exact event-time observation: \(T=Y\). A record with \(\Delta=0\) is right-censored: \(C=Y\) and \(T>Y\). The inequality is strict because the convention above assigns an event at a tie to the observed-event category.
How the Pair Encodes the Record
The encoding follows from comparing the two possible times. If the event happens no later than the end of observation, the minimum is \(T\), and the event is observed. If observation ends first, the minimum is \(C\), and the event has not been observed. This gives the rule directly:
This case-by-case form also explains what can be decoded. From \((y,1)\), we know that the event occurred at time \(T=y\); the end-of-observation time is not determined by the pair, except that it must be at least \(y\). From \((y,0)\), we know that observation ended at \(C=y\), while the event time exceeds \(y\). Its exact value is not supplied by the pair. In particular, a censored record does not say that the event occurred at \(y\), nor does it say that the event never occurs.
Worked Example: Encoding Event and Study-Closure Times
A hypothetical study records durations in months from the start of coverage to a specified event. For each policy, observation ends either at the event or at the study's administrative closing date. Convert the following underlying times to observed pairs: (a) \(T=8,\ C=11\); (b) \(T=15,\ C=10\); and (c) \(T=6,\ C=6\).
Model and assumptions. These are deterministic time records, with both times measured in months from the same origin. No probability distribution, interest rate, or mortality assumption is needed to construct the pairs. Use \(Y=\min(T,C)\) and the convention \(\Delta=I(T\leq C)\).
(a) Event before closure. The observed time is \(Y=\min(8,11)=8\). Since \(8\leq11\), the event is observed and \(\Delta=1\). The pair is \((8,1)\). It records an exact event at month 8.
(b) Closure before event. The observed time is \(Y=\min(15,10)=10\). Since \(15\leq10\) is false, \(\Delta=0\). The pair is \((10,0)\). It records that observation ended at month 10 and that the event time is later than month 10; it does not reveal that event time as month 15.
(c) Tie. The observed time is \(Y=\min(6,6)=6\). The convention counts an event at the observation endpoint as observed, so \(6\leq6\) and \(\Delta=1\). The pair is \((6,1)\).
Check. Each first component is the smaller of the two times. The second component is 1 in (a) and (c), where \(T\leq C\), and 0 in (b), where \(C<T\). The method avoids interpreting the second component as a censoring indicator that equals 1 when censored; here 1 specifically means the event was observed.
Decoding Observed Pairs
When a data set provides only \((Y,\Delta)\), decoding means stating the information warranted by that pair and not filling in the missing time by assumption. This is the reverse of encoding, but it is not a full reconstruction of both \(T\) and \(C\). For an observed event, the event time is known and the censoring time need not be. For a censored observation, the censoring time is known and the event time is only bounded below.
Worked Example: Interpreting a Set of Recorded Pairs
A hypothetical portfolio study reports the following elapsed times in days: \((12,1)\), \((19,0)\), \((7,0)\), and \((19,1)\). For each record, state whether the event time is known, and state the information available about the event and observation end.
Model and assumptions. Each pair follows the definition above, with time measured in days from the same study-entry point and \(\Delta=1\) indicating an observed event. No distributional model or interest or mortality assumption is used.
Record 1: \((12,1)\). The event was observed at \(T=12\) days. The end of observation was at least day 12, but its exact time is not identified by this pair.
Record 2: \((19,0)\). Observation ended at \(C=19\) days before the event. Thus \(T>19\) days. The exact event time is unknown; it could be any time later than day 19 under the information in this record.
Record 3: \((7,0)\). Observation ended at \(C=7\) days, with \(T>7\) days. This is right-censored even though its observed time is smaller than the other times. A short observed duration does not by itself indicate an event.
Record 4: \((19,1)\). The event was observed at \(T=19\) days. Although the first component matches record 2, the meaning differs because the indicator differs: record 4 gives an exact event time, while record 2 gives only a lower bound on the event time.
Check. There are two observed events, at days 12 and 19, and two right-censored records, at days 19 and 7. The pair \((19,0)\) does not mean an event at day 19; the indicator is essential to interpreting the first component.
Derivation: Why the Minimum and Indicator Work
The construction is justified by splitting all possible records according to which time comes first. If \(T\leq C\), then the event occurs while observation is available. The observed time is therefore \(\min(T,C)=T\), and the event indicator is 1. If \(C<T\), observation stops before the event; the observed time is \(\min(T,C)=C\), and the event indicator is 0. These cases are exhaustive, with equality assigned to the first case:
Consequently, given a pair \((y,\delta)\), the information can be expressed precisely as follows:
These statements describe what is known from the pair alone. They do not assert that every possible value of the unknown time is equally plausible, nor do they specify a statistical model for that time. Such additional conclusions require assumptions and methods beyond the basic representation.
Worked Example: Checking the Time Origin and Units
A hypothetical study follows two devices from installation and records the time until failure, in days. Device A fails 30 days after installation, and observation would otherwise have continued to day 45. Device B has not failed when the study closes 6 weeks after installation. Convert the records to pairs measured in days and interpret them.
Model and assumptions. Let \(T\) be time to failure and \(C\) be the end of observation, both measured in days from installation. For Device A, \(T=30\) and \(C=45\). For Device B, \(C=6\times7=42\) days and \(T>42\). These records use no distribution, interest rate, or mortality assumption.
Device A. \(Y=\min(30,45)=30\), and \(30\leq45\), so \(\Delta=1\). Its pair is \((30,1)\); failure was observed at day 30.
Device B. Since the device was still operating at study closure, \(T>42\) and \(C=42\). Thus \(Y=\min(T,42)=42\), and \(T\leq42\) is false, so \(\Delta=0\). Its pair is \((42,0)\); failure had not been observed by day 42.
Check. The units match in both pairs, and the converted closure time is \(6\times7=42\) days. Using 6 as the observed time while recording the unit as days would mix weeks and days and misstate the duration.
Common Exam Trap: Treating Every Observed Time as an Event
A frequent error is to read the first component \(Y\) as an exact event time for every record. Consider four hypothetical records: \((4,1)\), \((9,0)\), \((3,1)\), and \((5,0)\), with time measured in months. The observed times sum to \(4+9+3+5=21\), so their arithmetic average is \(21/4=5.25\) months. That calculation is the average of the four observed times; it is not the average of four known event times.
The incorrect interpretation labels all four as exact events at months 4, 9, 3, and 5. But the second and fourth records have \(\Delta=0\): their observation ended at months 9 and 5, respectively, before the events. The event times for those records are only known to exceed 9 and 5 months. Therefore, the four exact event times—and their average—cannot be recovered from these pairs alone. The correction is to identify the two exact event records, \((4,1)\) and \((3,1)\), and the two censored records, \((9,0)\) and \((5,0)\), without substituting censoring times for event times.
Check Your Understanding
For each problem, use the convention \(\Delta=I(T\leq C)\). All times are measured from the stated origin.
- A hypothetical contract has event time \(T=17\) months and observation-end time \(C=21\) months. Give \((Y,\Delta)\) and interpret it. (Answer: \((17,1)\); the event was observed at month 17.)
- A hypothetical device has not failed by its observation end at \(C=28\) days, and its eventual failure time is \(T=35\) days. Give the recorded pair and state the information it conveys. (Answer: \((28,0)\); observation ended at day 28 and \(T>28\) days.)
- Decode the hypothetical record \((14,1)\), where time is measured in weeks. What is known about \(T\) and \(C\)? (Answer: \(T=14\) weeks and \(C\geq14\) weeks; the exact value of \(C\) is not identified.)
- Decode \((22,0)\), where time is measured in days. Is the event known to have occurred on day 22? (Answer: No; \(C=22\) days and \(T>22\) days.)
- A hypothetical event occurs exactly when observation ends, at \(T=C=9\) months. Under the convention in this tutorial, give the pair and explain the classification. (Answer: \((9,1)\); equality is assigned to the observed-event case because \(T\leq C\).)