Handling Tied Events And Censoring Times

Why Tied Times Need a Convention

Survival and loss records often report a time at which an event occurred or observation ended. Several records can share the same recorded time. This can happen because times are naturally discrete, because records are rounded to days or months, or because genuinely simultaneous events occur. A further complication arises when an event and one or more censorings have the same recorded time.

The risk set at a time is determined by which individuals are still under observation just before that time. Therefore, a consistent rule for handling records tied at that time is essential. If tied censorings are removed too soon, the event denominator is reduced, even though those individuals were still under observation up to the recorded time.

All scenarios and records below are hypothetical. No parametric distribution, interest rate, or mortality assumption is needed: the calculations use only the recorded observation times and event indicators.

Definition: For individual \(i\), let \(T_i\) be the event time and \(C_i\) the right-censoring time. The observed time is \(Z_i=\min(T_i,C_i)\), and the event indicator is \(\delta_i=1\) if the event is observed at \(Z_i\), and \(\delta_i=0\) if the observation is censored there. At a recorded time \(t\), the risk set immediately before \(t\) is \(R(t)=\{i:Z_i\ge t\}\). Write \(n(t)\) for its size, \(e(t)\) for the number of events recorded at \(t\), and \(c(t)\) for the number of censorings recorded at \(t\).

This tutorial uses the following bookkeeping convention: all individuals with \(Z_i=t\) are counted in the risk set just before \(t\); events at \(t\) are counted before censorings at \(t\) are removed. In particular, an event at \(t\) is compared with \(n(t)\), which includes the individuals censored at \(t\). After processing the tied time, all event and censor records at that time leave the risk set:

$$ n(t^+)=n(t)-e(t)-c(t). $$

Here \(n(t^+)\) denotes the number remaining immediately after time \(t\). This is a convention for organizing records; it does not assert that an event physically happened before a censoring when their recorded times are equal. If the data are rounded, their true ordering within that interval may be unknown. The essential requirement is to state the convention and apply it consistently.

Grouped Updates at a Tied Time

When several events share a recorded time, treat them as a group. The number at risk is the count just before that time, not a count repeatedly reduced after each tied event. When censorings share the time as well, they are included in that same pre-time risk set under the convention above, and are removed only after the event count is recorded.

For a product-limit calculation, the grouped event fraction is \(e(t)/n(t)\), and the factor for remaining event-free through \(t\) is \(1-e(t)/n(t)\). If \(\widehat S(t-)\) is the estimate just before \(t\), the update is:

\widehat S(t)=\widehat S(t-)\left(1-\frac{e(t)}{n(t)}\right).

This update is useful for illustrating the effect of tied records. Its interpretation as a survival estimate relies on appropriate censoring assumptions; the risk-set bookkeeping itself is simply a description of how the observed records are organized.

Justification: Immediately before \(t\), the convention counts \(n(t)\) individuals as at risk. Of these, \(e(t)\) have an observed event at \(t\), leaving \(n(t)-e(t)\) who do not have an observed event at that time. The grouped fraction without an event is therefore \((n(t)-e(t))/n(t)=1-e(t)/n(t)\). Multiplying the estimate just before \(t\) by this conditional fraction gives the stated update. Censorings at \(t\) are removed after this event fraction is formed, so they do not reduce its denominator.

Worked Example: Events and Censorings Tied at Two Times

A hypothetical study follows eight units from time zero, measured in days. At day 2, two events and one censoring are recorded. At day 4, one event and two censorings are recorded. At day 6, one event and one censoring are recorded. Find the risk set immediately before each recorded time, the number remaining afterward, and the product-limit estimate after each time, starting from \(\widehat S(0)=1\).

Method and convention. Use the grouped risk-set update, counting all records at a time in the risk set before removing any of them. There is no assumed parametric distribution.

Day 2. All eight units are initially at risk, so \(n(2)=8\). There are \(e(2)=2\) events and \(c(2)=1\) censoring. The number remaining is \(8-2-1=5\). The event-free factor is \(1-2/8=6/8=0.75\), so \(\widehat S(2)=1(0.75)=0.75\).

Day 4. The five remaining units form the risk set, so \(n(4)=5\). There are \(e(4)=1\) event and \(c(4)=2\) censorings. The number remaining is \(5-1-2=2\). The event-free factor is \(1-1/5=4/5=0.80\), giving \(\widehat S(4)=0.75(0.80)=0.60\).

Day 6. The two remaining units are at risk, so \(n(6)=2\). There are \(e(6)=1\) event and \(c(6)=1\) censoring. The number remaining is \(2-1-1=0\). The event-free factor is \(1-1/2=0.50\), so \(\widehat S(6)=0.60(0.50)=0.30\).

Check. The records account for \(2+1+1+2+1+1=8\) units, and the risk-set sizes move from 8 to 5 to 2 to 0. The final estimate is \(0.30\). The common mistake avoided here is removing the day-2 censoring before calculating the day-2 event fraction.

Censor-Only Times and Multiple Ties

A recorded time can contain censorings but no events. Such a time changes the size of later risk sets but does not create an event-factor update. It should still appear in a risk-set table: omitting it can make the next event denominator too large.

Worked Example: A Censor-Only Time Before a Tied Event Group

A hypothetical portfolio study records ten units, with times measured in months. At month 1, two events and one censoring occur. At month 3, there are two censorings and no events. At month 5, three events and one censoring occur. At month 8, one event occurs. Calculate the risk set at each time and the product-limit estimate after each event time.

Method. Process each recorded time in order, including the censor-only time in the risk-set updates. There is no assumed parametric distribution.

At month 1, \(n(1)=10\), \(e(1)=2\), and \(c(1)=1\). Thus \(n(1^+)=10-2-1=7\), and the estimate is \(1(1-2/10)=0.80\).

At month 3, \(n(3)=7\), \(e(3)=0\), and \(c(3)=2\). Thus \(n(3^+)=7-0-2=5\). There is no event factor, so the estimate remains \(0.80\).

At month 5, \(n(5)=5\), \(e(5)=3\), and \(c(5)=1\). Thus \(n(5^+)=5-3-1=1\). The event-free factor is \(1-3/5=2/5=0.40\), so the estimate becomes \(0.80(0.40)=0.32\).

At month 8, \(n(8)=1\), \(e(8)=1\), and \(c(8)=0\). Thus \(n(8^+)=1-1=0\), and the event-free factor is \(1-1/1=0\). The estimate becomes \(0.32(0)=0\).

Check. The event and censor counts total \(2+1+2+3+1+1=10\), matching the number of units. The estimate stays unchanged at month 3 because no event occurs there. The common mistake avoided is skipping the censor-only time and using seven, rather than five, as the month-5 risk set.

Why the Tie Convention Changes the Result

If censorings at \(t\) are removed before events at \(t\), the calculation treats those censored individuals as unavailable for the event comparison. That can reduce the denominator and increase the estimated event fraction. The following comparison shows the difference directly.

Worked Example: Comparing Two Rules at a Shared Time

In a hypothetical study, six units are at risk immediately before day 10. Two events and two censorings are recorded on day 10. The estimate just before day 10 is \(0.75\). Calculate the update using the convention in this tutorial, then show the result of the alternative rule that removes the censorings first.

Method. First use the event-before-censor bookkeeping convention. Then calculate the alternative only to show why the convention matters. There is no assumed parametric distribution.

Under the stated convention, both censoring records are included in \(n(10)=6\). The event-free factor is \(1-2/6=4/6=2/3\). Therefore, \(\widehat S(10)=0.75(2/3)=0.50\).

If the two censorings are incorrectly removed first, the event denominator becomes \(6-2=4\). The resulting event-free factor is \(1-2/4=1/2\), giving \(0.75(1/2)=0.375\). This differs from \(0.50\) because the alternative rule changed who was counted at risk for the tied event time.

Check. Under the stated convention, the count after day 10 is \(6-2-2=2\), regardless of the order used to record the two groups. The correct update is \(0.50\). The common mistake is to present the alternative denominator as though it followed the stated convention.

Practical Checks and Common Traps

A compact table helps keep the timing convention visible. For every distinct recorded time, list \(n(t)\), \(e(t)\), and \(c(t)\) before calculating anything. Then check that \(n(t)-e(t)-c(t)\) equals the next risk-set size. This catches many bookkeeping errors before they affect an estimate.

Using a post-time denominator. The event fraction uses the risk set immediately before the tied time, not the count remaining after events or censorings.
Removing tied censorings too early. Under the convention used here, censorings at \(t\) remain in \(n(t)\) when the event fraction is formed.
Ignoring a censor-only time. It contributes no event factor, but its censorings reduce later risk sets.
Mixing conventions. Switching between event-before-censor and censor-before-event handling across times makes the calculations inconsistent.
Assuming tied recorded times reveal exact order. A shared day or month may reflect rounding. Unless finer timing information is available, the records do not establish the within-time ordering.

The risk-set convention is a data-handling rule, not a substitute for examining how times and event indicators were recorded. If a study defines an event as occurring only after the observation cutoff, or uses a different boundary rule, its records must be interpreted according to that definition. When the data provide only grouped times, report the convention used rather than implying that the unobserved ordering is known.

Key takeaway: At each tied time, form the risk set from all records observed up to that time, count the events using that pre-time risk set, and then remove both events and censorings. A clear and consistent convention keeps tied-time calculations reproducible.

Check Your Understanding

For each problem, use the event-before-censor bookkeeping convention unless the question asks for a comparison.

Seven units are at risk at day 2. One event and two censorings are recorded then. How many remain immediately afterward? (Answer: 4)
Twelve units are initially at risk. At time 1 there are three events and one censoring; at time 4 there are two events and two censorings. Give the risk-set sizes immediately before times 1 and 4, and the number remaining after time 4. (Answer: \(n(1)=12\), \(n(4)=8\), and 4 remain after time 4.)
Ten units are initially at risk. At time 2 there are two events and two censorings; at time 5 there is one event and no censorings. Starting from \(\widehat S(0)=1\), find the estimate after time 5. (Answer: \((1-2/10)(1-1/6)=2/3=0.6667\).)
Eight units are at risk at time 6, with two events and three censorings tied there. Find the event-free factor under this tutorial’s convention and under the rule that removes censorings first. (Answer: \(0.7500\) under this convention; \(0.6000\) if censorings are removed first.)
Fifteen units are initially at risk. At time 1 there are two events and one censoring; at time 2 there are no events and three censorings; at time 4 there are four events and two censorings; at time 7 there is one event and two censorings. Give the risk sets just before times 2, 4, and 7, and the product-limit estimate after time 7. (Answer: \(n(2)=12\), \(n(4)=9\), \(n(7)=3\), and \(\widehat S(7)=(13/15)(5/9)(2/3)=26/81=0.3210\).)

What You'll Learn