Why Entry Time Changes the Calculation
In left-truncated survival data, a participant is observed only after reaching an entry time later than the time origin. For example, a participant may enter a study at age 60, while the event-time scale begins at birth. The participant was not under observation before age 60, and—if eligibility required being event-free at entry—could not have experienced the event before then and still joined the observed cohort.
This changes the denominator of every cumulative-hazard increment. At an event time, the risk set must contain only people who have entered and remain event-free immediately before that time. Counting people before they entered treats them as observed and at risk when they were not. In multiple imputation, an entry-aware cumulative-hazard summary can be important when survival information is used as a predictor for a missing covariate.
The strict inequality at entry makes the timing convention explicit: a participant recorded as entering exactly at an event time is not counted as at risk just before that event. If a data system uses different conventions for simultaneous entry and event timestamps, those conventions must be resolved from the study’s timing rules before calculating the risk sets. The calculations below use one shared time scale, not time reset to zero at each participant’s entry.
Risk Sets and Nelson–Aalen Increments
Let \(t_1<t_2<\cdots<t_K\) denote the distinct observed event times. Define \(Y(t_j)\) as the number of participants at risk immediately before \(t_j\), and \(d_j\) as the number of observed events at \(t_j\). With delayed entry,
The Nelson–Aalen increment at \(t_j\) is \(d_j/Y(t_j)\), provided \(Y(t_j)>0\). The cumulative-hazard estimate through time \(t\) adds these increments over event times no later than \(t\):
When multiple events occur at the same time, this standard tied-event form uses the total number of events at that time divided by the risk set just before that time. Censorings at that time remain in the denominator because they were event-free immediately before it; they do not add to \(d_j\). A participant who entered after the event time is in neither its risk set nor its event count.
Worked Calculations
Worked Example: Two Event Times With Delayed Entry
Consider five hypothetical participants. All times are measured from the same origin.
| Participant | Entry \(L_i\) | Exit \(U_i\) | Exit status |
|---|---|---|---|
| A | 0 | 2 | Event |
| B | 1 | 3 | Event |
| C | 0 | 3 | Censored |
| D | 2 | 3 | Event |
| E | 0 | 4 | Censored |
At time \(2\), A is at risk because \(0<2\leq2\). B, C, and E are also at risk; D is not, because \(L_D=2\) and entry must precede the event time. Thus \(Y(2)=4\), and \(d_2=1\). The increment is
At time \(3\), B, C, D, and E are at risk: each entered before 3 and has an exit time at or after 3. A exited at 2 and is no longer at risk. There are two events, B and D, so \(Y(3)=4\), \(d_3=2\), and the increment is \(2/4=0.5000\). Therefore \(\widehat{H}(3)=0.2500+0.5000=0.7500\). C’s censoring at 3 does not count as an event, but C is included in the risk set immediately before 3.
Worked Example: The Error From Ignoring Entry
Suppose four hypothetical participants have these records: A enters at 0 and has an event at 3; B enters at 0 and is censored at 4; C enters at 2.5 and has an event at 3; D enters at 4 and has an event at 5. At time 3, A, B, and C are at risk. D has not yet entered, so \(Y(3)=3\), while \(d_3=2\). The correct increment is
If an analyst incorrectly counts D as at risk from time zero, the denominator becomes 4 and the increment is calculated as \(2/4=0.5000\). The difference is not a minor bookkeeping choice: the incorrect calculation assigns D exposure to a time before observation began. This example shows the arithmetic consequence; whether the entry-adjusted estimate has a valid population interpretation still depends on the assumptions about entry and censoring.
Worked Example: Using the Correct Time Scale for a Cumulative-Hazard Summary
Return to the first example. The cumulative hazard at absolute time 3 is \(0.7500\), using the two event-time increments at 2 and 3. Participant B entered at time 1 and had an event at time 3. On the shared time scale, the relevant event time is 3, and the cumulative-hazard summary at that time is \(\widehat{H}(3)=0.7500\). Participant D entered at 2 and also had an event at 3; the same absolute-time summary is used for D.
It would be a different calculation to reset each participant’s clock to zero at entry and evaluate a hazard at elapsed follow-up time 1 for B or time 1 for D. That calculation requires an analysis defined on time since entry, rather than the common origin-time scale used above. The two summaries are not interchangeable: they use different time scales and can imply different risk sets. In an imputation workflow, the time scale for a cumulative-hazard predictor must match the survival analysis and the interpretation of its other predictors.
Why the Increment Has This Form
The denominator has a direct risk-set interpretation. Over a very short time interval after \(t_j\), only participants in the risk set can experience an observed event. If the event hazard over that interval is approximately \(dH(t_j)\) for each person at risk, the expected number of events is approximately \(Y(t_j)dH(t_j)\). Equating this local expected count to the observed count \(d_j\) motivates the increment \(d_j/Y(t_j)\):
Adding the local estimates over event times gives the Nelson–Aalen sum. Delayed entry changes this reasoning only through the people eligible to contribute at each time: the count \(Y(t_j)\) must exclude anyone who has not yet entered. It is not valid to repair an incorrect risk set after computing the cumulative sum; the entry rule belongs inside each increment.
Diagnostics, Alternatives, and Limitations
A practical check is to inspect the risk-set membership at each distinct event time, not just the final cumulative-hazard column. Verify that every included person entered earlier than the event time and had not exited before it. Also check that event counts do not exceed risk-set sizes, that the risk set is nonzero at every event time, and that a participant is not counted repeatedly because a start-stop data record was split into intervals. In interval data, the records represent periods of risk; the person’s contribution at a given event time must still follow the stated entry and exit rules.
A risk set of zero at an event time signals an inconsistency in the records or timing convention: the observed event cannot be assigned an increment using \(d_j/Y(t_j)\). Review time origins, entry and exit values, event indicators, and the treatment of tied timestamps rather than silently dropping the event. A sharp change in risk-set size when a group enters may be correct, but it is a reason to verify that entry times are on the intended scale.
The main alternative is to use a survival calculation that ignores left truncation by treating everyone as at risk from the time origin. That is inappropriate when participants only become observable after entry and can distort the denominators. The Nelson–Aalen estimator is also distinct from the product-limit Kaplan–Meier estimator: both require appropriate risk sets, but they combine event information differently. Choosing between them does not remove the need to handle delayed entry correctly. The Nelson–Aalen sum estimates cumulative hazard; it should not be described as a probability of failure.
In MICE, an entry-adjusted cumulative-hazard summary can help represent survival information when imputing a missing covariate, but it is a summary, not a complete model of the event process. Its usefulness depends on the event-time scale, entry and censoring assumptions, and the imputation model’s other predictors. The estimate can also be unstable late in follow-up when few participants remain at risk. Entry adjustment does not by itself fix informative truncation, informative censoring, misspecified time scales, or an imputation model that omits important survival-related information.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A participant enters at time 5 and is event-free through time 8. Under the convention in this tutorial, are they in the risk set at time 5? At time 6?
- At an event time, the risk set contains 8 people and 2 events occur. What is the Nelson–Aalen increment?
- At time 7, one participant is censored and another has an event. Does the participant censored at 7 count in the denominator immediately before 7? Explain.
- Why is a participant who enters at time 9 excluded from the risk set at time 7, even if their event occurs at time 12?
Solutions. (1) No at time 5, because entry must be strictly earlier than the event time; yes at time 6 if the participant remains under observation and event-free. (2) The increment is \(2/8=0.2500\). (3) Yes. The participant was at risk immediately before time 7; censoring does not count as an event. (4) The participant had not entered by time 7, so the record provides no at-risk observation at that earlier time. Their later event does not make them part of earlier risk sets.