Why Delayed Entry Changes the Calculation
In some studies, a person’s time origin precedes the date they can first be observed. For example, a study may measure time since diagnosis, but enroll people only at a later clinic visit. Someone who experiences an event before that visit cannot enter the observed cohort. This is delayed entry, also called left truncation.
A person who has not entered observation is not yet part of the observed risk set. Consequently, the risk-set denominator at each event time must include only people who have entered by then and have not already had an event or been censored. Using the ordinary Aalen–Johansen calculation with everyone counted from time zero can produce incorrect increments.
The calculation still targets cumulative incidence from a common time origin, such as diagnosis. But that interpretation requires assumptions about how people enter observation. Correctly constructing the risk sets is necessary; it does not by itself make entry noninformative.
Definitions and Risk-Set Convention
Let \(t_1<t_2<\cdots\) denote distinct observed event times, pooling events of all causes at each time. Under the convention used here, a subject is at risk immediately before \(t_k\) if they have entered strictly before \(t_k\) and their observed endpoint is at or after \(t_k\):
The strict inequality \(L_i<t_k\) excludes a person whose recorded entry time is exactly the event time. The inequality \(t_k\leq X_i\) includes someone censored at \(t_k\) in the risk set immediately before that time. This convention treats event and censoring times as applying after the just-before-time risk set is formed. If entry and event timestamps can coincide, the data definition must specify their ordering; do not silently switch conventions.
Let \(d_{kj}\) be the number of observed events from cause \(j\) at \(t_k\), and let \(d_k=\sum_j d_{kj}\) be the total number of events at that time. Censored subjects contribute to \(r_k\) if their censoring time is \(t_k\), but not to \(d_k\).
The Delayed-Entry Aalen–Johansen Estimator
The Aalen–Johansen estimator uses the risk-set size \(r_k\) to estimate the conditional event increment at each event time. The estimated all-cause event-free survival just before \(t_k\) is denoted by \(\widehat S(t_k-)\). The delayed-entry formulas are:
Here \(\widehat F_j(t)\) estimates the probability of experiencing cause \(j\) by time \(t\), before any competing event. At a tied event time, \(d_k\) includes events from every cause: all cause-specific increments use the same risk-set denominator, and the survival multiplier uses the total event count. If there are no events at a time, there is no event-time increment; censoring changes later risk sets, not the survival estimate at that censoring time.
Why These Risk Sets Give the Estimator
The key calculation is the conditional event increment. Among subjects event-free and observable just before \(t_k\), \(d_{kj}/r_k\) estimates the probability of a cause \(j\) event at \(t_k\). The probability of remaining event-free through that time is estimated by \(1-d_k/r_k\). Delayed entry changes which subjects supply these denominators, not the form of the increments.
Count subjects with \(L_i<t_k\leq X_i\). This represents subjects observed and event-free immediately before \(t_k\).
The cause \(j\) increment is \(d_{kj}/r_k\); the all-cause increment is \(d_k/r_k\).
Multiply the previous survival estimate by \(1-d_k/r_k\), and add \(\widehat S(t_k-)d_{kj}/r_k\) to the cause \(j\) cumulative incidence.
This justification relies on the risk set representing the target population’s event-free subjects at each time. In particular, delayed entry must not selectively admit people with different subsequent event risks, after accounting for any covariates used to define the analysis. A standard sufficient condition is that the entry time \(L_i\) is independent of the event time and cause \((T_i,J_i)\) in the target population. Conditional independence given measured covariates can be used when the analysis accounts for those covariates. Censoring must also be independent of future event outcomes among subjects at risk, at least conditional on the covariates used in the analysis. There must be adequate entry support and observations at the times being estimated; otherwise the target risk set is not represented by the observed data.
Worked Example: Updating Two Cause-Specific Estimates
Suppose a hypothetical study has six subjects. All times are measured from diagnosis. The table records entry time, observed endpoint, and endpoint status.
| Subject | Entry \(L_i\) | Endpoint \(X_i\) | Status |
|---|---|---|---|
| A | 0 | 2 | Cause 1 |
| B | 0 | 3 | Cause 2 |
| C | 1 | 4 | Censored |
| D | 2 | 4 | Cause 1 |
| E | 3 | 5 | Cause 2 |
| F | 0 | 5 | Censored |
At time \(2\), the risk set is A, B, C, and F, so \(r_1=4\). D is excluded because entry at time 2 is not strictly before the event time. There is one cause 1 event. Therefore, \(\widehat S(2)=1-1/4=0.75\), and \(\widehat F_1(2)=1/4=0.25\).
At time \(3\), the risk set is B, C, D, and F, so \(r_2=4\). B has a cause 2 event. Thus the cause 2 increment is \(0.75(1/4)=0.1875\). The updated estimates are \(\widehat F_2(3)=0.1875\) and \(\widehat S(3)=0.75(1-1/4)=0.5625\).
At time \(4\), C, D, E, and F are at risk, so \(r_3=4\). C is censored at 4 and is included in the risk set; D has a cause 1 event. The increment for cause 1 is \(0.5625(1/4)=0.140625\). Therefore, \(\widehat F_1(4)=0.25+0.140625=0.390625\), and \(\widehat S(4)=0.5625(1-1/4)=0.421875\).
At time \(5\), E and F are at risk, giving \(r_4=2\). E has a cause 2 event; F is censored at 5 and remains in the just-before-time risk set. The cause 2 increment is \(0.421875(1/2)=0.2109375\). Hence \(\widehat F_2(5)=0.1875+0.2109375=0.3984375\), and \(\widehat S(5)=0.421875(1-1/2)=0.2109375\). The calculations use all observed subjects who have entered by each event time, not just subjects enrolled at time zero.
Worked Example: Handling Tied Events After Delayed Entry
In a second hypothetical data set, A enters at 0 and has a cause 1 event at 4; B enters at 1 and has a cause 2 event at 4; C enters at 0 and is censored at 6; D enters at 4 and is censored at 6; and E enters at 2 and has a cause 2 event at 3.
At time 3, A, B, C, and E are at risk, so \(r_1=4\). The cause 2 event gives \(\widehat F_2(3)=1/4\) and \(\widehat S(3)=3/4\).
At time 4, A, B, and C are at risk. D is excluded because \(L_D=4\). There are two tied events, one of each cause, so \(r_2=3\), \(d_{21}=1\), and \(d_{22}=1\). The cause 1 increment is \((3/4)(1/3)=1/4\), and the cause 2 increment is also \((3/4)(1/3)=1/4\). The updated estimates are \(\widehat F_1(4)=1/4\), \(\widehat F_2(4)=1/2\), and \(\widehat S(4)=(3/4)(1-2/3)=1/4\). The total probability remains \(1/4+1/2+1/4=1\).
Assumptions, Diagnostics, and Alternatives
The estimator is useful for estimating cumulative incidence from the common time origin when entry and censoring are noninformative for the event process, subject to adequate support. If entry depends on prognosis—for example, people with worsening health are more likely to enroll sooner—the observed risk sets can have a different cause-specific event distribution from the target population’s risk sets. The ordinary delayed-entry estimator then need not estimate the target-population cumulative incidence. A different estimand, such as risk conditional on being enrolled, may be relevant, but it must be defined and interpreted explicitly.
Several practical checks help detect problems before interpreting the estimate:
- Audit the time origin and units. Entry, event, and censoring times must share the same origin and units. Time since enrollment cannot be mixed with time since diagnosis.
- Check ordering and eligibility. Each included subject should have \(L_i<X_i\). Investigate recorded events before entry and specify how exact entry-event ties are handled.
- Recompute risk sets at event times. Confirm that every counted subject satisfies \(L_i<t_k\leq X_i\), and verify \(d_k\leq r_k\).
- Inspect support and late risk sets. If few eligible subjects remain at later times, the estimate relies on little information. If no members of the target population can have entered by a time, the target risk set is not supported by the data.
- Check censoring patterns. If censoring is related to future event risk and the analysis does not account for that relationship, estimated increments can be biased.
With no delayed entry, everyone begins at time zero and the risk-set construction reduces to the ordinary right-censored Aalen–Johansen calculation. Treating delayed entrants as though they had been observed from time zero is not an equivalent alternative: it places people in risk sets before they were observed. Conversely, excluding everyone who enters after time zero discards observed information and changes the analyzed population. When entry is informative, simply changing the risk-set formula does not solve the problem; analysis may require an explicit model for the entry mechanism or a clearly restricted target population.
Worked Example: The Cost of Counting a Person Before Entry
In a hypothetical study, A enters at 0 and has a cause 1 event at time 2. B enters at 0 and remains under observation past time 2. C enters at 1 and remains under observation past time 2. D enters exactly at time 2 and remains under observation afterward.
Under the stated convention, the time-2 risk set is A, B, and C, so \(r=3\). The cause 1 increment is \(1/3\). D is excluded at that instant because \(L_D=2\) is not less than 2. If an analyst incorrectly counts all four subjects, the increment becomes \(1/4\). That denominator includes a person who was not yet observed just before the event, so it does not represent the defined risk set. The difference is not a rounding issue; it changes the estimated event increment.
Common Mistakes and Interpretation
- Counting a person before entry. Being event-free in the data after enrollment does not mean a subject was observed or eligible at earlier times. Apply \(L_i<t_k\) at every event time.
- Treating entry as censoring. Entry does not remove a subject from a risk set; it marks the first time they can begin contributing to one. Censoring removes a subject at the endpoint.
- Dropping censoring ties from the denominator. With the just-before-time convention, a subject censored at \(t_k\) is included in \(r_k\), though not in the event count.
- Assuming correct risk sets guarantee an unbiased target estimate. The estimator also needs appropriate entry and censoring assumptions and sufficient support. Risk-set bookkeeping cannot correct informative selection into observation.
- Changing the time origin at enrollment. If the scientific question concerns time since diagnosis, resetting each person’s clock to enrollment estimates a different time scale and does not implement delayed entry on the diagnosis scale.
Check Your Understanding
Work through each question, then compare your reasoning with the solutions below.
- At an event time \(t=5\), subjects have \((L_i,X_i)\) values \((0,5)\), \((2,7)\), \((5,8)\), and \((1,4)\). Which subjects are in the risk set?
- Just before an event time, \(\widehat S(t-)=0.8\), the risk set has 8 subjects, and there are 1 cause 1 event and 2 cause 2 events. Find both CIF increments and the updated event-free survival.
- Why can the delayed-entry Aalen–Johansen estimator fail to estimate the target-population CIF if entry time depends on prognosis?
- At a tied event time, explain why the survival multiplier uses the total number of events across causes rather than the count for just one cause.
Solutions. (1) The first two subjects are at risk: both have \(L_i<5\) and \(X_i\geq5\). The third is excluded because entry is exactly 5, and the fourth has already reached an endpoint before 5. (2) Here \(d_1=1\), \(d_2=2\), and \(d=3\). The increments are \(0.8(1/8)=0.1\) for cause 1 and \(0.8(2/8)=0.2\) for cause 2. Updated survival is \(0.8(1-3/8)=0.5\). (3) Prognosis-dependent entry can make observed risk sets differ in event risk from the target population’s event-free risk sets. The denominators then need not estimate the target conditional event probabilities. (4) Event-free survival means no event of any cause has occurred. All cause-specific events remove subjects from that state, so the multiplier is \(1-d/r\), with \(d\) counting events of every cause.