Why Delayed Entry Changes the Fine–Gray Risk Set
In a delayed-entry study, a participant becomes observable only after the time origin. For example, follow-up measured from diagnosis may begin only when a patient enrolls, or age may be the time scale while people join a study at different ages. A participant who enters at time \(L_i\) contributes no observed information before \(L_i\). This restriction is straightforward in an ordinary survival risk set, but Fine–Gray regression has a feature that makes it more delicate: people who experience a competing event remain represented in the subdistribution risk set.
Fine–Gray regression models the subdistribution hazard for a particular event cause \(k\). Its risk set is linked to the cumulative incidence function for that cause, rather than simply to the people who are currently event-free. With delayed entry, the analyst must respect entry time and censoring while retaining the intended contribution of observed competing events. Adding an entry time to a standard right-censored Fine–Gray analysis is not, by itself, a valid adjustment.
A delayed entrant cannot contribute before \(L_i\). In addition, an individual censored before \(t\) cannot be counted as observed and event-free through \(t\). By contrast, if a competing event was observed at \(T_i<t\), the participant may contribute to the pseudo risk set after that event, with a censoring adjustment. These are distinct contributions and require different conditions.
Risk-Set Contributions and Their Assumptions
For a schematic censoring-adjusted contribution at time \(t\), let \(G_i(u)\) denote the probability of remaining uncensored through \(u\), conditional on the relevant covariates and entry history. If censoring is common across participants conditional on the modeled information, write \(G(u)\). A representative contribution is
The first term represents a participant who entered by \(t\), remains event-free through \(t\), and was observed through \(t\). The second represents someone who entered before an observed competing event, experienced that event before \(t\), and was observed at least through the competing event. The factor \(G_i(t)/G_i(T_i)\) adjusts that retained contribution under the censoring assumptions described below. In particular, the second term does not require \(C_i\ge t\): censoring after the competing event does not erase the fact that the competing event was observed.
This expression is a schematic risk-set contribution, not a universal software recipe. Valid implementations can differ in how they estimate censoring distributions, handle tied times, define weights, and calculate variance. With delayed entry, \(G_i\) may need to condition on \(L_i\) and covariates; using a single marginal \(G\) when censoring varies with those factors can be inappropriate.
The assumptions behind an estimator must match the data and the target estimand. At minimum, the analyst needs a coherent time origin and event coding, a model for the subdistribution hazard that is adequately specified, and a valid treatment of entry and censoring. The entry process must satisfy an independent or quasi-independent truncation condition appropriate to the estimator, possibly conditional on covariates. In practical terms, after conditioning as required by the method, entry must not selectively hide event histories in a way the analysis fails to model. There must also be adequate support: relevant covariate and time patterns in the target population must have a chance of being observed after entry, and censoring probabilities used in weights must not be zero over the times being analyzed.
The censoring adjustment also relies on censoring being independent of the event process conditional on the information used to estimate \(G_i\), or on a suitable method explicitly handling departures from that assumption. These assumptions are not repaired merely by including entry time as a covariate. Later material treats censoring assumptions and inverse-probability censoring weights in greater detail; here the key point is that the risk-set adjustment is only defensible when its censoring model and entry mechanism are appropriate.
Worked Example: Checking the Two Contributions
Suppose the target cause is \(k\), the time of interest is \(t=8\) years, and, for illustration, \(G(4)=0.80\) and \(G(8)=0.60\). Consider three hypothetical participants.
Participant A: \(L_A=2\), \(T_A=10\), and \(C_A=6\). The participant has not had an event by year 8, but was censored at year 6. The first indicator is zero because \(C_A\not\ge 8\). The second is also zero because there was no competing event before year 8. The contribution is \(0\). Counting this person as event-free at year 8 would incorrectly treat unobserved follow-up as observed.
Participant B: \(L_B=1\), \(T_B=4\), \(J_B\ne k\), and \(C_B=6\). The competing event was observed because \(C_B\ge T_B\), and it occurred before year 8. The retained contribution is \(G(8)/G(4)=0.60/0.80=0.75\). Censoring at year 6 does not make the contribution zero; the competing event at year 4 is known.
Participant C: \(L_C=3\), \(T_C=10\), and \(C_C=11\). This participant entered by year 8, remained event-free through year 8, and was observed through year 8. The first contribution is \(1\), and the second is \(0\), so the total is \(1\).
The contrast between A and B is important: a person censored before \(t\) without a known event by \(t\) is not observed in the event-free component, while a person with an observed competing event can remain represented through the censoring adjustment.
Why the Weight Has This Form
The retained competing-event contribution uses a ratio of censoring survival probabilities. Under the relevant conditional independent-censoring assumption, the probability of remaining uncensored through \(t\), given that censoring was avoided through the competing-event time \(T_i\), is
This follows from the definition \(G_i(u)=P(C_i\ge u\mid Z_i,L_i)\) and the fact that \(C_i\ge t\) implies \(C_i\ge T_i\) when \(t>T_i\). Dividing the joint probability \(P(C_i\ge t\mid Z_i,L_i)\) by \(P(C_i\ge T_i\mid Z_i,L_i)\) gives the ratio. It is a conditional survival probability, not an inverse probability. Its role is to adjust the retained pseudo-risk contribution as censoring accumulates after the competing event.
The entry condition is separate from this ratio. The indicator \(L_i\le T_i<t\) ensures that the participant had entered observation by the time the competing event occurred and that the event preceded \(t\). The first component likewise requires \(L_i\le t\). A censoring weight cannot create information about a participant before entry, nor recover event histories that the entry mechanism makes unobservable without suitable assumptions and an appropriate estimator.
Specify whether time is measured from diagnosis, age, or another origin, and define the population to which the subdistribution-hazard model refers.
For every participant, retain \(L_i\) and ensure that no risk-set contribution is assigned before that time.
Require observation through \(t\) for the event-free component; for a retained competing event, require observation through \(T_i\) and apply the appropriate censoring adjustment.
Check the truncation and censoring conditions, positivity, and software documentation or validation examples for delayed-entry support.
Worked Risk-Set Calculation
A useful way to audit an implementation is to hand-check a small risk set at a target-cause event time. Suppose a hypothetical target-cause event occurs at \(t=8\), and the fitted covariate is binary, \(Z_i\in\{0,1\}\). Let \(\beta=\log(2)\), so a participant with \(Z_i=1\) has relative subdistribution hazard \(\exp(\beta)=2\). Assume the censoring survival values are \(G(4)=0.80\), \(G(6)=0.72\), and \(G(8)=0.60\).
Worked Example: Auditing a Weighted Pseudo Risk Set
Consider these hypothetical records at year 8: A has \(L=1\), a target event at 8, \(Z=1\), and is observed through the event; B has \(L=2\), remains event-free through 8, \(Z=0\), and is observed through 8; C has \(L=1\), a competing event at 4, \(Z=1\), and \(C=6\); D has \(L=1\), a competing event at 6, \(Z=0\), and \(C=8\); E enters at 9; and F is censored at 7 without a prior event.
At year 8, A contributes with weight \(1\) and covariate multiplier \(2\). B contributes \(1\times 1\). C was observed through the competing event, so its retained weight is \(G(8)/G(4)=0.60/0.80=0.75\), and its covariate multiplier is \(2\). D contributes \(G(8)/G(6)=0.60/0.72=0.8333\), with multiplier \(1\). E has not entered by year 8 and contributes zero. F was censored before year 8 without a known event and contributes zero to this risk set.
The weighted denominator at \(\beta=\log(2)\) is therefore \(2+1+(0.75)(2)+(0.8333)(1)=5.333\), to four significant figures. For this single target event, the corresponding illustrative partial-likelihood factor is \(\exp(\beta Z_A)/5.333=2/5.333=0.3750\). This calculation is a hand-check of the risk-set construction, not a complete model fit or a variance estimate. A software risk set that counts F as event-free, includes E, or drops C solely because \(C_C<8\) does not match the stated construction.
Entry Conditions, Software, and Alternatives
A valid delayed-entry analysis needs more than a risk-set formula. The entry mechanism must allow the intended population or conditional estimand to be identified. If people with early events are systematically less likely to enter, simply analyzing observed entrants can distort the event process. A method that accounts for independent truncation can address the selection induced by delayed observation under its assumptions; it cannot recover an unobserved portion of the population when entry depends on unmeasured event risk in an unmodeled way. Similarly, poor support near early times can make estimation unstable even when formal independence assumptions are plausible.
Worked Example: Deciding Whether an Analysis Is Supported
Suppose a hypothetical study uses age as its time scale. Of 100 people in the source population, 25 experience the target event before their possible enrollment age, 15 have a competing event before enrollment, and 60 enter while event-free. Enrollment ages range from 50 to 70 years, and the analysis is intended to describe subdistribution hazards from age 40.
The 60 observed entrants provide no direct records of the 40 people with events before enrollment. A standard right-censored Fine–Gray fit that treats all 60 as observed from age 40 would fabricate pre-entry follow-up. A fit that merely starts each entrant's record at enrollment is also not automatically valid for the stated age-40 target: it needs a method whose truncation assumptions and risk-set construction identify that target, with adequate support over the relevant ages. If the scientific estimand is instead explicitly restricted to a population and time scale for which the delayed-entry method is valid, that target must be stated and justified. The numbers alone do not establish that entry is independent of event history.
Before fitting, an analyst should confirm that the software accepts entry times specifically for Fine–Gray regression, not just for a cause-specific Cox model. Then inspect documentation or a validation example to verify that entry gates both event-free and retained competing-event contributions, that censoring adjustment is applied as intended, and that standard errors correspond to the implemented estimator. If these features are absent or unclear, the output should not be treated as a valid delayed-entry Fine–Gray analysis.
The main alternative is to fit cause-specific hazard models that accommodate delayed entry and then combine the cause-specific results to estimate cumulative incidence. This targets cumulative-incidence behavior through a different modeling route; it is not equivalent to fitting a Fine–Gray proportional subdistribution hazards model, and its coefficients have a different interpretation. The choice depends on the scientific question and the plausibility of each model. A standard Fine–Gray fit that ignores entry is not a sound shortcut to either approach.
Diagnostics should focus on whether the required data structure and assumptions are credible. Check event and entry timelines, confirm that each observed event occurs after entry, tabulate entry across the analysis time range, and look for times or covariate strata with near-zero censoring survival. Compare software risk-set contributions with hand calculations on a small set of records, including a competing event censored between its event time and \(t\). Investigate sensitivity to plausible entry and censoring models when those choices materially affect estimates. A smoothly converged optimizer does not demonstrate that delayed entry was handled correctly.
Common Mistakes and Limitations
- Counting censored participants as event-free after censoring. If \(C_i<t\) and no event was observed by \(t\), the event-free contribution is zero; observed follow-up does not extend beyond censoring.
- Dropping every competing-event participant once censored. A competing event observed at \(T_i\) remains eligible for the retained component after \(T_i\), even if censoring occurs before \(t\). Apply the stated adjustment rather than requiring \(C_i\ge t\).
- Ignoring entry time. A participant cannot contribute before \(L_i\). Starting everyone at the common origin invents observed risk time for delayed entrants.
- Assuming any survival package's entry option applies to Fine–Gray regression. Support for left truncation in a cause-specific Cox fit does not establish support for the Fine–Gray pseudo risk set or its variance.
- Treating weighting as a cure for selection. A censoring ratio does not correct an invalid truncation assumption, lack of support, or unmeasured dependent entry.
Delayed-entry Fine–Gray estimation is therefore more limited than a routine right-censored fit. Its validity depends on the estimand, entry mechanism, censoring model, and exact software implementation. When these cannot be defended, a different estimand or modeling strategy may be more credible than reporting a numerically precise but unsupported subdistribution-hazard ratio.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- At time \(t\), a participant entered before \(t\), has no event by \(t\), and was censored at \(t-1\). What is the event-free risk-set contribution?
- A competing event occurred at time 5, was observed, and censoring occurred at time 7. At \(t=9\), with \(G(5)=0.90\) and \(G(9)=0.63\), what is the retained contribution?
- Why is a participant who enters after \(t\) excluded from the pseudo risk set at \(t\)?
- What does a delayed-entry option for a cause-specific Cox model fail to establish about a Fine–Gray analysis?
- Give one reason why an entry-aware risk-set formula may still fail to identify the intended population model.
Solutions. (1) Zero: censoring before \(t\) means the participant was not observed event-free through \(t\). (2) The observed competing event is retained with weight \(G(9)/G(5)=0.63/0.90=0.70\); censoring at 7 does not itself remove the known competing event. (3) The participant was not yet observable at \(t\), so including them would assign a contribution before entry. (4) It does not establish that Fine–Gray pseudo risk sets, censoring adjustments, or variance calculations handle delayed entry correctly. (5) Entry may depend on event history in an unmodeled way, or the target times may lack adequate entry or censoring support, so the target is not identified under the assumed method.