Delayed entry: build the risk set when observation starts late
A company first observed at age four has survived to age four, but the registry did not necessarily observe its first four years. Treating that company as if it had been followed from formation adds guaranteed-survival time to the data. Delayed entry requires admitting it to each event-time denominator only after its actual observation begins.
The deliverable is an age-based entry, exit and event ledger, together with the risk set at every observed event. That bookkeeping is necessary for a valid analysis. It does not by itself reconstruct companies that failed before they could enter the registry or prove the assumptions needed to generalize beyond observed entrants.
Distinguish the time origin from observation entry
Suppose the scientific clock is company age from a prespecified formation event. For each company, retain the formation date, first eligible observation date and final event or censoring date. Convert all three to the same age scale. A calendar-year enrollment date and an age in years cannot be compared without that conversion.
Let L be the age at entry and R the observed exit age. In counting-process notation the observed interval is (L,R]. At event age t, a company contributes to the risk set if L < t ≤ R. The open-left, closed-right convention matters at ties: a company entering exactly at t is not included in that interval’s risk set at t, whereas an observed endpoint at t remains represented just before the event.
The R survival Surv documentation, in the time 2 and counting-process descriptions, specifies this interval convention. A software object encodes the intended records; it cannot discover an unknown observation start from an exit date alone.
Work through the event-time denominators
Assume this hypothetical first-exit dataset, measured in company-years of age. An event ends further risk, while a censoring endpoint records the end of observation without the target event.
| Company | Entry age L | Exit age R | Event at R? | Observed interval |
|---|---|---|---|---|
| A | 0 | 3 | yes | (0,3] |
| B | 2 | 4 | no | (2,4] |
| C | 4 | 5 | yes | (4,5] |
At age 3, the risk set contains A and B. C has not entered. With one event among two at risk, the product-limit survival step is 1−1/2=0.5.
At age 4, B reaches its censoring endpoint and C reaches its entry age. There is no event in the table at 4, so neither the departure nor the entry creates a survival-curve drop. Under the stated interval convention, B is represented at age 4 and C becomes eligible immediately after 4. Keep this ordering explicit if an additional event shares that recorded age.
At age 5, C is the only company at risk and has the event. The product-limit factor is zero, so the illustrative curve reaches zero. This extreme final step reflects a single remaining observation; it is not reliable evidence that every company in the population fails by age 5.
For these three records, observed exposure is 3+(4−2)+(5−4)=6 company-years. Incorrectly starting every record at zero would create 3+4+5=12 company-years, adding six unobserved pre-entry years. It would also put C into the age 3 risk set and give a wrong first survival factor 1−1/3=2/3 instead of 1/2.
The survival authors’ multi-state vignette, section 2.1, constructs transition probabilities from the subjects at risk just before a transition and explains the product-limit special case. It supplies the event-time accounting; the company records here are an original calculation, not a fitted real registry.
Left truncation is not simply a missing early measurement
In a left-truncated registry, companies exiting before their eligible entry time can be absent entirely. This differs from a known company whose event time is only known to fall before a threshold, and from right censoring after an observed follow-up period. Those situations carry different information and require different data representations.
If the database began covering a company at age 4, a researcher may know from another record that it existed earlier. Existence alone does not establish complete event ascertainment over ages 0–4. Adding the earlier period requires evidence that the company was eligible and that the relevant event history was actually observed under the study definition.
Likewise, relabeling age 4 as “time zero” changes the question. Time since registry entry can be a legitimate clock, but it describes post-entry survival, not automatically survival from formation. Keep the original age origin when the intended comparison concerns company life cycles.
State what risk-set adjustment can identify
Sondhi’s original left-truncation paper, introduction and section 2.1, distinguishes independent entry and event times from conditional independence given baseline variables. Standard delayed-entry adjustment needs an appropriate truncation-independence condition and suitable censoring assumptions; the intervals alone do not establish them.
For a simple marginal analysis, think of entry timing as independent of the event process under the specified model and observation scheme, with adequate observable support. If troubled companies seek registry coverage just before an exit, that condition is questionable. A correctly coded (start,end] dataset can then still produce a biased population interpretation.
Conditional independence is a different claim: entry and event times may become independent within suitable baseline strata. Estimates conditional on those strata do not automatically give the original population average. Recovering a marginal population distribution may require an appropriate external baseline distribution or additional weighting assumptions. Do not substitute the observed entrants’ covariate mix for a population mix without examining the selection.
A test that fails to find entry-time dependence also does not certify independence. Its power, modeling choices and unmeasured factors remain relevant. Treat diagnostics as evidence about plausibility and discrepancies, not a mechanical identification certificate.
Audit interval construction and duplicate rows
| Check | Evidence to preserve | If unresolved |
|---|---|---|
| Time origin | formation or other initiating event definition | do not mix incompatible age clocks |
| Entry | first eligible observable date and provenance | retain unknown entry; do not default to zero |
| Exit | event/censor date and reason | separate absent events from lost coverage |
| Risk set | explicit membership at every event age | reconcile start/end tie conventions |
| Row structure | IDs and nonoverlapping intervals | repair duplicate or overlapping exposure |
| Generalization | truncation/censoring assumptions and target mix | restrict the interpretation explicitly |
Splitting a company’s observation into several intervals for changing covariates must not create several companies. Adjacent intervals such as (2,3] and (3,4] preserve one continuous observation; overlapping intervals can double-count risk unless the chosen model intentionally represents different states. Zero-length intervals need investigation rather than silent expansion by a small arbitrary time.
When a source lists only a year, ties can reflect date coarsening rather than actual simultaneous entry and event. Preserve the known resolution and specify a consistent convention or suitable analysis. Do not rearrange dates merely to obtain a larger denominator or avoid a software warning.
Preserve the unobservable part of the target
The ledger tells readers which observed companies could contribute at each age. It does not reveal how many never reached entry, what their event times were, or whether their baseline mix matched the observed sample. An external formation cohort can help investigate that gap, but it must use compatible eligibility and endpoint definitions.
If historical coverage is unavailable, report the delayed-entry analysis with its actual support and assumptions. Avoid claiming a complete historical survival distribution just because the software returns a curve starting at one. Where no companies are observed at risk, the data provide no event-time denominator; a smooth fitted line through that region is a model extrapolation.
Update the risk sets when entry dates, event status, entity identifiers or coverage histories are corrected. Separately update the population interpretation if new evidence changes the truncation assumptions or target covariate distribution. Preserve the old ledger so reviewers can see whether a change came from repaired arithmetic, new observation time or a stronger claim about the unobserved population.
Keep the source’s formation date, coverage date and notice-download date as separate fields. A provider backfilling an old notice today does not necessarily extend reliable coverage continuously to formation. Verify what the backfill searched and which events it could miss before changing entry. That distinction lets a genuine historical-coverage improvement update the ledger without treating every newly discovered old document as proof of complete early observation.
FAQ
Can a company be counted before its first registry observation?
Only if additional evidence establishes eligible, adequately observed history for that earlier period. Surviving until entry alone is insufficient.
Does correct delayed-entry coding remove every selection problem?
No. It repairs risk-set timing. Population inference still depends on truncation, censoring and target-population assumptions.