Censoring and Left-Truncation Audit¶

Audit — instantiates Survival-Conditioned Persistence Forecasting

Reconstructs the failures and delayed entrants missing from a survivor sample, so a persistence forecast is not silently biased by who happened to be observed.

Before you can ask how much longer a thing will last, you have to trust the record of how long things like it have lasted — and that record almost always over-represents survivors. Censoring and Left-Truncation Audit repairs the observation process rather than the model: it hunts down the cases that never entered the sample and the cases whose lifetimes are only partly seen, and makes both explicit before any curve is fit. Its defining move is to treat "the data we have" as itself the product of a selection process — failures that were deleted, still-alive cases whose lifetimes are known only to exceed their current age, and entities that entered the sample only after surviving long enough to be noticed. Fix the observation process here and every downstream estimate inherits an honest denominator; skip it and "old things last" becomes a self-confirming artifact.

Example¶

An analyst wants to know how long a mutual fund typically survives before it closes, using a commercial database of funds currently on offer. The audit finds three distortions baked into that sample. First, right-censoring: most listed funds are still open, so their true lifetime is unknown — only that it exceeds their present age; treating current age as final biases the estimate down. Second, survivorship deletion: funds that liquidated or were quietly merged away have been purged from the live database, so the failures that matter most are invisible. Third, left-truncation: a fund typically enters the database only once it clears an asset threshold, years after launch, so early deaths that never reached that threshold were never eligible to be counted.

The audit rebuilds the risk set: it recovers delisted and merged funds from archival snapshots, decides that "merged into another fund" counts as an exit rather than a silent survival, and conditions on each fund's entry age so late entrants don't masquerade as having survived from birth. The naive database answer — an average life measured only over healthy survivors — collapses toward a much shorter, censoring-corrected figure, and, more importantly, arrives with its observation window and entry rule written down.

How it works¶

The audit is a pass over the observation model, not the lifetime model. It separates and handles four distinct ways the record lies:

Right-censoring — cases still alive at the cutoff; kept as "survived at least this long," never as completed lifetimes.
Left-truncation / delayed entry — cases observable only after surviving to an entry condition; handled by conditioning on entry age so they join the risk set late instead of anchoring it.
Survivorship deletion — failed cases erased from registries, archives, or markets; recovered as far as the trail allows and flagged where it goes cold.
Exit ambiguity — merges, dormancy, relaunches, and functional-but-unsupported states that blur "still alive" versus "exited"; resolved by a written exit rule.

It does not estimate a hazard, choose a distribution, or issue a forecast — it hands the corrected risk set and exit definition to the mechanisms that do.

Tuning parameters¶

Exit-event strictness — how narrowly "exit" is drawn (hard liquidation only, versus any loss of function or support). Stricter definitions cut phantom censoring but can miss slow deaths.
Truncation depth — how far back delayed entrants are reconstructed. Deeper reconstruction removes more survivor tilt but costs archival digging.
Failure-recovery effort — how hard vanished failures are hunted across archives and secondary records; more effort shrinks survivorship bias with sharp diminishing returns.
Censoring treatment — whether still-alive cases are retained as censored observations or dropped; dropping them biases longevity down, ignoring the censoring biases it up.
Recovered-vs-residual split — how much unrecoverable loss is carried forward as stated uncertainty rather than assumed to be zero.

When it helps, and when it misleads¶

Its strength is that it targets the archetype's single most common evidential failure — fitting the survivor set while the failures are invisible — and turns "old things last" back into a testable claim by restoring the missing denominator. It is the one step that can reveal a striking longevity pattern to be mostly an artifact of who got recorded.

Its failure modes are quieter. You cannot audit what left no trace, so a thorough-looking reconstruction can still understate bias when whole classes of failure vanished cleanly — the audit's own completeness is unfalsifiable past a point. It is also easily run backwards: a survivorship correction softened just enough to spare a favored long-life estimate. And it over-corrects if maintained or relaunched cases are miscoded as deaths. The discipline is to write down the observation window and entry rule, carry the naive and corrected estimates side by side, and treat unrecoverable failures as a stated floor on bias rather than a zero. The canonical warning is survivorship bias — the reason to reinforce the bombers where the returning planes were unscathed.^[1]

How it implements the components¶

This audit fills the observation-integrity components — the ones that decide whether the survival record can be trusted at all:

selection_and_censoring_check — its primary output: a risk set with right-censoring, left-truncation, and survivorship deletion each identified and handled.
observed_survival_age_record — it validates and corrects this record, flagging still-alive cases, delayed entrants, and erased failures rather than taking the raw sample at face value.
exit_event_definition — it pins down what counts as an observed exit versus an unobserved (censored) one, because ambiguous exits are exactly what manufacture phantom censoring.

It does not fit the lifetime distribution or read hazard shape (Hazard-Shape Diagnostic, Lifetime Distribution Comparison), test whether the environment stays comparable (Stationarity Test), or decide whether the subject is eligible at all (Non-Aging Eligibility Review).

Instantiates: Survival-Conditioned Persistence Forecasting — it supplies the censoring-corrected risk set every later step depends on.
Sibling mechanisms: Survival or Time-to-Event Analysis · Hazard-Shape Diagnostic · Lifetime Distribution Comparison · Age-Conditioned Remaining-Life Table · Reference-Class Forecasting · Non-Aging Eligibility Review · Stationarity Test · Change-Point Detection · Rolling Window Comparison · Forecast Backtesting · Historical or Holdout Coverage Backtest · Periodic Durability Inspection · Lindy Decision-Horizon Review

Notes¶

The audit is a dependency, not a verdict: it never says how long anything lasts, only whether the survival record is fit to be read. Because every downstream estimate inherits its residual, unrecoverable bias, the audit's stated uncertainty has to travel with the data — a forecast built on an audited-but-still-thin failure record is only as clean as that record, and should say so.

References¶

[1] Survivorship bias — reasoning from the cases that remained visible while the failures dropped out of sight. The canonical illustration is Abraham Wald's WWII analysis of returning bombers: armor belonged where the survivors were not hit, because planes hit there did not come back to be measured. The same logic governs any lifetime sample assembled from the things still around to be sampled. ↩