Observation Patterns for Interval-Censored Events

Why Inspection Patterns Matter

In many studies, an event is not observed continuously. A participant may have a negative examination at one visit and a positive examination at the next, but the event time itself is not recorded. The data then locate the event within a span of time rather than at a single instant. Correctly representing that span is essential: writing down an interval preserves what was observed, while assigning an unsupported single time can change the data.

This tutorial concerns observation patterns for a single event time. It explains how inspection times and event-status results determine whether the time is exact, left-censored, right-censored, or interval-censored. It does not assume that the event time is uniformly distributed within an observed interval.

The word persistent matters. These definitions apply when the event, once it has occurred, remains detectable as having occurred. A one-time diagnosis that remains part of the record can have this property; a temporary symptom that can appear and disappear does not. In the latter case, a sequence of positive and negative visits cannot be translated by the rules below without a model of the changing state.

Inspection times must be recorded on a common time scale and with a stated origin. A time origin might be study entry or birth. If a participant is known to be event-free at a baseline time \(a\), then \(T>a\). That baseline information supplies a lower bound even if no later negative visit is available.

From Visit Results to an Event-Time Set

For a persistent event, the observed results have a simple pattern: zero or more negative inspections followed by positive inspections. The last negative inspection gives the lower bound; the first positive inspection gives the upper bound. With the convention that a negative result at \(L\) implies \(T>L\), and a positive result at \(R\) implies \(T\leq R\), the combined information is \((L,R]\).

The parentheses and bracket encode the observations, not a claim about the event-time distribution. The left endpoint is excluded because the participant was still event-free at that time. The right endpoint is included because a positive result at that time is consistent with an event occurring exactly then. For a continuously distributed event time, the probability of an exact endpoint may be zero, but the convention still makes the observation pattern unambiguous.

In a study with a known event-free origin \(a\), a first positive result at \(R\) and no later or earlier informative negative inspection usually give \((a,R]\), a left-censored time relative to the study's observation history. If the event-free status at \(a\) is known, it is more precise to report that finite lower bound. In everyday use, “left-censored” often refers to the fact that the event occurred before the first inspection that detected it.

Right censoring is different from an interval that ends at the last study visit. If event-free status is confirmed at \(L\), but there is no subsequent event observation, the data say \(T>L\). They do not say that the event occurred before some known future time. A scheduled visit at which the participant is lost to follow-up is not itself evidence of event-free status at that visit.

Worked Translations

Why the Interval Follows From the Observations

The translation can be justified by intersecting the constraints imposed by individual inspections. Let \(N\) be the set of times at which the event was not yet present and \(P\) the set of times at which it was present. Under the persistent-event definition, each negative visit at \(v\in N\) imposes \(T>v\), and each positive visit at \(v\in P\) imposes \(T\leq v\). Therefore the data require

when both sets are nonempty. Thus the consistent event times are \((\max(N),\min(P)]\). Earlier negative results and later positive results do not tighten those two bounds, because they are weaker than the last negative and first positive constraints. This also explains why the observation pattern should be checked before it is summarized by a single interval.

If there is no positive visit, the upper-bound constraint is absent and the result is right-censored after the last known-negative time. If there is no observed negative visit before a positive visit, the lower bound comes only from the known event-free origin, if there is one. If an exact event date is available, it provides more information than the inspection-based bounds.

Assumptions, Diagnostics, and Alternatives

The translation is valid when inspection times and results are correctly recorded, the event definition is consistent across visits, and a positive result represents occurrence by that time while a negative result represents nonoccurrence through that time. It also requires the event to be persistent for the simple negative-then-positive pattern to apply. These are assumptions about measurement and the event definition; they do not assert that visits occurred at random or that the inspection schedule is independent of event risk.

For this task, the inspection schedule can be translated into bounds without making a model for the event-time distribution. Later statistical analyses may require additional assumptions about how observation times were chosen or how loss to follow-up relates to event risk. Those assumptions cannot be inferred merely from a correctly constructed interval.

A useful diagnostic is to inspect each participant's full sequence, not only a final summary field. A positive result followed by a negative result is incompatible with a persistent event under error-free measurement. Possible explanations include data-entry errors, a false positive, a false negative, a changing or reversible condition, or an incorrect event definition. Do not silently force such a record into \((L,R]\); resolve the discrepancy or use a framework that represents the actual observation process.

The main alternative observation pattern is exact-time recording, often from a reliable event record. It identifies \(T=t\), rather than a range. Right-censored data provide only a lower bound, while interval-censored data provide both bounds. Treating interval-censored times as if they were right-censored discards the observed upper bound; treating them as exact at the first positive visit substitutes detection time for event time.

Midpoint substitution is another tempting shortcut. Replacing \((L,R]\) by \((L+R)/2\) gives a convenient number but asserts information not contained in the observations. It can distort summaries and analyses, especially when interval widths vary or event times are concentrated unevenly within intervals. The interval itself is the faithful record. The midpoint is justified only under a separately stated approximation or model, not by the inspection pattern alone.