Inspection Schedules and Coarsened Event Times

How inspection schedules coarsen event times

When an event is detected only at scheduled inspections, its exact time is usually unknown. A negative inspection followed by a positive one locates the event between two visits; it does not show that the event occurred at the later visit. The inspection schedule therefore determines the resolution of the event-time information and, under suitable assumptions, the likelihood contribution for each subject.

This tutorial focuses on how fixed, irregular, and subject-specific schedules produce observed intervals, and on the assumptions needed to use those intervals in an event-time likelihood. The likelihood contribution comes from the probability of the interval actually observed, not from assigning the event to an inspection date.

The open left endpoint and closed right endpoint follow the convention that a negative result at \(L_i\) establishes \(T_i>L_i\), while a positive result at \(R_i\) establishes \(T_i\le R_i\). For a continuous event-time distribution, changing endpoint inclusion does not change interval probabilities, but the convention is useful for consistent data recording. If inspection results can be wrong, or if an event can resolve before an inspection, this simple interval representation needs a model for measurement or state history as well.

From schedule to observed interval

A fixed schedule uses the same prespecified visit times for everyone, such as inspections at months 3, 6, 9, and 12. An irregular schedule has unequal gaps between visits, which may occur even when each subject follows a prespecified plan. A subject-specific schedule allows visit times to differ across subjects, for example because of enrollment dates or individually assigned follow-up. These descriptions concern the visit times; they do not by themselves establish whether the schedule is statistically noninformative.

Given a schedule and the recorded results, locate the last negative and first positive inspections. If the event is detected at the first visit, use the left-censored interval. If it is never detected before follow-up ends, use a right-censored observation. The visit times give the boundaries; they do not generally reveal where inside an interval the event occurred.

For a continuous event time with cumulative distribution function \(F(t\mid X_i)\), the probability of an interval observation is

The subtraction works because the event \(T_i\le R_i\) consists of the disjoint possibilities \(T_i\le L_i\) and \(L_i<T_i\le R_i\). A right-censored observation at \(L_i\) contributes \(S(L_i\mid X_i)=P(T_i>L_i\mid X_i)=1-F(L_i\mid X_i)\). These are probabilities for the coarsened observations; they do not require imputing an event time within an interval.

Likelihood contributions and schedule assumptions

For independent subjects, if the schedule is fixed by design or is independent of the event time conditional on \(X_i\), the event-time likelihood can condition on the observed schedule. For an interval observation, its contribution is the interval probability above; for right censoring at \(L_i\), it is \(S(L_i\mid X_i)\). If the event time is exact, its contribution is a density, but exact event times are not the usual result of periodic inspection.

For competing causes with known event cause \(J_i=k\), define \(F_k(t\mid X_i)=P(T_i\le t,J_i=k\mid X_i)\). A detected cause-\(k\) event in \((L_i,R_i]\) contributes \(F_k(R_i\mid X_i)-F_k(L_i\mid X_i)\). The schedule still determines the interval, while the cause identifies which cause-specific cumulative incidence probability applies.

A key distinction is whether the schedule is externally set or responds to recorded outcomes. If visits are scheduled in advance, or are independent of event-time information conditional on \(X_i\), conditioning on the schedule is generally appropriate. Subject-specific schedules can also be handled this way; differing visit times alone do not make a schedule informative.

In contrast, suppose a clinician schedules an earlier next inspection after a concerning result and a later one after a reassuring result. The next visit time depends on recorded history. In this adaptive setting, it is not generally valid to claim that the marginal schedule distribution \(p(V_i\mid X_i)\) is free of the event-time parameter: the recorded history affecting future visits has a distribution governed by that parameter. A safer representation uses the sequential schedule mechanism conditional on the history available when each decision is made.

Here \(Y_{i,<j}\) denotes the recorded history before inspection \(j\), and \(V_{i,<j}\) the earlier visit times. This factorization makes the timing mechanism conditional on the history actually available at the decision point. If that mechanism is known and depends only on recorded history and covariates, its conditional factors may contain no additional event-time parameter. But the distribution of the recorded history itself still depends on the event-time model. Therefore one must not replace this sequential representation with a claim that the marginal schedule distribution is parameter-free. If visit timing depends on unrecorded clinical information related to the latent event time, the schedule may be informative and requires additional modeling or sensitivity analysis.

Subject-specific schedules and a competing-risk contribution

Justification, diagnostics, and alternatives

The likelihood contribution follows directly from the information recorded. A positive inspection at \(R_i\), preceded by a negative inspection at \(L_i\), says that the event belongs to \((L_i,R_i]\). Under the assumed event-time model, the probability of this observation is the integral of the event-time density over that interval:

The right-censoring contribution follows similarly:

These expressions are justified when the interval or censoring time is a valid summary of the observed event information and the schedule assumptions make conditioning on inspection times appropriate. A schedule can be technically fixed yet still be associated with event risk if, for example, different risk groups were assigned different schedules and those groups are not included in \(X_i\). Conversely, an irregular or person-specific schedule is not automatically invalid if its assignment is independent of the latent event time conditional on the variables modeled.

Useful diagnostics begin with the inspection process itself. Tabulate visit gaps and interval widths, examine whether schedules differ by observed covariates or recorded outcomes, and compare reasons for missed or added visits across risk groups. Verify that each interval has a preceding negative result and a subsequent positive result, or that a right-censored record truly has no positive result through its stated censoring time. Check for impossible sequences, such as a negative inspection after a reliably detected irreversible event. For adaptive schedules, document which information was available at each scheduling decision and whether relevant decision variables were recorded.

A common mistake is to treat every first positive inspection as an exact event time. That substitutes \(R_i\) for an unknown \(T_i\in(L_i,R_i]\), systematically assigning events to visit dates and discarding the interval information. The correction is to use the interval probability, \(F(R_i)-F(L_i)\), under the appropriate schedule assumptions. Another mistake is to assume that a known, outcome-dependent scheduling rule makes the marginal schedule distribution independent of the event-time parameter. It does not: recorded histories influence future visits and themselves have event-model-dependent probabilities.

The main alternatives depend on the observation design. Exact event-time methods apply when the event time is actually measured, while ordinary right-censoring methods apply when only event-free follow-up through a time is known. Current-status data are a special case with one inspection per subject: each observation states only whether the event has occurred by that inspection. Treating visit times as discrete event times is sometimes used as an approximation, but it changes the data model and can distort event-time inference when inspection gaps are appreciable. When the schedule is outcome-adaptive or potentially informative, a joint or sequential model for event and observation processes may be required; the simple conditional interval likelihood alone does not resolve that problem.

Even under valid schedule assumptions, interval observations provide limited information about events within wide gaps. Different event-time distributions can assign similar probabilities to the observed intervals, particularly when visits are sparse. The likelihood can estimate only what the inspection pattern and model jointly support; it cannot recover each subject's exact event time without additional assumptions.