Maximal Intersections in Interval-Censored Data

Why Maximal Intersections Matter

With interval-censored data, an event is known to have occurred in an observation set, but its exact time is not recorded. The interval likelihood therefore depends on how much probability the fitted distribution assigns to each subject’s observation set. A support representation that keeps every possible time point is unwieldy; a coarse representation that merges times with different observation compatibility can be wrong. Maximal intersections provide a data-driven set of regions for expressing the likelihood without retaining irrelevant distinctions.

The introductory material on the Turnbull estimator establishes the interval-probability likelihood. Here the focus is the construction and meaning of its support regions. Assume independent subjects have event times governed by a common distribution \(F\), and that each observed interval is a correctly recorded set containing that subject’s event time. The basic likelihood is appropriate when the interval-observation mechanism is ignorable for inference about \(F\), such as when inspection schedules are fixed by design or otherwise independent of the event time under the model. If the observation process is informative, it generally needs its own model.

For example, an observation recorded as \((L_i,R_i]\) includes times greater than \(L_i\) and at most \(R_i\). Left-censored and right-censored observations can be written using an appropriate unbounded endpoint, such as \((-\infty,R_i]\) or \((L_i,\infty)\). Use the endpoint convention in the data consistently: a shared endpoint may belong to one interval and not another, changing whether their intersection is empty. If the collection is finite, any collection with a nonempty common intersection can be extended to a maximal one by adding compatible observations until none remain.

Finding the Maximal Intersections

A practical construction is to identify groups of observation sets that can all contain one common event time, then extend each group until no further observation set can be added. The intersection of each resulting maximal group is a support region. Two distinct maximal groups cannot have a common time: if they did, their union would also have a common intersection, contradicting maximality. This is why maximal intersections form nonoverlapping regions, subject to the chosen endpoint convention.

The intersection can be calculated by taking the largest lower endpoint and the smallest upper endpoint, then checking whether the resulting set is nonempty under the endpoint conventions. For intervals \((L_i,R_i]\), a common intersection requires \(\max_{i\in J}L_i<\min_{i\in J}R_i\). With other open or closed endpoints, equality at a boundary must be checked directly rather than resolved by this strict-inequality shortcut.

How These Regions Enter the Turnbull Likelihood

Write \(p_j\) for the probability mass assigned to maximal intersection \(M_j\), with \(p_j\geq 0\) and \(\sum_j p_j=1\). Define \(a_{ij}=1\) when \(M_j\subseteq A_i\), and \(a_{ij}=0\) otherwise. In this construction, a maximal intersection corresponds to a maximal compatible collection: its members’ observation sets contain the entire region, and intervals outside that collection do not contain its points. The interval probability for observation \(i\) is represented by the sum of the masses in regions compatible with it. The likelihood is

This representation does not say that an event time is known to be a particular point within a region. It says that, for the interval likelihood, the region’s probability mass has the same observation-compatibility pattern. The fitted distribution may be represented computationally by placing a mass at a representative time in the region, but that representation is not evidence that the true event-time distribution has an atom at that time.

Why the Support Reduction Is Justified

The key argument is that mass on a time compatible with a nonmaximal collection can be moved to a time compatible with a maximal extension without reducing any interval probability. For any time \(t\), let \(S(t)=\{i:t\in A_i\}\), the set of observations containing it. Because there are finitely many observations, \(S(t)\) can be extended to a maximal compatible collection \(J\). Every time in \(M_J\) is contained in all observation sets indexed by \(J\), including every set in \(S(t)\). Moving mass from \(t\) to a point in \(M_J\) therefore cannot remove that mass from the probability of an interval that contained \(t\); it may add compatibility with other intervals in \(J\).

A time outside every observation set contributes to none of the interval probabilities. Assigning probability mass there instead of to a compatible region can only waste probability under this likelihood. Thus, for interval-probability observations, there is a likelihood maximizer supported on the maximal intersections. This is a support-reduction result: it does not claim that every maximizer is unique, nor that the data identify where within a region the event-time distribution places its mass.

This justification depends on treating the observations as sets and using their interval probabilities. An exact event time is different. For a continuous event-time distribution, \(P(T=t)=0\), so an exact observation contributes a density (or an appropriate mixed-likelihood contribution), not the probability of the singleton \(\{t\}\). A singleton can be used as an observation set only if the model explicitly treats the time as an atom and uses the probability of that atom. Do not insert an exact time into the interval-probability likelihood by writing \(F(\{t\})\) under a continuous model.

Diagnostics, Alternatives, and Limitations

Before using the resulting incidence structure, check that every observation has at least one compatible support region, that the endpoint rules were applied consistently, and that each proposed collection really is maximal. A row of zeros in the incidence structure usually signals a construction or coding error: no candidate region can satisfy that observation. A support region assigned to an observation that does not contain it is also an error. Repeated or identical incidence columns are possible; they indicate regions that the likelihood cannot distinguish through these observations, not necessarily a calculation mistake.

A useful check is to list, for each region, exactly which observation indices contain it. If two proposed maximal intersections share a time, their collections can be combined, so they were not both maximal. If a region can be extended by another observation set while retaining a nonempty intersection, the collection was not maximal either. These checks are especially important with rounded visit times, where changing a strict endpoint to an inclusive one can change the overlap pattern.

Maximal intersections are one way to express the Turnbull likelihood’s relevant support. A finer partition at all interval endpoints can be used instead, but may create multiple regions with identical incidence patterns; those distinctions do not change the interval likelihood. An arbitrary time grid is simpler to implement in some settings but can omit useful regions or impose an unsupported restriction. Parametric survival models offer a different alternative by specifying a distributional form, which can yield smoother estimates but depends on that model specification. The maximal-intersection representation avoids choosing a parametric shape, but it does not recover precise event times within a region.

The construction also does not repair nonignorable inspection or censoring, measurement errors in interval endpoints, or an incorrectly specified observation history. With covariates, multiple event types, or panel-observed state changes, the likelihood and support structure must reflect those features; the simple common-distribution interval likelihood above is not a substitute for those models. Maximal intersections organize the available interval information, but they cannot create information the observation schedule did not record.