Interval-censored support resolution times

Last updated: 5 Oct 20265 min read

tutorial

An interval-censored event is known to have occurred between two inspections, while its exact time is unknown; neither endpoint is an observed event time.

Preserve the observation interval

A batch system checks case status every few days. A case unresolved at day two but resolved when next checked at day five has a resolution time in (2,5]. Labelling it as a day-five resolution creates an artificial delay; labelling it as day two creates an artificial improvement. Store the last confirmed unresolved age and first confirmed resolved age. An open case last checked at day seven is right-censored beyond seven. Right censoring and interval censoring carry different information.

Handle interval boundaries consistently

The teaching ledger has several (left,right] resolution intervals and one right-censored case (7,infinity). A case resolved at the first inspection may be left-censored within (0,first inspection]. If an event timestamp is exact, keep it as exact rather than broadening it to an arbitrary daily window. Reject a negative age, a zero-width interval in this simplified model, or a right endpoint earlier than its left. A system that changed inspection cadence can change interval widths without changing the underlying resolution process.

Use a likelihood that respects what was seen

For an illustrative exponential resolution model with rate lambda, the probability of resolution in (L,R] is exp(-lambda L) minus exp(-lambda R). The probability of still being unresolved beyond L is exp(-lambda L). Summing the log of the appropriate contribution for each case yields a likelihood that uses interval information without inventing exact event times. The grid search below is deliberately small and parametric; it is not a general nonparametric interval-censored survival estimator.

Separate model fit from business interpretation

A constant hazard assumption can be implausible when new cases are easy and old cases become increasingly difficult. Inspect whether predicted inspection-state shares match observed ones by interval and queue. A fitted rate is conditional on the inspection schedule and inclusion rule; it should not be compared with a Kaplan–Meier curve built from invented exact times. If the goal is a six-day service claim, report an interval-aware estimate and uncertainty under a method that supports the observation pattern.

Build a verification route

For production, use an implementation designed for interval censoring and verify its supported interval conventions. Keep raw inspections so the bounds can be rebuilt after a status correction. Avoid replacing infinity with a giant made-up timestamp in an export. The review project blocks model release when interval cases are silently forced into the exact-time path.

Implementation

from math import expm1, log

def interval_resolution_log_likelihood(observations, resolution_rate):
    if resolution_rate <= 0:
        raise ValueError("rate must be positive")
    total = 0.0
    for last_unresolved_day, first_resolved_day in observations:
        if last_unresolved_day < 0:
            raise ValueError("negative observation age")
        if first_resolved_day is None:
            total -= resolution_rate * last_unresolved_day
            continue
        if first_resolved_day <= last_unresolved_day:
            raise ValueError("invalid resolution interval")
        interval_width = first_resolved_day - last_unresolved_day
        total += -resolution_rate * last_unresolved_day
        total += log(-expm1(-resolution_rate * interval_width))
    return total

inspections = [(0, 2), (2, 5), (4, 7), (7, None), (0, 3)]
candidate_rates = [step / 1000 for step in range(20, 501, 2)]
best_rate = max(candidate_rates, key=lambda rate:
                interval_resolution_log_likelihood(inspections, rate))
assert 0.02 <= best_rate <= 0.5
assert interval_resolution_log_likelihood(inspections, best_rate) >= (
    interval_resolution_log_likelihood(inspections, 0.1))

Performance and operating cost

With N observations and G candidate rates, the explicit grid is O(N × G) time and O(N) input space. A general interval-censored fit needs more careful optimization and uncertainty calculation. The dominant data risk is losing inspection bounds during extraction.

Common Mistakes

Do not substitute the right inspection time as the exact resolution time.
Do not call a right-censored case an interval-censored resolution.
Do not present a constant-rate grid search as a model-free survival curve.