Interval censoring arises in survival analysis when an event time T is not observed exactly but is only known to lie in an interval (L,R] — typically because the subject is inspected at discrete times and the event is known only to have occurred between two inspections. It generalizes right censoring (the interval is (L,∞)) and left censoring (the interval is (0,R]), and it degrades both estimation efficiency and the tidy structure of right-censored inference (Anderson-Bergman 2017).
Key Ideas
Current status vs. mixed case.Current status data comes from a single inspection per subject, so every observation is purely left- or right-censored (only whether the event has happened yet is known). Mixed case interval censoring comes from repeated inspections and yields a mix of left-, right-, uncensored, and genuinely interval-censored observations. The observation/inspection times are assumed independent of the event.
NPMLE (Turnbull estimator). The nonparametric maximum likelihood estimator (Turnbull 1976) generalizes the Kaplan-Meier survival curve to interval-censored data; it places mass only on a set of "Turnbull intervals" and is computed by EM-type / iterative convex minorant algorithms. It is preferred for diagnostics but is inefficient: an n1/3 convergence rate for current-status data instead of the usual n1/2.
Semiparametric regression loses its shortcut. Under right censoring, Cox's partial likelihood separates the regression coefficients from the baseline hazard; under interval censoring this separation fails, so semiparametric proportional-hazards and proportional-odds fits estimate baseline and regression jointly and rely on the bootstrap for standard errors.
Model tiers. Analysts choose among nonparametric (Turnbull, assumption-light but inefficient), semiparametric (proportional hazards / proportional odds / accelerated failure time with an unspecified baseline shape), and fully parametric (efficient but sensitive to the baseline family) — trading robustness against efficiency.
Diagnostics and imputation. Because parametric baselines are hard to check on interval data, the semiparametric/nonparametric fits serve as diagnostic references, and multiple imputation of the bracketed event times is a common device for downstream analysis.
How It Works
Encode each observation as an interval (Li,Ri] (with Ri=∞ for right-, Li=0 for left-censored).
Write the likelihood in terms of the survival function: each subject contributes S(Li)−S(Ri), the probability mass the event falls in its interval.
Fit the chosen model — NPMLE (Turnbull) for an assumption-free survival curve, a semiparametric PH/PO model for covariate effects with a flexible baseline, or a parametric AFT/PH/PO model for efficiency.
Assess fit by comparing parametric baselines to the nonparametric estimate; obtain uncertainty by bootstrap for the semiparametric fits.
Why It Matters
Ubiquitous observation scheme. Panel inspections, periodic medical exams, and reliability testing all produce interval-censored times; treating them as exact (imputing midpoints) biases estimates.
Where the standard toolkit breaks. It is the case that defeats the two conveniences of right-censored survival analysis — the n-efficient nonparametric estimator and the partial-likelihood separation — so it needs its own estimators and software.
Practical tooling. Reliable implementations (e.g., the R package icenReg) put NPMLE, semiparametric, and parametric interval-censored regression within reach for applied work.
Open Questions
Efficiency of the NPMLE. The slow (n1/3) convergence for current-status data limits precision; improving efficiency or building better smoothed alternatives is ongoing.
Spline baselines and knot selection. Flexible spline baselines improve fit but knot placement remains an open, somewhat ad hoc choice.
Inference beyond the bootstrap. Analytic standard errors for semiparametric interval-censored regression (avoiding costly resampling) are not fully settled.