Concept Architecture
Concept
Theoretically, Interval Censoring is a form of censoring in survival analysis in which the exact time of an event is unknown but is known to have occurred within a specified time interval. It arises when individuals are assessed periodically rather than continuously, so the event is observed only to have occurred between two examination times. In health economics, interval censoring is encountered in studies with scheduled follow-up visits and is important for accurately estimating survival, disease progression and transition times used in economic evaluations.
Mathematically, interval censoring is represented by defining lower and upper bounds for the unknown event time. Rather than observing an exact failure time, the likelihood contribution for each individual is the probability that the event occurred within the observed interval. Model parameters are estimated using maximum likelihood or Bayesian methods that explicitly account for interval-censored observations.
In practice, interval-censored survival data are analysed using specialised survival analysis methods rather than standard Kaplan-Meier estimation, which assumes exact or right-censored event times. Parametric, semi-parametric and non-parametric interval-censored models are fitted using statistical software, and the resulting survival estimates are used to inform long-term health economic models.
Purpose
Used to analyse survival data when event times are known only to occur within observation intervals, improving estimation of survival, disease progression and long-term outcomes in health economic evaluations.
Mathematical Formulae
Primary Formula
L? = P(L? < T � R?) = S(L?) ? S(R?)
where:
- T = event time
- L? = lower observation time
- R? = upper observation time
- S(t) = survival function
Supporting Formulae
Likelihood for n independent observations:
L = ????� [S(L?) ? S(R?)]
Model parameters are estimated using:
?? = arg max L(?)
Related Mathematical Methods
- Maximum likelihood estimation
- Survival analysis
- Parametric survival modelling
- Turnbull estimator
- Cox proportional hazards model for interval-censored data
- Bayesian survival modelling
Example
Patients in a chronic disease study are reviewed every six months. A patient has no evidence of disease progression at the 12-month visit but has progressed by the 18-month visit. The exact progression time is unknown and is therefore treated as interval-censored between 12 and 18 months when estimating survival for a cost-effectiveness model.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| IF | =IF(AND(EventTime>LowerBound,EventTime<=UpperBound),1,0) | Identify observations falling within censoring intervals during data preparation. |
| EXP | =EXP(-H2) | Calculate survival probabilities from cumulative hazards. |
| LN | =LN(B2-C2) | Calculate log-likelihood components for interval probabilities in simplified analyses. |
| Solver | Minimise negative log-likelihood | Estimate interval-censored survival model parameters. |
VBA (Optional)
Automate preparation of interval-censored datasets, likelihood calculations and comparison of alternative survival models.
Sources
- Sun J. The Statistical Analysis of Interval-censored Failure Time Data.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Turnbull BW. The empirical distribution function with arbitrarily grouped, censored and truncated data.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials – extrapolation with patient-level data — Nicholas R. Latimer, TSD 14 ed., 2013 (NICE Decision Support Unit (University of Sheffield))
The reference guidance on survival analysis for economic evaluation: fitting standard parametric models (exponential, Weibull, Gompertz, log-logistic, log-normal) to censored trial data and extrapolating to estimate lifetime survival benefit, with a process guide for model selection and justification.
Frequently Asked Questions (6)
What is interval censoring?
A form of censoring in which an event's exact time is unknown but known to fall within a specific interval, such as between visits.
Source: Kalbfleisch & Prentice 2002
What is known about an event under interval censoring?
Under interval censoring the event is known to have occurred, but only that it happened somewhere between two observation points, not the exact moment. This is common when a condition is detected only at scheduled visits, so a recurrence found at one appointment could have begun any time since the last. What is known is a window, not a time, which ordinary methods assuming exact times cannot use directly. Special techniques account for the event lying anywhere in the interval. Collett (2015) describes this.
Source: Collett 2015
How does interval censoring arise?
Interval censoring arises when the status of individuals is checked only at discrete times, such as periodic clinic visits or scheduled tests, so that an event detected at a visit is known to have occurred since the previous one but its exact time is not observed. For example, disease progression found at a scan is known to lie between the last clear scan and the current one. This intermittent observation, common in studies with periodic follow-up, produces interval-censored event times.
Source: Kalbfleisch & Prentice 2002
How is interval-censored data analysed?
Interval-censored data are analysed using methods that account for the event time lying within an interval rather than being known exactly, since standard methods assuming exact times are not directly applicable. Approaches include specialised non-parametric estimators of the survival function for interval-censored data and parametric or regression models that use the interval information in the likelihood. These methods incorporate that the event occurred within a known interval, giving valid estimates of survival despite the exact times being unobserved.
Source: Collett 2015
How does interval censoring differ from right censoring?
Interval censoring and right censoring differ in what is known about the event time. In right censoring, the event has not occurred by the last observation, so the event time is known only to exceed it. In interval censoring, the event has occurred but its exact time is unknown, known only to lie within an interval between observations. Right censoring gives a lower bound on the event time, while interval censoring gives both a lower and upper bound, so different methods are needed for each.
Source: Kalbfleisch & Prentice 2002
Why does interval censoring require special methods?
Interval censoring requires special methods because standard survival analysis assumes exact event times or right censoring, and treating an interval-censored event as occurring at a single point, such as the interval's end, can bias estimates. Proper methods use the information that the event occurred within a known interval, incorporating this into the analysis so that survival is estimated correctly. Ignoring the interval nature of the censoring, by imputing a specific time, distorts the estimated timing of events, which the appropriate methods avoid.
Source: Collett 2015
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 21 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/interval-censoring
- Term code
- HE-EM-SM-035
Stable URI · Machine-readable · Resolvable · CC BY 4.0