Concept Architecture
Concept
Theoretically, Left Censoring is a form of censoring in survival analysis in which the event of interest has already occurred before the individual first enters observation, but the exact event time is unknown. Rather than observing the occurrence of the event, only an upper bound on the event time is available. In health economics, left censoring arises in studies where disease onset, biomarker progression or other clinical events occur before enrolment, requiring specialised statistical methods to avoid biased survival estimates.
Mathematically, left censoring is represented by observing only that the event time is less than or equal to a known censoring time. The likelihood contribution for a left-censored observation is therefore the cumulative distribution function evaluated at the censoring time rather than the probability density associated with an exact event time. Model parameters are estimated using maximum likelihood or Bayesian methods that explicitly accommodate left-censored observations.
In practice, left-censored data are analysed using specialised survival models because standard Kaplan-Meier estimation assumes right censoring. Parametric, semi-parametric and non-parametric methods capable of handling left-censored observations are implemented in statistical software to estimate survival distributions, disease progression and long-term outcomes for health economic analyses.
Purpose
Used to analyse survival data in which events occur before observation begins, enabling unbiased estimation of survival, disease progression and long-term outcomes in health economic evaluation.
Mathematical Formulae
Primary Formula
L? = P(T � C?) = F(C?)
where:
- T = event time
- C? = left-censoring time
- F(C?) = cumulative distribution function
Supporting Formulae
Relationship between cumulative distribution and survival:
F(t) = 1 ? S(t)
Maximum likelihood estimation:
?? = arg max L(?)
Related Mathematical Methods
- Maximum likelihood estimation
- Survival analysis
- Parametric survival modelling
- Turnbull estimator
- Interval censoring methods
- Bayesian survival modelling
Example
A study investigates the time to seroconversion following infection, but patients enter the study only after routine screening. A participant is already seropositive at enrolment, indicating that the event occurred before the first observation. The event time is therefore treated as left-censored and incorporated into the likelihood when estimating the survival distribution for subsequent health economic modelling.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| IF | =IF(EventTime<=CensorTime,1,0) | Identify left-censored observations during data preparation. |
| NORM.DIST | =NORM.DIST(CensorTime,Mean,SD,TRUE) | Illustrate cumulative probabilities for exploratory analyses. |
| LN | =LN(B2) | Calculate log-likelihood components during parameter estimation. |
| Solver | Minimise negative log-likelihood | Estimate parameters of left-censored survival models. |
VBA (Optional)
Automate identification of left-censored observations, likelihood calculations and estimation of survival models accommodating left-censored data.
Sources
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Lawless JF. Statistical Models and Methods for Lifetime 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)
Frequently Asked Questions (6)
What is left censoring?
A form of censoring in which an event is known to have occurred before an observed time, but its exact prior timing is unknown.
Source: Kalbfleisch & Prentice 2002
What is known about an event under left censoring?
Under left censoring the event of interest is known to have already happened by the time observation begins, but exactly when it occurred beforehand is unknown. A patient found at their first examination to have already developed a condition illustrates this, since the onset lies somewhere before that visit, at an unrecorded time. What is known is an upper bound on the event time rather than the time itself. This is the mirror image of right censoring, where the event lies somewhere after the last observation. Klein and Moeschberger (2003) describe it.
Source: Klein & Moeschberger 2003
How does left censoring arise?
Left censoring arises when the event of interest has already occurred by the time observation begins, so its exact timing before that is unknown. For example, if a study measures the age at which a condition first appeared and some participants already have it at enrolment, their onset time is left-censored, known only to be before enrolment. It occurs whenever individuals enter observation already having experienced the event, so only an upper bound on the event time is known.
Source: Kalbfleisch & Prentice 2002
How does left censoring differ from right censoring?
Left censoring and right censoring differ in what is known about the event time. In left censoring, the event has already occurred by the first observation, so its time is known only to be earlier, giving an upper bound. In right censoring, the more common form, the event has not occurred by the last observation, so its time is known only to be later, giving a lower bound. The two involve opposite kinds of incomplete information and require different handling in analysis.
Source: Collett 2015
How is left-censored data analysed?
Left-censored data are analysed using methods that account for the event time being known only to lie before a certain point, incorporating this information into the likelihood rather than treating the censoring time as the event time. Specialised estimators and parametric or regression models handle left censoring, sometimes alongside right and interval censoring in a general framework. These methods use the upper bound on the event time appropriately, giving valid estimates despite the exact earlier timing being unobserved.
Source: Kalbfleisch & Prentice 2002
Why is left censoring less common than right censoring?
Left censoring is less common than right censoring because most survival studies follow individuals forward from a defined starting point, such as diagnosis or enrolment, so events after that are observed or right-censored, while events before entry are usually excluded by design. Left censoring arises mainly when individuals enter already having experienced the event, which is avoided in prospective studies with a clear start. Consequently, right censoring dominates in typical survival data, and left censoring occurs in more specific situations.
Source: Kalbfleisch & Prentice 2002
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/left-censoring
- Term code
- HE-EM-SM-043
Stable URI · Machine-readable · Resolvable · CC BY 4.0