Concept Architecture
Concept
Theoretically, Truncation is a statistical mechanism in which observations falling outside a specified range are completely excluded from the sampled population and therefore cannot be observed. Unlike censoring, truncated observations are absent from the dataset rather than partially observed. The concept arises in probability theory and statistical inference because sampling restrictions alter the underlying probability distribution, requiring likelihood-based methods that explicitly account for the truncated sampling process.
Mathematically, truncation is represented by conditioning a probability distribution on the event that a random variable lies within specified bounds. The resulting truncated probability density or mass function is obtained by dividing the original distribution by the probability of inclusion within the truncation interval. Maximum likelihood estimation is commonly used to estimate model parameters under truncation by constructing likelihood functions based on the truncated distribution.
In practice, truncation is addressed by specifying the truncation limits during statistical modelling and fitting truncated probability models or regression methods. In health economics, truncation occurs when analyses include only individuals meeting predefined eligibility criteria, when delayed entry creates left-truncated survival data, or when expenditure datasets exclude observations below or above reporting thresholds. Appropriate estimation methods are required to avoid biased parameter estimates.
Purpose
Used to model data generated under sampling mechanisms in which observations outside predefined limits are unobservable, enabling unbiased estimation of statistical distributions, regression models and survival models affected by truncated sampling.
Mathematical Formulae
Primary Formula
For lower truncation at a:
f?(x) = f(x) / [1 ? F(a)], for x > a
For double truncation between a and b:
f?(x) = f(x) / [F(b) ? F(a)], for a < x < b
Supporting Formulae
Likelihood for n truncated observations:
L(?) = ?? f(x?; ?) / [F(b; ?) ? F(a; ?)]
Log-likelihood:
ln L(?) = ? ln f(x?; ?) ? n ln[F(b; ?) ? F(a; ?)]
Related Mathematical Methods
- Maximum Likelihood Estimation
- Truncated Regression
- Tobit Models (contrast with censoring)
- Survival Analysis with Left Truncation
- Conditional Probability
- Truncated Probability Distributions
Example
A health expenditure study includes only patients whose annual costs exceed �500 because smaller claims are not recorded. The observed costs therefore follow a lower-truncated distribution.
If X ~ N(�2,000, �600�) and only observations with X > �500 are observed, parameter estimation must be based on the truncated normal likelihood rather than the ordinary normal distribution to avoid biased estimates of mean healthcare costs.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| NORM.DIST | =NORM.DIST(A2,2000,600,FALSE)/(1-NORM.DIST(500,2000,600,TRUE)) | Calculates a lower-truncated normal density. |
| NORM.DIST | =NORM.DIST(500,2000,600,TRUE) | Calculates the truncation probability. |
| LN | =LN(B2) | Computes log-likelihood contributions for maximum likelihood estimation. |
| SUM | =SUM(C2:C101) | Sums log-likelihood values for parameter estimation. |
VBA (Optional)
Automate maximum likelihood estimation for truncated distributions by iteratively optimising parameter estimates while accounting for specified truncation limits.
Sources
- Greene WH. Econometric Analysis.
- Cameron AC, Trivedi PK. Microeconometrics: Methods and Applications.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (3)
Library
Publications
1
Good Practices for Real-World Data Studies of Treatment and/or Comparative Effectiveness: Recommendations from the Joint ISPOR-ISPE Special Task Force on Real-World Evidence in Health Care Decision Making — Berger, Sox, Willke, Brixner, Eichler, Goettsch, Madigan, Makady, Schneeweiss, Tarricone, Wang, Watkins & Mullins, Vol. 20, No. 8 ed., 2017 (Value in Health)
The joint ISPOR-ISPE recommendations on good procedural practice for real-world data studies (observational studies and registries) used to inform healthcare decisions — study registration, replicability and stakeholder involvement — the reference for RWE credibility in HTA.
Journal ArticleView source →
Frequently Asked Questions (6)
What is truncation?
A situation in survival or longitudinal analysis where certain observations are systematically excluded based on a variable's value, such as required registration.
Source: Kalbfleisch & Prentice 2002
Why are truncated individuals missing entirely from the data?
Truncation excludes certain individuals from a study altogether based on a variable's value, so that, unlike a censored person who is observed for part of the time, a truncated person never appears in the data at all. In left truncation, those who had the event before qualifying to enter, such as before registering with a registry, are simply absent. Because they are invisible rather than partially observed, analysis must adjust for the selective way the sample was formed. Their complete absence is what distinguishes truncation from censoring. Klein and Moeschberger (2003) describe this.
Source: Klein & Moeschberger 2003
What are the types of truncation?
The types of truncation are left truncation, where individuals are included only if they survive or reach a certain point, so those who experienced the event before that point are excluded; and right truncation, where individuals are included only if the event has occurred by a certain point, so those who would experience it later are missed. Left truncation is common, arising when entry to a study requires surviving to registration. Each type selects a subset of individuals based on the timing of events or entry, and each must be handled appropriately in analysis to avoid bias.
Source: Kalbfleisch & Prentice 2002
How does truncation differ from censoring?
Truncation differs from censoring in whether the individual is observed at all: in truncation, individuals outside the inclusion condition are entirely excluded from the sample, so nothing is known about them, whereas in censoring, the individual is observed but their event time is only partially known, such as knowing it exceeds the last follow-up. So truncation is a selection issue affecting who is in the sample, while censoring is an incomplete-information issue for those included. Both must be handled in survival analysis, but they are distinct and require different adjustments.
Source: Kalbfleisch & Prentice 2002
How is truncation handled in analysis?
Truncation is handled by using methods that account for the selective inclusion, such as adjusting the risk set for left truncation so individuals contribute only from the time they enter observation, reflecting that they had to survive to be included. For right truncation, methods condition on the event having occurred by the truncation time. These adjustments correct for the fact that the sample is a selected subset, avoiding the bias that would result from ignoring the truncation. So handling truncation requires analytic methods that incorporate the inclusion condition into the estimation.
Source: Collett 2015
Why must truncation be accounted for?
Truncation must be accounted for because the truncated sample is not representative of the full population, being systematically selected based on survival or event timing, so treating it as a complete sample biases estimates, for example overestimating survival under left truncation by excluding those who experienced the event early. Ignoring truncation would give misleading results. By adjusting for the inclusion condition, analysis corrects this selection, yielding valid estimates. Because truncation systematically excludes part of the population, accounting for it is necessary for unbiased survival or longitudinal analysis.
Source: Kalbfleisch & Prentice 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 19 Nov 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/truncation
- Term code
- HE-ES-CTM-100
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