Risk Modelling and Survival Analysis · Estimation procedures for lifetime distributions

Censoring and Truncation in Lifetime Data Explained

Q: What is the difference between censoring and truncation?

With censoring, the life is in your data but you only partly know its lifetime. With truncation, a life is in your data only if it meets a condition, so other lives are never seen. This is why truncation needs a conditional likelihood.

Q: Which type of censoring is most common in mortality studies?

Right censoring is the most common. It occurs when a life is still alive at the end of the study or leaves the study alive, for example by withdrawing. Left censoring is rare in mortality work.

Q: What is informative censoring?

Censoring is informative if the reason a life is censored is related to its risk of death. For example, lives withdraw because they become seriously ill. Standard estimators assume non-informative censoring, so informative censoring can bias the results.

Q: How do I handle left truncation in a likelihood?

Condition on survival to the entry time x. A death at t contributes f(t) ÷ S(x), and a life censored at c contributes S(c) ÷ S(x). In a constant hazard model this means counting exposure only from entry.

Updated 11 October 2026 · Fact-checked

Censoring means you know a life was observed but its exact time of death is only partly known, for example it is known only to exceed some age. Truncation means lives are only observed if they meet a condition, so some lives are never seen. Censored lives are in the data; truncated lives may be absent.

Understand Censoring and Truncation in Lifetime Data

In a mortality study you rarely see every life from birth to death. Studies start and end on fixed dates. People join late, leave early or move away. So the data on lifetimes is often incomplete. Censoring and truncation are the two ways data can be incomplete.

Censoring means you observe the life, but you only have partial information about the lifetime. The life is in your data set. You know something about when death occurred, but not exactly.

Right censoring: you know the lifetime is greater than some value c. The life is still alive at the end of the study, or leaves the study (for example surrenders a policy or withdraws) while alive. This is the most common type in mortality work.
Left censoring: you know the lifetime is less than some value. The event happened before observation began, but you do not know when. It is rare in mortality studies.
Interval censoring: you know the lifetime lies between two values a and b, but not where. This happens when lives are checked only at fixed dates, for example at annual policy renewal.

Truncation means lives are only included in the study if they satisfy a condition. A life that fails the condition is not observed at all, and you may not even know it existed.

Left truncation: a life enters observation only if it survives to some age or date. For example, a pension scheme study includes only people who reach age 60 and join. Lives that died before 60 are not in the data. You must condition on survival to the entry age.
Right truncation: a life is included only if the event has occurred by some date. For example, a study of deaths recorded in a registry up to a given date. Lives who have not yet died are not in the data.

The key difference: a censored life is in the data with partial information. A truncated life is in the data only because it met the entry condition; others are missing. This changes the likelihood. A censored life contributes the probability of the partial information. A left-truncated life contributes a probability conditional on surviving to the entry time.

Censoring can also be classed by how it arises. Type I censoring happens at a fixed time, such as the study end. Type II censoring happens when a fixed number of deaths has occurred. Random censoring happens when each life has its own censoring time, such as a withdrawal. Censoring is non-informative if the reason for censoring is independent of the lifetime, given the model. Then standard methods are valid. It is informative if censoring is related to the risk of death, for example people withdraw because they are ill. Then standard methods can give biased results.

Key rules to remember

Right-censored observation: Contribution to likelihood = S(c) = P(T > c); Use when you only know the life survived beyond c. In terms of the hazard: exp(−∫₀^c μ(t) dt).
Uncensored (death) observation: Contribution to likelihood = f(t) = S(t) × μ(t); Use when the exact time of death t is observed.
Left-censored observation: Contribution to likelihood = F(c) = P(T < c) = 1 − S(c); Use when you know the event occurred before c but not when.
Interval-censored observation: Contribution to likelihood = S(a) − S(b) = P(a < T ≤ b); Use when death is known to fall between a and b.
Left-truncated observation (death at t, entry at x): Contribution to likelihood = f(t) ÷ S(x); Condition on survival to entry time x. If instead the life is right-censored at c, the contribution is S(c) ÷ S(x).
Right-truncated observation (included only if T ≤ r): Contribution to likelihood = f(t) ÷ F(r); Condition on the event having happened by r.
Constant hazard with right censoring: μ̂ = d ÷ v, where d = number of deaths and v = total time observed; Total time includes time lived by censored lives. This is the MLE when the hazard is constant.

How to solve Censoring and Truncation in Lifetime Data questions

Use this method for any question that asks you to classify observations, explain the effect on estimation, or write a likelihood.

1List the study start and end dates, and the entry condition. Note who is included and who is not.
2For each life, ask: is the life in the data set? If it was excluded because of an entry condition, it is a truncation issue. If it is in the data with partial information, it is censoring.
3For lives in the data, decide what you know about the lifetime: exact (death observed), greater than c (right censored), less than c (left censored), or between a and b (interval censored).
4Check the entry time. If lives enter observation at age x only because they survived to x, treat this as left truncation and condition on survival to x.
5Write each life's likelihood contribution using the formulas: f(t) for death, S(c) for right censoring, with division by S(x) for left truncation.
6Multiply the contributions to get the likelihood L, take logs, and differentiate if asked for an estimate.
7State whether censoring is informative or non-informative, and say what bias would arise if it is informative.
8Check the answer: the exposure used must run from entry time to exit time, not from birth.

Quickest way: Three-question classification

When to use it: Use this for MCQs and short classification parts where you must name the type of incompleteness.

Question 1: Is this life actually in the data? If the life is missing because of a selection rule, it is truncation.
Question 2: If the life is in the data, is the exact time of death known? If not, it is censoring.
Question 3: What do you know? Greater than c means right censored. Less than c means left censored. Between a and b means interval censored.
For truncation, ask which end: lives enter only if they survive to a point (left truncation) or only if the event has already occurred (right truncation).
For likelihood parts, write the contribution as: death f(t), right censored S(c), then divide by S(entry age) if there is left truncation.

Common mistakes in Censoring and Truncation in Lifetime Data

Using the words censoring and truncation as if they mean the same thing.
Both describe incomplete data, so they feel alike.
Fix: Ask whether the life is in the data. Censored lives are observed with partial information. Truncated lives are included only if they meet a condition, and others are never seen.
Ignoring left truncation and counting exposure from birth.
Students think of a life table that starts at age 0.
Fix: Count exposure only from the age at which the life entered observation. Condition the likelihood on survival to that age by dividing by S(entry age).
Treating a life that withdraws as a death.
The life leaves the data, so it feels like an event.
Fix: A withdrawal while alive is right censoring. It adds exposure time but no death to the numerator.
Leaving out the time lived by censored lives when estimating a constant hazard.
Students sum only the times of the lives that died.
Fix: Use total time observed for all lives, deaths and censored, in the denominator: μ̂ = d ÷ v.
Saying all censoring can be ignored.
Students remember that standard estimators allow for censoring and over-generalise.
Fix: Standard methods are valid for non-informative censoring. If lives leave because they are ill or healthy, censoring is informative and the estimates can be biased. State this assumption.
Calling a study-end cut-off left censoring.
The word end suggests the left or last side.
Fix: If the life is alive at the end of the study, the lifetime is greater than the observed time. That is right censoring.

Worked examples

Example 1

A mortality study runs from 1 January 2020 to 31 December 2023. Classify each of the following lives: (a) a policyholder who is alive on 31 December 2023; (b) a policyholder who cancels his policy in 2021 and is lost to follow-up; (c) a life aged 60 who joins a pension scheme in 2021 and is included in the data only because she reached 60; (d) a person who is known to have died sometime between the annual checks in 2021 and 2022.

Show the solution

(a) The life is in the data. The lifetime is known to be greater than the observed time. This is right censoring (Type I, at the end of the study).
(b) The life is in the data until cancellation. After that the lifetime is only known to exceed the observed time. This is right censoring, random, due to withdrawal.
(c) The life is included only because she survived to 60. Lives dying before 60 are not in the data. This is left truncation at age 60. Her likelihood must be conditional on survival to 60.
(d) Death is known to lie in an interval between two checks. This is interval censoring.

Answer: (a) right censored (Type I); (b) right censored (random); (c) left truncated at 60; (d) interval censored.

Example 2

Five lives are observed from age 60. Lives A and B die at 62 and 64. Life C withdraws alive at 63. Lives D and E are alive at the end of the study at 65. Assuming a constant force of mortality μ, find the maximum likelihood estimate and state the likelihood contribution of life C.

Show the solution

Time lived from age 60: A = 2, B = 4, C = 3, D = 5, E = 5.
Total time observed v = 2 + 4 + 3 + 5 + 5 = 19.
Number of deaths d = 2.
Each death contributes f(t) = μ e^(−μt) divided by S(60) conditioned out, so measured from entry. Life C, censored, contributes e^(−3μ).
Likelihood L = μ² × e^(−μ × 19), since all times add up to the total exposure.
Log-likelihood: ln L = 2 ln μ − 19μ.
Differentiate: 2 ÷ μ − 19 = 0, so μ̂ = 2 ÷ 19.
Check this is a maximum: second derivative is −2 ÷ μ², which is negative.

Answer: Life C contributes e^(−3μ). The MLE is μ̂ = 2 ÷ 19 ≈ 0.105.

Exam tips

Begin every answer by stating whether each life is censored or truncated, with a one-line reason. This earns marks even if the later calculation slips.
Write the likelihood contribution of each type of life in a short list before multiplying. Examiners award method marks for each correct contribution.
When the question mentions that lives enter at a certain age, think left truncation and condition on survival to that age.
For discussion questions, always state the assumption that censoring is non-informative, and say what goes wrong if it is not.
In MCQs, the wrong options are often the swapped terms. Re-read the definition: partial information means censoring, missing lives means truncation.

Practice questions from Estimation procedures for lifetime distributions

Censoring and Truncation in Lifetime Data: frequently asked questions

What is the difference between censoring and truncation?

With censoring, the life is in your data but you only partly know its lifetime. With truncation, a life is in your data only if it meets a condition, so other lives are never seen. This is why truncation needs a conditional likelihood.

Which type of censoring is most common in mortality studies?

Right censoring is the most common. It occurs when a life is still alive at the end of the study or leaves the study alive, for example by withdrawing. Left censoring is rare in mortality work.

What is informative censoring?

Censoring is informative if the reason a life is censored is related to its risk of death. For example, lives withdraw because they become seriously ill. Standard estimators assume non-informative censoring, so informative censoring can bias the results.

How do I handle left truncation in a likelihood?

Condition on survival to the entry time x. A death at t contributes f(t) ÷ S(x), and a life censored at c contributes S(c) ÷ S(x). In a constant hazard model this means counting exposure only from entry.