# What is the difference between censoring and truncation?

In the book Statistical Models and Methods for Lifetime Data , it is written :

Censoring: When an observation is incomplete due to some random cause.
Truncation: When the incomplete nature of the observation is due to a systematic selection process inherent to the study design.

What is meant by "systematic selection process inherent to the study design" in the definition of truncation?

What is the difference between censoring and truncation?

• Take a look at the answer here. – Dimitriy V. Masterov Mar 30 '15 at 14:44
• Censoring: "We have had an observation in that region somewhere but we don't know what it is". Truncation: "Observation? What observation?" – Glen_b Mar 31 '15 at 0:47
• Where are your definitions quoted from? – Glen_b Mar 31 '15 at 0:49
• @Glen_b I've edited my question. – ABC Apr 1 '15 at 10:12

Definitions vary, and the two terms are sometimes used interchangeably. I'll try to explain the most common uses using the following data set: $$1\qquad 1.25\qquad 2\qquad 4 \qquad 5$$

Censoring: some observations will be censored, meaning that we only know that they are below (or above) some bound. This can for instance occur if we measure the concentration of a chemical in a water sample. If the concentration is too low, the laboratory equipment cannot detect the presence of the chemical. It may still be present though, so we only know that the concentration is below the laboratory's detection limit.

If the detection limit is 1.5, so that observations that fall below this limit is censored, our example data set would become: $$<1.5\qquad <1.5\qquad 2\qquad 4 \qquad 5,$$ that is, we don't know the actual values of the first two observations, but only that they are smaller than 1.5.

Truncation: the process generating the data is such that it only is possible to observe outcomes above (or below) the truncation limit. This can for instance occur if measurements are taken using a detector which only is activated if the signals it detects are above a certain limit. There may be lots of weak incoming signals, but we can never tell using this detector.

If the truncation limit is 1.5, our example data set would become $$2\qquad 4 \qquad 5$$ and we would not know that there in fact were two signals which were not recorded.

• So, on this use of the terms, "censored" is misleading if we think in term of the non-technical uses of the word? i.e. in this statistical sense, it means something like "vague" or "only known to fall within some range", rather than in something like the non-technical sense--i.e. suppressed or removed, as when a book is removed from stores because of its content. – Mars Dec 18 '15 at 18:16
• For a concrete example of truncation, car insurance companies never hear about accidents where the damage is less than the deductible, because people don't report there. This is left truncation; we never see data on these incidents at all. For an example of right censoring, when a sick patient decides to stop seeing their doctor, or moves to a different city, then all that is known is that they were alive on the day they left, but we don't know when they died. – David White Jan 20 '17 at 17:51
• @Mars: I agree that it sounds backwards from modern non-technical usage where "censoring" is removing all trace of, and "truncating" is removing details. But in statistics "Censoring" is used in the more-old-fashioned non-technical sense where a censor could remove but not eliminate any trace of something: black boxes or blurs placed over offensive parts of a photo or video, bleeps that cover profanity on the radio, or soldiers' letters to home or classified document releases where the censored (more modern term "redacted") parts are blacked out. – Wayne Jan 24 '17 at 20:56
• Imagine I measure the time lapse between two kind of event events. But I can only record event for 1 year. Will time be censored or truncated? – skan Feb 1 '18 at 16:09

Just as a perspective from another field (programming), censoring and truncating are two distinct operations.

When working with a sensitive dataset, for example social security numbers and telephone numbers, I might censor it or have it censored prior to access being granted:

123-12-1234 => 999-99-9999
567-56-5678 => 999-99-9999
(906) 123-4567 => (000) 000-0000


This allows the rest of the application to operate as it normally would, with similar data structures, but with no real informational content or dissemination of private information.

Truncation, by contrast, is typically just cutting off remaining values after a certain point. To work on an application, I don't need hundreds of thousands of records, perhaps I only need ~50 of each which makes the data access much faster and the data sets smaller.

A similar variant of truncation is when inserting a value into a column or datatype of limited length or precision:

abcdefghijklmnopqrstuv => abcdef
10.23412421345 => 10.23
10.92455311 => 10

• +1 It's important to know that censoring and truncation can have completely different meanings outside of statistics! – MånsT Mar 31 '15 at 8:34