Instatistics,censoring is a condition in which thevalue of ameasurement orobservation is only partially known.

For example, suppose a study is conducted to measure the impact of a drug onmortality rate. In such a study, it may be known that an individual's age at death isat least 75 years (but may be more). Such a situation could occur if the individual withdrew from the study at age 75, or if the individual is currently alive at the age of 75.

Censoring also occurs when a value occurs outside the range of ameasuring instrument. For example, a bathroom scale might only measure up to 140 kg, after which it rolls over 0 and continues to count up from there. If a 160 kg individual is weighed using the scale, the observer would only know that the individual's weight is 20mod 140 kg (in addition to 160kg, they could weigh 20kg, 300kg, 440kg, and so on).

The problem of censored data, in which the observed value of some variable is partially known, is related to the problem ofmissing data, where the observed value of some variable is unknown.

Censoring should not be confused with the related idea oftruncation. With censoring, observations result either in knowing the exact value that applies, or in knowing that the value lies within aninterval. With truncation, observations never result in values outside a given range: values in the population outside the range are never seen or never recorded if they are seen. Note that in statistics, truncation is not the same asrounding.

• Left censoring – a data point is below a certain value but it is unknown by how much.
• Interval censoring – a data point is somewhere on an interval between two values.
• Right censoring – a data point is above a certain value but it is unknown by how much.
• Type I censoring occurs if an experiment has a set number of subjects or items and stops the experiment at a predetermined time, at which point any subjects remaining are right-censored.
• Type II censoring occurs if an experiment has a set number of subjects or items and stops the experiment when a predetermined number are observed to have failed; the remaining subjects are then right-censored.
• Random (ornon-informative)censoring is when each subject has a censoring time that isstatistically independent of their failure time. The observed value is the minimum of the censoring and failure times; subjects whose failure time is greater than their censoring time are right-censored.

Interval censoring can occur when observing a value requires follow-ups or inspections. Left and right censoring are special cases of interval censoring, with the beginning of the interval at zero or the end at infinity, respectively.

Estimation methods for using left-censored data vary, and not all methods of estimation may be applicable to, or the most reliable, for all data sets.^[1]

A common misconception with time interval data is to class asleft censored intervals when the start time is unknown. In these cases, we have a lower bound on the timeinterval; thus, the data isright censored (despite the fact that the missing start point is to the left of the known interval when viewed as a timeline!).

Special techniques may be used to handle censored data. Tests with specific failure times are coded as actual failures; censored data are coded for the type of censoring and the known interval or limit. Special software programs (oftenreliability oriented) can conduct amaximum likelihood estimation for summary statistics, confidence intervals, etc.

One of the earliest attempts to analyse a statistical problem involving censored data wasDaniel Bernoulli's 1766 analysis ofsmallpox morbidity and mortality data to demonstrate the efficacy ofvaccination.^[2] An early paper to use theKaplan–Meier estimator for estimating censored costs was Quesenberry et al. (1989),^[3] however this approach was found to be invalid by Lin et al.^[4] unless all patients accumulated costs with a common deterministic rate function over time, they proposed an alternative estimation technique known as the Lin estimator.^[5]

Reliability testing often consists of conducting a test on an item (under specified conditions) to determine the time it takes for a failure to occur.

• Sometimes a failure is planned and expected but does not occur: operator error, equipment malfunction, test anomaly, etc. The test result was not the desired time-to-failure but can be (and should be) used as a time-to-termination. The use of censored data is unintentional but necessary.
• Sometimes engineers plan a test program so that, after a certain time limit or number of failures, all other tests will be terminated. These suspended times are treated as right-censored data. The use of censored data is intentional.

An analysis of the data from replicate tests includes both the times-to-failure for the items that failed and the time-of-test-termination for those that did not fail.

An earlier model forcensored regression, thetobit model, was proposed byJames Tobin in 1958.^[6]

Thelikelihood is the probability or probability density of what was observed, viewed as a function of parameters in an assumed model. To incorporate censored data points in the likelihood the censored data points are represented by the probability of the censored data points as a function of the model parameters given a model, i.e. a function of CDF(s) instead of the density or probability mass.

The most general censoring case is interval censoring:, where${\displaystyle F(x)}$ is the CDF of the probability distribution, and the two special cases are:

• left censoring:

• right censoring:

For continuous probability distributions:

Suppose we are interested in survival times,${\displaystyle T_1,T_2,...,T_n}$, but we don't observe${\displaystyle T_i}$ for all${\displaystyle i}$. Instead, we observe

${\displaystyle (U_i,{\delta }_i)}$, with${\displaystyle U_i=T_i}$ and${\displaystyle {\delta }_i=1}$ if${\displaystyle T_i}$ is actually observed, and

${\displaystyle (U_i,{\delta }_i)}$, with and${\displaystyle {\delta }_i=0}$ if all we know is that${\displaystyle T_i}$ is longer than${\displaystyle U_i}$.

When${\displaystyle T_i>U_i,U_i}$ is called thecensoring time.^[7]

If the censoring times are all known constants, then the likelihood is

${\displaystyle L=\prod _{i,{\delta }_i=1}f(u_i)\prod _{i,{\delta }_i=0}S(u_i)}$

where${\displaystyle f(u_i)}$ = the probability density function evaluated at${\displaystyle u_i}$,

and${\displaystyle S(u_i)}$ = the probability that${\displaystyle T_i}$ is greater than${\displaystyle u_i}$, called thesurvival function.

This can be simplified by defining thehazard function, the instantaneous force of mortality, as

${\displaystyle \lambda (u)=f(u)/S(u)}$

so

${\displaystyle f(u)=\lambda (u)S(u)}$.

Then

${\displaystyle L=\prod _i\lambda (u_i{)}^{{\delta }_i}S(u_i)}$.

For theexponential distribution, this becomes even simpler, because the hazard rate,${\displaystyle \lambda }$, is constant, and${\displaystyle S(u)=\exp (-\lambda u)}$. Then:

${\displaystyle L(\lambda )={\lambda }^k\exp (-\lambda \sum u_i)}$,

where${\displaystyle k=\sum {\delta }_i}$.

From this we easily compute${\displaystyle \hat{\lambda }}$, themaximum likelihood estimate (MLE) of${\displaystyle \lambda }$, as follows:

${\displaystyle l(\lambda )=\log (L(\lambda ))=k\log (\lambda )-\lambda \sum u_i}$.

Then

${\displaystyle dl/d\lambda =k/\lambda -\sum u_i}$.

We set this to 0 and solve for${\displaystyle \lambda }$ to get:

${\displaystyle \hat{\lambda }=k/\sum u_i}$.

Equivalently, themean time to failure is:

${\displaystyle 1/\hat{\lambda }=\sum u_i/k}$.

This differs from the standard MLE for theexponential distribution in that the censored observations are considered only in the numerator.

• Blower, S. (2004), D, Bernoulli's ""An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it"(PDF). Archived fromthe original(PDF) on 2017-08-08. Retrieved2019-06-25. (146 KiB)",Reviews of Medical Virology,14: 275–288
• Bradley, L. (1971).Smallpox Inoculation: An Eighteenth Century Mathematical Controversy. Nottingham.ISBN 0-902031-23-6.{{cite book}}: CS1 maint: location missing publisher (link)
• Mann, N. R.; et al. (1975).Methods for Statistical Analysis of Reliability and Life Data. New York: Wiley.ISBN 047156737X.
• Bagdonavicius, V., Kruopis, J., Nikulin, M.S. (2011),"Non-parametric Tests for Censored Data", London, ISTE/WILEY,ISBN 9781848212893.

• "Engineering Statistics Handbook", NIST/SEMATEK,[1]

• Statistical data types
• Survival analysis
• Reliability engineering
• Unknown content

• Articles with short description
• Short description is different from Wikidata
• CS1 maint: location missing publisher