Censored variable

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In statistics, a censored variable is a variable that is not fully measured or observed because of floor or ceiling effects providing a cut off in the range, beyond which data cannot be quantified.

For example, suppose a study is conducted to measure the impact of a drug on mortality. In such a study, it may be known that an individual's age at death is at least 75 years. 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 a measuring instrument. For example, a bathroom scale might only measure up to 300 lbs. If a 350 lb individual is weighed using the scale, the observer would only know that the individual's weight is at least 300 lbs.

Types

• 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 (or non-informative) censoring is when each subject has a censoring time that is statistically 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.

Censoring should not be confused with the related idea truncation. With censoring, observations result either in knowing the exact value that applies, or in knowing that the value lies within an interval. 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 as rounding.

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

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.

Left-censored data, is observed, for example, in environmental analytical data where trace concentrations of chemicals may indeed be present in an environmental sample (e.g., groundwater, soil) but are "non-detectable," i.e., below the detection limit of the analytical instrument or laboratory method. 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.

Epidemiology

One of the earliest attempts to analyse a statistical problem involving censored data was Daniel Bernoulli's 1766 analysis of smallpox morbidity and mortality data to demonstrate the efficacy of vaccination.

Analysis

Special techniques may be used to handle censored data.

Bibliography

• 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 itPDF (146 KiB)", Reviews of Medical Virolology, 14: 275–288
• Bradley, L. (1971) Smallpox Inoculation: An Eighteenth Century Mathematical Controversy, Nottingham
• 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.

See also

• Survival analysis
• Kaplan–Meier estimator
• Data analysis
• Reliability (statistics)
• Imputation (statistics)
• Censored regression model