Censoring (statistics)

In statistics, engineering, economics, and medical research, censoring is a condition in which the value of a measurement or observation is only partially known.

For example, suppose a study is conducted to measure the impact of a drug on mortality rate. In such a study, it may be known that an individual's age at death is at 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 a measuring instrument. For example, a bathroom scale might only measure up to 300 pounds (140 kg). If a 350 lb (160 kg) individual is weighed using the scale, the observer would only know that the individual's weight is at least 300 pounds (140 kg).

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.

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.

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.

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 as left censored intervals where the start time is unknown. In these cases we have a lower bound on the time interval, thus the data is right censored (despite that fact that the missing start point is to the left of the known interval when viewed as a timeline!).

Analysis

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 (often reliability oriented) can conduct a maximum likelihood estimation for summary statistics, confidence intervals, etc.

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.^[2] An early paper to use the Kaplan-Meier estimator for estimating censored costs was Quesenberry et al. (1989),^[3] however this approach was later found in 1997 by Lin et al.^[4] to be invalid 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]

Operating life testing

Example of five replicate tests resulting in four failures and one suspended time.

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.

Censored regression

An earlier model for censored regression was proposed by James Tobin in 1958.^[6]

References

↑ Helsel, D. Much ado about next to Nothing: Incorporating Nondetects in Science, Ann. Occup. Hyg., Vol. 54, No. 3, pp. 257-262, 2010
↑ Bernoulli D. (1766) "Essai d’une nouvelle analyse de la mortalité causée par la petite vérole. Mem. Math. Phy. Acad. Roy. Sci. Paris, reprinted in Bradley (1971) 21 and Blower (2004)
↑ C P Quesenberry, Jr, B Fireman, R A Hiatt, and J V Selby, A survival analysis of hospitalization among patients with acquired immunodeficiency syndrome, Am J Public Health 1989
↑ Lin DY, Feuer EJ, Etzioni R, Wax Y., Estimating medical costs from incomplete follow-up data, Biometric 1997
↑ Wijeysundera HC, Wang X, Tomlinson G, Ko DT, Krahn MD, Techniques for estimating health care costs with censored data: an overview for the health services researcher, Clinicoecon Outcomes Res. 2012
↑ Tobin, James (1958). "Estimation of relationships for limited dependent variables". Econometrica 26 (1): 24–36. doi:10.2307/1907382. JSTOR 1907382.

External links

"Engineering Statistics Handbook", NIST/SEMATEK,

Statistics

Descriptive statistics

Continuous data

Location	Mean arithmetic geometric harmonic Median Mode

Dispersion	Range Standard deviation Coefficient of variation Percentile Interquartile range

Shape	Variance Skewness Kurtosis Moments L-moments

Count data

Index of dispersion

Summary tables

Dependence

Statistical graphics

Data collection

Study design	Effect size Standard error Statistical power Sample size determination

Survey methodology	Sampling stratified cluster Opinion poll Questionnaire

Controlled experiments	Design control optimal Controlled trial Randomized Random assignment Replication Blocking Factorial experiment

Uncontrolled studies	Observational study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Confidence interval Testing hypotheses Power

Unbiased estimators	Mean unbiased minimum-variance Median unbiased

Biased estimators	Maximum likelihood Method of moments Minimum distance Density estimation

Parametric tests	Likelihood-ratio Wald Score

Specific tests

Z (normal) Student's t-test F Shapiro–Wilk Kolmogorov–Smirnov

Goodness of fit	Chi-squared G Sample source (Anderson–Darling) Sample normality (Shapiro–Wilk) Skewness / kurtosis normality (Jarque–Bera) Model comparison (Likelihood-ratio) Model quality (Akaike criterion)

Signed-rank	1-sample (Wilcoxon) 2-sample (Mann–Whitney U) 1-way anova (Kruskal–Wallis)

Bayesian inference

Correlation	Pearson product–moment Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality

Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey

Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)

Frequency domain	Spectral density estimation Fourier analysis Wavelet

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time

Hazard function	Nelson–Aalen estimator

Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics

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