Source: Principles of Statistics I, Texas A&M University
Tags: censored data, right censoring, left censoring, Type I censoring, Type II censoring, random censoring, interval censoring, truncated data, incomplete data, survival analysis, reliability, time to failure
Censored data arises when some observations in a study are incomplete, meaning we have partial but not full information about certain experimental units. The main types are right censoring (we know the value exceeds some bound), left censoring (the value is below some bound), and random censoring (units leave the study unpredictably). Correctly identifying and handling censoring is essential for valid statistical inference.
Censored data
Observations for which the exact value is unknown, but partial information (such as a bound) is available. The experimental unit was observed, but the measurement is incomplete.
Right censoring
We observe a value for a unit but only know the true value is greater than the recorded value. We have a lower bound on the true observation.
Left censoring
We observe a value for a unit but only know the true value is less than the recorded value. We have an upper bound on the true observation.
Type I censoring (fixed time)
A form of right censoring where the experiment is terminated at a preselected time t_c. Any unit that has not experienced the event by t_c is censored. The censoring time is fixed; the number of observed events is random.
Type II censoring (fixed number of failures)
A form of right censoring where the experiment is terminated after a preselected number m of events (failures) have been observed. The number of observed events is fixed; the time to termination is random.
Random censoring
Units leave the study at unpredictable times for reasons unrelated to the event of interest (e.g. patient drops out, equipment breaks). For each unit, we observe the minimum of the event time and the censoring time.
Interval censoring
The event is known to have occurred within a specific interval, but the exact time is not recorded. Common when units are inspected at scheduled times.
Truncated data
Distinct from censoring. With truncation, we would not even know a potential observation existed if its value fell in a certain range. The observation is entirely missing from the dataset, not just partially recorded.
Censoring indicator (delta)
A variable, typically denoted δ_i, that flags whether an observation is censored or uncensored. Commonly δ_i = 1 if uncensored and δ_i = 0 if censored (though coding conventions can vary by software).
Right censoring occurs whenever we know a unit's true event time is greater than its recorded value. Type I and Type II censoring are both special cases of right censoring.
Typical examples of event times that might be right-censored:
Time to failure of an electrical device
Time to occurrence of a tumour in a laboratory animal exposed to a toxin
Time until a patient's blood pressure reaches a target level after treatment
Stress level at which an alloy specimen fractures
Place n units on test with event times T_1, T_2, ..., T_n. The study terminates at a preselected time t_c.
Any unit whose event has not occurred before t_c is censored. The ordered observed event times satisfy:
T_(1) ≤ T_(2) ≤ ... ≤ T_(m) ≤ t_c < T_(m+1) ≤ ... ≤ T_(n)
Key characteristics:
t_c (the stopping time) is fixed and chosen in advance
m (the number of observed events) is random
For the n − m censored units, we only know T_j > t_c
If t_c is set too small, nearly all values will be censored, leaving very little usable data
Place n units on test and terminate the experiment when the m-th (preselected) unit fails.
The ordered observed event times satisfy:
T_(1) ≤ T_(2) ≤ ... ≤ T_(m) < T_(m+1) ≤ ... ≤ T_(n)
Key characteristics:
m (the number of observed events) is fixed and chosen in advance
T_(m) (the stopping time) is random
For the remaining n − m units, we know T_(j) > T_(m)
If the device is very reliable and m is set too large, it may take a very long time for the study to end
We observe T_i for m units, but for the remaining n − m units, we only know their true values are less than their recorded values. We have an upper bound.
Examples:
Units are inspected at scheduled times and some fail before the first inspection. The exact failure time is unknown, but it is less than the first inspection time.
A measuring device cannot register values below a threshold. For instance, an ultrasonic noise recorder may not capture frequencies below 20 kHz. Any sound below that threshold is left-censored: we know it exists but only that its value is below 20 kHz.
A single study can have both left and right censoring. If units fail before the first inspection, that is left censoring; if units are still functioning after the last inspection, that is right censoring.
Individual units fail to have their event time recorded because of reasons unrelated to the event itself:
A patient stops returning to the medical centre
The testing machine breaks down before the specimen fractures
A patient is removed from the study due to side effects
A lab operator stops working
Budget cuts reduce the study size
Lab equipment fails during recording
Formally, for each unit i we observe Y_i = min(T_i, C_i), where T_i is the true event time and C_i is the time at which unit i leaves the study. The censoring indicator δ_i = 1 if T_i ≤ C_i (event observed) and δ_i = 0 otherwise.
Type I censoring is a special case of random censoring where C_i = t_c for all n units. In most applications, T and C are assumed independent.
Each censored unit has its measured value recorded only as falling within an interval, with no specific value known. This arises naturally in studies where units are inspected at scheduled times: the actual event time is unknown, only that it occurred between two consecutive inspection times.
Truncation is fundamentally different from censoring. With censoring, we know a unit exists but lack its exact value. With truncation, we would not even be aware the unit existed if its value fell in a certain range.
Example: ultrasonic inspection of titanium alloys for cracks. In a lab test, specimens with known flaws are inspected and signals below a threshold τ_L are recorded as left-censored (we know the flaw exists but the signal is below τ_L). In production, a flaw below τ_L is not detected at all, so the number of such flaws is unknown. Those undetected observations are left-truncated, not censored.
⚠️ Distinguish between fixed and random quantities: in Type I censoring, the stopping time is fixed and the number of events is random. In Type II censoring, the number of events is fixed and the stopping time is random. This is a common exam question.
⚠️ Be aware that the textbook (Tamhane/Dunlop) labels these types the opposite way round from the handout. The handout's Type I (fixed time) is the textbook's Type II, and vice versa. Check which convention your exam uses.
⚠️ Random censoring is the most general case. Type I censoring is a special case with C_i = t_c for all units.
⚠️ Truncation and censoring are different concepts. With censoring, the unit is in the dataset but incomplete. With truncation, the unit is entirely absent from the dataset.
⚠️ Left censoring and right censoring can coexist in the same study.
Q: What is the key difference between Type I and Type II censoring?
A: In Type I censoring, the experiment ends at a fixed, preselected time t_c and the number of observed events m is random. In Type II censoring, the experiment ends after a fixed, preselected number of events m and the termination time T_(m) is random.
Q: A clinical trial monitors patients for 5 years after treatment. Some patients are still alive at the end of the study. What type of censoring is this?
A: Type I (right) censoring, because the study terminates at a fixed time (5 years) and the surviving patients have event times that exceed the study's end.
Q: A patient drops out of a study because they moved to another city. What type of censoring applies to their observation?
A: Random censoring, because the reason for leaving the study is unrelated to the event of interest.
Q: What is the formal expression for the observed data under random censoring?
A: We observe (Y_i, δ_i) where Y_i = min(T_i, C_i) and δ_i = I(T_i ≤ C_i). Here T_i is the event time, C_i is the censoring time, and δ_i = 1 if the event was observed (uncensored) and 0 if censored.
Q: How does truncation differ from censoring?
A: With censoring, the unit is known to exist but its exact value is not fully observed (we have a bound). With truncation, the unit's existence is unknown if its value falls in a certain range, so it is entirely absent from the data.
censored data, right censoring, left censoring, Type I censoring, Type II censoring, random censoring, interval censoring, truncated data, survival data, incomplete data, time-to-event, reliability testing, fixed censoring time, fixed number of failures, censoring indicator, delta indicator, dropout, loss to follow-up, STAT Handout 07, Tamhane Dunlop