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Definitions Censoring Truncation Quiz Independence Inference
Censoring and truncationST3242: Introduction to Survival Analysis
Alex Cook
August 2008
ST3242 : Censoring and truncation 1/22
Definitions Censoring Truncation Quiz Independence Inference
Definitions
Censoring
Censoring is when an observation is incomplete due to somerandom cause. The cause of the censoring must beindependent of the event of interest if we are to use standardmethods of analysis.
Truncation
Truncation is a variant of censoring which occurs when theincomplete nature of the observation is due to a systematicselection process inherent to the study design.
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Definitions Censoring Truncation Quiz Independence Inference
Definitions
Censoring
Censoring is when an observation is incomplete due to somerandom cause. The cause of the censoring must beindependent of the event of interest if we are to use standardmethods of analysis.
Truncation
Truncation is a variant of censoring which occurs when theincomplete nature of the observation is due to a systematicselection process inherent to the study design.
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Definitions Censoring Truncation Quiz Independence Inference
Examples
Examples of censored data
Lung cancer patients are recruited to a study to test the effectof a drug on their survival from lung cancer.
A takes part in the study until her death at time TA. Hersurvival time is uncensored.
B takes part in the study until time TB . He then leavesthe study. His survival time is censored: we know it is atleast TB but we dont know it precisely.
C takes part in the study until time TC . She then is hitby a car and dies. Her survival time with regard to theevent of interest, namely death through lung cancer, isalso censored: we know it is at least TC .
ST3242 : Censoring and truncation 3/22
Definitions Censoring Truncation Quiz Independence Inference
Examples
Examples of censored data
Lung cancer patients are recruited to a study to test the effectof a drug on their survival from lung cancer.
A takes part in the study until her death at time TA. Hersurvival time is uncensored.
B takes part in the study until time TB . He then leavesthe study. His survival time is censored: we know it is atleast TB but we dont know it precisely.
C takes part in the study until time TC . She then is hitby a car and dies. Her survival time with regard to theevent of interest, namely death through lung cancer, isalso censored: we know it is at least TC .
ST3242 : Censoring and truncation 3/22
Definitions Censoring Truncation Quiz Independence Inference
Examples
Examples of censored data
Lung cancer patients are recruited to a study to test the effectof a drug on their survival from lung cancer.
A takes part in the study until her death at time TA. Hersurvival time is uncensored.
B takes part in the study until time TB . He then leavesthe study. His survival time is censored: we know it is atleast TB but we dont know it precisely.
C takes part in the study until time TC . She then is hitby a car and dies. Her survival time with regard to theevent of interest, namely death through lung cancer, isalso censored: we know it is at least TC .
ST3242 : Censoring and truncation 3/22
Definitions Censoring Truncation Quiz Independence Inference
Types of censoring
Right censored data
Commonest form of censoring is right censoring.
Subjects followed until some time, at which the event hasyet to occur, but then takes no further part in the study.
This may be because:
the subject dies from another cause, independently ofthe cause of interest;the study ends while the subject survives; orthe subject is lost to the study, by dropping out, movingto a different area, etc.
ST3242 : Censoring and truncation 4/22
Definitions Censoring Truncation Quiz Independence Inference
Types of censoring
Right censored data
If our data contain only uncensored and right-censored data,we can represent all individuals by the triple (i , ti , i):
i indexes subjects,
ti is the time at which the death or censoring eventoccurs to individual i , and
i is an indicator: i = 1 if i is uncensored and i = 0 ifcensored.
Most methods in the course apply to right censored data only.
ST3242 : Censoring and truncation 5/22
Definitions Censoring Truncation Quiz Independence Inference
Types of censoring
Right censored data
l
0 2 4 6 8 10
12
34
56
time
inde
xl xl xl l
l xl l
l x
ST3242 : Censoring and truncation 6/22
Definitions Censoring Truncation Quiz Independence Inference
Types of censoring
Left censored data
Left censoring is much rarer.
Event of interest already occurred at the observationtime, but it is not known exactly when.
Examples of left censoring include:
infection with a sexually-transmitted disease such asHIV/AIDS;onset of a pre-symptomatic illness such as cancer; andtime at which teenagers begin to drink alcohol.
ST3242 : Censoring and truncation 7/22
Definitions Censoring Truncation Quiz Independence Inference
Types of censoring
Interval censored data
Interval censoring: exact time event occurs is not knownprecisely, but an interval bounding this time is known.
If interval is very short, e.g. 1d, it is common to ignorethis form of censoring and pick one end point of theinterval consistently.
Examples of interval censoring include:
infection with a sexually-transmitted disease such asHIV/AIDS with regular testing (e.g. annually); andfailure of a machine during the Chinese New Year.
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Definitions Censoring Truncation Quiz Independence Inference
Types of truncation
Left truncated data
Left truncation: due to structure of the study design, wecan only observe those individuals whose event time isgreater than some truncation threshold.
This threshold may be the same for all individuals or maybe random.
For example, in actuarial life tables, it is common toignore those who die in the womb.
Another example: imagine you wish to study how longpeople who have been hospitalised for a heart attacksurvive taking some treatment at home. The start time istaken to be the time of the heart attack. Only thoseindividuals who survive their stay in hospital are able tobe included in the study.
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Definitions Censoring Truncation Quiz Independence Inference
Types of truncation
Right truncated data
Right truncation: only individuals with event times lessthan some threshold are included in the study.
If you ask a group of smoking school pupils at what agethey started smoking, you neccessarily have truncateddata, as individuals who start smoking after leavingschool are not included in the study.
Similarly, if a drug developed in 1990 is compared with aplacebo to analyse the survival times of (dead) cancerpatients, all times to death must be 18y.
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Definitions Censoring Truncation Quiz Independence Inference
Types of censoring and truncation
Quiz: what kind of censoring or truncation?
Leukmia patients are given a drug or a placebo. Survivaltime is the duration from remission to relapse. The study endsat 52 weeks with some patients yet to relapse.
Answer:
Right censoring
ST3242 : Censoring and truncation 11/22
Definitions Censoring Truncation Quiz Independence Inference
Types of censoring and truncation
Quiz: what kind of censoring or truncation?
Leukmia patients are given a drug or a placebo. Survivaltime is the duration from remission to relapse. The study endsat 52 weeks with some patients yet to relapse.
Answer:
Right censoring
ST3242 : Censoring and truncation 11/22
Definitions Censoring Truncation Quiz Independence Inference
Types of censoring and truncation
Quiz: what kind of censoring or truncation?
The age at which children are able to count from 110 atschool. Some children are already able to count before joiningschool.
Answer:
Left censoring
ST3242 : Censoring and truncation 12/22
Definitions Censoring Truncation Quiz Independence Inference
Types of censoring and truncation
Quiz: what kind of censoring or truncation?
The age at which children are able to count from 110 atschool. Some children are already able to count before joiningschool.
Answer:
Left censoring
ST3242 : Censoring and truncation 12/22
Definitions Censoring Truncation Quiz Independence Inference
Types of censoring and truncation
Quiz: what kind of censoring or truncation?
Death times at a retirement community.
Answer:
Left truncation
ST3242 : Censoring and truncation 13/22
Definitions Censoring Truncation Quiz Independence Inference
Types of censoring and truncation
Quiz: what kind of censoring or truncation?
Death times at a retirement community.
Answer:
Left truncation
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Definitions Censoring Truncation Quiz Independence Inference
Types of censoring and truncation
Quiz: what kind of censoring or truncation?
Early in the AIDS outbreak, patients with AIDS were recruitedto study the time from infection with HIV to development ofAIDS. At the time of the study, many people were infectedwith HIV but had not yet developed symptoms of AIDS.
Answer:
Right truncation
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Definitions Censoring Truncation Quiz Independence Inference
Types of censoring and truncation
Quiz: what kind of censoring or truncation?
Early in the AIDS outbreak, patients with AIDS were recruitedto study the time from infection with HIV to development ofAIDS. At the time of the study, many people were infectedwith HIV but had not yet developed symptoms of AIDS.
Answer:
Right truncation
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Definitions Censoring Truncation Quiz Independence Inference
Types of censoring and truncation
Quiz: what kind of censoring or truncation?
NUS students are asked the age at which they first triedmarijuana.
Some answer never.
Some answer with an exact age.
Some report using it but forget when.
Answer:
Right censoring
No censoring!
Left censoring
ST3242 : Censoring and truncation 15/22
Definitions Censoring Truncation Quiz Independence Inference
Types of censoring and truncation
Quiz: what kind of censoring or truncation?
NUS students are asked the age at which they first triedmarijuana.
Some answer never.
Some answer with an exact age.
Some report using it but forget when.
Answer:
Right censoring
No censoring!
Left censoring
ST3242 : Censoring and truncation 15/22
Definitions Censoring Truncation Quiz Independence Inference
An alternative formulation
Let. . .
X be the set of possible failure times, |X | = nxi be the potential failure time of individual iyi be the potential censoring/truncation time of individual iT be the set of observed failure/censoring times, |T | = m ntj be the failure time of individual j
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Definitions Censoring Truncation Quiz Independence Inference
Importance of independence
Vital that cause of censoring or truncation beindependent of failure.
If not the methods in the course cannot be used.
Example of violation of independence
Patients are recruited following operation for lung cancer.Survival is time after diagnosis to death from lung cancer.Some patients may die of other cancers before dying of lungcancer. Those dying of other cancers more likely to have diedfrom lung cancer.
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Definitions Censoring Truncation Quiz Independence Inference
Importance of independence
Vital that cause of censoring or truncation beindependent of failure.
If not the methods in the course cannot be used.
Example of violation of independence
Patients are recruited following operation for lung cancer.Survival is time after diagnosis to death from lung cancer.Some patients may die of other cancers before dying of lungcancer. Those dying of other cancers more likely to have diedfrom lung cancer.
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Definitions Censoring Truncation Quiz Independence Inference
If in doubt. . .
Try to recast the problem in some way. E.g. redefinefailure as death from any cancer.
Alternatively, try sensitivity analysis. Imagine all deathsdue to other cancer are:
independent censoring eventsdeaths due to lung cancer
Then compare consistency of results.
ST3242 : Censoring and truncation 18/22
Definitions Censoring Truncation Quiz Independence Inference
If in doubt. . .
Try to recast the problem in some way. E.g. redefinefailure as death from any cancer.
Alternatively, try sensitivity analysis. Imagine all deathsdue to other cancer are:
independent censoring eventsdeaths due to lung cancer
Then compare consistency of results.
ST3242 : Censoring and truncation 18/22
Definitions Censoring Truncation Quiz Independence Inference
Fitting to censored data
Although censored and truncated data are only partiallyobserved, we do not throw them away. Excluding them:
increases the variance in our estimates; and
leads to biased results.
Fortunately, for parametric models, it is easy to construct thelikelihood function in the presence of censoring.
ST3242 : Censoring and truncation 19/22
Definitions Censoring Truncation Quiz Independence Inference
Fitting to censored data
Likelihood
If T is distributed according to some known family ofdistributions with unknown parameters , then:
density function is f (t|)distribution function is F (t|) = t0 f ( |) d
ST3242 : Censoring and truncation 20/22
Definitions Censoring Truncation Quiz Independence Inference
Fitting to censored data
Likelihood
Let R be the set of right-censored, L of left-censored, Iof interval-censored and U of uncensored observations.Denote the data by D = (R,L, I,U , T ) whereT = {ti : i = 1, . . . , n}.The log-likelihood function with no truncation is
log L(D|) = . . . (1)
The log-likelihood function with truncation is
log L(D|) = . . . (2)
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Definitions Censoring Truncation Quiz Independence Inference
Conclusions
In this section, we introduced the concepts of censoringand truncation.
These are very frequently occuring phenomena in survivalanalysis.
The most common is right censoring.
In the next section we introduce the survival and hazardfunctions, which are popular ways to describe survivaldata in the presence of censoring.
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DefinitionsCensoringTruncationQuizIndependenceInference
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