censoring and truncation

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Definitions Censoring Truncation Quiz Independence Inference Censoring and truncation ST3242: Introduction to Survival Analysis Alex Cook August 2008 ST3242 : Censoring and truncation 1/22

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censoring and truncation

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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 .

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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 .

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  • 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.

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  • 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.

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  • 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

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  • 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.

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  • 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

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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

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  • 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

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  • 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

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  • 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?

    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

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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

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  • 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.

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  • 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.

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  • 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

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  • 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