if you want peace, you must first have peace of mind.
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04/19/23survival analysis1
If you want peace, you must first have peace of mind. To have peace of mind, you must first act according to reason.
With reason, you will have peace of mind, and then the whole family will be at peace.
04/19/23survival analysis2
Survival Analysis
Censoring and Truncation
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Abbreviated Outline
Mechanisms that can lead to incomplete observation of a survival time are discussed.
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Difficulty of Survival Analysis
The possibility that some individuals may not be observed for the full time to failure.
Two mechanisms that can lead to incomplete observation of failure time are censoring and truncation.
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Censoring and Truncation
A censored observation arises when the exact failure time is unknown, but can only be determined to lie within a certain interval.
A truncated observation is one which is unobservable due to a selection process inherent in the study design.
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Typical Censoring Mechanisms
Right censoringType I censoringType II censoringRandom censoring
Left censoring Double censoring (a data set
containing left & right censoring data) Interval censoring
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Right Censoring
Observation begins at the defined time origin and ceases before the event of interest is realized.The survival time is only known to
exceed a certain value.Incomplete nature of the observation
occurs in the right tail of the time axis.
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Notation
Yi = the survival time of subject i.
Y1, …, Yn are i.i.d. survival times.
Ci = the censor time of subject i (or say potential observation duration).
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Right Censoring
The information from subject i can be represented by
where Zi = min{ Yi, Ci } and
),( iiZ
ii
iii CY
CY
if0
if1
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Right Censoring: Type I
The censor times, Cis, are fixed. The event of interest is observed only
if it occurs prior to some prespecified time.
Y1, …,Yn are assumed to be independent of the mechanism generating the fixed censor times.
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Example: Diet-tumor Study
A laboratory investigator is interested in the relationship between diet and the development of tumors.
3 diet groups: low-fat, saturated-fat, unsaturated-fat diets
30 rates per group An identical amount of tumor cells were injected
into a foot pad of each rat, and the tumor-free times of the rats were recorded.
The study was terminated after 200 days.
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Example: Diet-tumor Study
Tumor-free times (days) for the low-fat group are as follows:140, 177, 50, 65, 86, 153, 181, 191, 77, 84, 87, 56, 66, 73, 119, 140and 200+ for the other 14 rats.“+” denotes a censored observation.
Q: What are Cis?
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Example: HIV+ Study
Subjects were enrolled from 1/1/1989 to 12/31/1991.
The study ended on 12/31/1995. The event of interest is death due to
AIDS or AIDS-related complications.
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Example: HIV+ Study
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Example: HIV+ Study
Study time: calendar time period
Patient time: the length of time period that a patient spends in the study
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Right Censoring: Type II
Arises when n subjects start on study at the same time, with the study terminating once r failures have been observed, where r is some pre-determined integer (r<n).
Experiments involving type II censoring are often used in testing of equipment life.
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Example
A life test of aircraft components cannot wait until all components have failed.
Others?
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Right Censoring: Random
Arises when other competing events cause subjects to be removed from the study.
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Right Censoring: Random
Some events which cause the subject to be randomly censored, with respect to the event of interest, includePatient withdrawal from a clinical trialDeath due to some cause other than
the one of interestMigration of human population
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Right Censoring: Random
The censoring times Ci are random variables assumed to be independent of each other and of the survival times Yi, i=1,…,n.
Often, the censoring scheme in biomedical studies is a combination of random and type I censoring.
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Example: Diet-tumor Study
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Left Censoring
Arises when the event of interest has already occurred for the individual before observation time.The survival time is only known to be
less than a certain value.Incomplete nature of the observation
occurs in the left tail of the time axis.
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Left Censoring
The observed data are
where Zi = max{ Yi, Ci } and
niZ ii ,...,2,1),,(
ii
iii CY
CY
if0
if1
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Left Censoring
Left censoring is common when the measurement apparatus has a low resolution threshold.
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Double Censoring
The observed data are
where Zi = max{ min{ Yi, ti },li } and
(ti: the right censor time)
(li: the left censor time)
niZ ii ,...,2,1),,(
ii
ii
iii
i
lY
tY
tYl
if1
if0
if1
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Example: Marijuana
Q: When did you first use marijuana?
Answer:
1. Exact age
2. I have never used it
3. Cannot recall when the first time was
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Example: Marijuana
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Interval Censoring
A more general type of censoring occurs when failure is known to occur only within an interval.
A generalization of left and right censoring.
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Example: Cancer Recurrence
Survival (failure) time is the time to recurrence of colorectal cancer, following surgical removal of primary tumor.
After surgery, patients are examined every 3 months to determine if cancer has recurred.
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Truncation
Truncation is a condition which screens out certain subjects so that the investigator will not be aware of their existence.
For truncated data, only subjects who satisfy the condition are observed by the investigator.
The condition is usually associated with a truncation time
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Truncation Time
Truncation time for individual i, denoted ti, is the time of the occurrence of the event truncating individual i.
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Left Truncation
Arises when the condition is that the truncation time must occur prior to the event of interest.
Only individuals with Yi > ti are observed.
Left truncated data are rarely seen in medical research; it is often due to the threshold of an apparatus.
Example: astronomical data
With a given telescope, we can only detect a very distant stellar object which is brighter than some limiting flux — the object is left-truncate if it lies beyond detection by our telescope – we cannot tell if the object is even there if we cannot see it.
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Example: Channing House
Channing House is a retirement center in Palo Alto, CA
All the residence were covered by a health care program provided by the center
Ages at death of 462 individuals who were in residence during Jan 1964 to July 1975 are recorded
Ages at which individuals entered the retirement center are also recorded
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Example: Channing House
What is the time and condition of truncation?
The problem can be solved by revising our target population.
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Right Truncation
Arises when only individuals who have experienced the event of interest are included in the sample.
That is, ti = the end date of study and only individuals with Yi < ti are observed.
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Example: AIDS
Only those who developed AIDS were asked for their infection dates
Data: infection and induction times for 258 adults who were infected with AIDS virus and developed AIDS by 6/30/1986 Time in years infected by AIDS virus (from
4/1/1978) Waiting time to the development of AIDS
(from the date of infection)Q: What is the time and condition of truncation?
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