The greatest blessing in life is in giving and not taking.
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Transcript of The greatest blessing in life is in giving and not taking.
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The greatest blessing in life is in giving and not taking.
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Survival Analysis
Nonparametric Estimation of Basic Quantities (Sec. 5.4 & Ch. 6)
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Abbreviated Outline
Survival data are summarized through estimates of the survival function and hazard function.
Methods for estimating these functions from a sample of right-censored survival data are described.
These methods are nonparametric. Non-informative censoring is assumed.
Non-informative Censoring
The knowledge of a censoring time for an individual provides NO further information about this person’s likelihood of survival at a FUTURE time had the individual continued on the study.
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Nonparametric Methods
Distribution free: no assumptions about the underlying distribution of the survival times.
Less efficient than parametric methods if the survival times follow a theoretical distribution.
More efficient when no suitable theoretical distributions are known.
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Nonparametric Methods
Estimates obtained by nonparametric methods can be helpful in choosing a theoretical distribution, if the main objective is to find a parametric model for the data.
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Example: 6-MP
A case-control study Experimental drug: 6-mercaptopurine (6-
MP) for treating acute leukemia 11 American hospitals participated 42 patients with complete or partial
remission of leukemia were randomly assigned to either 6-MP or a placebo
21 patients per group Patients were followed until their leukemia
relapse or until the end of the study
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Example: 6-MP
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Kaplan-Meier Estimator
Also called product-limit estimator The standard estimator of the
survival function using right-censoring data
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Kaplan-Meier Estimator
Data: n individuals with observed survival
times: z1, z2, …, zn. Some of them may be right-censored. There may be > 1 individuals with the
same observed survival time. Let r be the number of distinct
uncensored survival times among zis.
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Kaplan-Meier Estimator
Sort distinct uncensored zis in ascending order:
Notation:
)()2()1( ... rttt
)(
)(
at timerisk at sindividual of #:
at time failures observed of #:
jj
jj
tn
td
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Example: 6-MP
Consider the 6-MP group:
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Kaplan-Meier Estimator
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Kaplan-Meier Estimator
Let tmax be the largest survival time.
For t > tmax,
censored. istmax ?
,uncensored istmax 0)(ˆ tS
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Example: 6-MP6-MP group
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Example: 6-MP
Placebo group
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Estimation beyond tmax
If tmax is censored, for t > tmax:
Efron (1967) suggests
Gill (1980) suggests
0)(ˆ tS
max)(ˆ)(ˆ tStS
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Understanding K-M Estimator
The K-M estimator was constructed by a reduce-sample approach.
The K-M estimator is an extension of the empirical survivor function.
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Standard Error
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Pointwise Confidence Interval
Under certain regularity conditions, the K-M estimator is:
A mle Consistent Asymptotically normal
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Pointwise Confidence Interval
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Example: 6-MP
95% C. I. for the 6-MP group:
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Potential Problem
If is close to 0 or 1, the resulting confidence limits could lie outside [0,1].
A possible solution: complementary log-log transformation
)(ˆ0uS
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Complementary Log-log
Reference: Collect, Sec. 2.2.3.
Comp. log-log transformation:
Find C.I. for first and then convert it back to .
)](ˆloglog[))(ˆ( 00 uSuSg
))(ˆ( 0uSg)(ˆ
0uS
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Complementary Log-log
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Complementary Log-log
By Delta Method:
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Example: 6-MP
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Life-table Estimate
Also called actuarial estimate For large data sets Grouping survival times into intervals The process is similar to the formation
of a frequency table and a histogram in elementary statistics.
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Life-table Estimate
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Life-table Estimate
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Life-table Estimate
Actuarial assumption: The censored survival times in Ij are uniformly distributed across Ij The average # of individuals at risk in Ij is:
2
** jjj
wnn
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Life-table Estimate
An actuarial estimate of pj is:
.1,...,2,1,1~*
kjn
wp
j
jj
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Life-table Estimate
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Life-table estimate
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Estimating the Cumulative Hazard Function
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Estimating the Cumulative Hazard Function
Nelson-Aalen estimate:
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K-M Estimate vs. N-A Estimate