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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