Transcript Document

If we can reduce our desire,
then all worries that bother us will disappear.
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Survival Analysis
Survival Analysis
Semiparametric Proportional
Hazards Regression (Part II)
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Survival Analysis
Inference for the Regression
Coefficients
Risk set at time y, R(y), is the set of
individuals at risk at time y.
 Assume survival times are distinct and
their order statistics are
t(1) < t(2) < … < t(r).
 Let X(i) be the covariates associated
with t(i).

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Survival Analysis
Partial Likelihood
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Survival Analysis
Partial Likelihood

The product is taken over subjects
who experienced the event.

The function depends on the ranking
of times rather than actual times 
robust to outliers in times
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Survival Analysis
Understanding the Partial
Likelihood

The partial likelihood is based on a
conditional probability argument.

The lost information include:
Censoring times & subjects in
between t(k-1) & t(k)
 Only one failure at t(k)
 No failures in between t(k-1) & t(k)

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Survival Analysis
Maximum Partial Likelihood
Estimate

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An estimate for b is obtained as the
maximiser of PLn(b), called the
maximum partial likelihood estimate
(MPLE).
Survival Analysis
Score Function
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Survival Analysis
Fisher Information Matrix
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Survival Analysis
Estimating Covariance
Matrix


Let bˆ be the MPLE of b, which can be
found using the Newton-Rhapson
method.
The covariance matrix of bˆ is estimated
by
 
var( bˆ )  I ( bˆ )
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1
Survival Analysis
Ties in Survival Times

The construction of partial likelihood is
under the assumption of no tied survival
times

However, real data often contain tied
survival times, due to the way times are
recorded.
How do such ties affect the partial likelihood?
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Survival Analysis
Example

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Consider the following survival data:
6, 6, 6, 7+, 8 (in months)
Survival Analysis
Ties in Survival Times

When there are both censored observations
and failures at a given time, the censoring is
assumed to occur after all the failures.

Potential ambiguity concerning which
individuals should be included in the risk set
at that time is then resolved.

Accordingly, we only need consider how
tied survival times can be handled.
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Survival Analysis
Ties in Survival Times
Let  j  exp(b x j ).
T
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Survival Analysis
Breslow Approximation
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Survival Analysis
Breslow Approximation

Counts failed subjects more than once
in the denominator, producing a
conservative bias.

Adequate if, for each k=1,…,r, dk is
small relative to size of risk set.
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Survival Analysis
Efron Approximation
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Survival Analysis
Efron Approximation

Approximation assumes that all possible
orderings of tied survival times are equally
likely.

Hertz-Picciotto and Rockhill (Biometrics 53,
1151-1156, 1997) presented a simulation
study which shows that Efron approximation
performed far better than Breslow
approximation
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Survival Analysis
Discrete Partial Likelihood
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Survival Analysis
Discrete Partial Likelihood
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Survival Analysis
Discrete Partial Likelihood



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The computational burden grows very
quickly.
Gail, Lubin and Rubinstein (Biometrika 68,
703-707, 1981) develop a recursive
algorithm that is more efficient than the
naive approach of enumerating all subjects.
If the ties arise by the grouping of
continuous survival times, the partial
likelihood does not give rise to a consistent
estimator of b.
Survival Analysis