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If we can reduce our desire, then all worries that bother us will disappear.

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Survival Analysis

Semiparametric Proportional Hazards Regression (Part II)

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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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Partial Likelihood

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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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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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Maximum Partial Likelihood Estimate

An estimate for is obtained as the maximiser of PLn(), called the maximum partial likelihood estimate (MPLE).

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Score Function

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Fisher Information Matrix

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Estimating Covariance Matrix Let be the MPLE of , which can be

found using the Newton-Rhapson method.

The covariance matrix of is estimated by

1)ˆ()ˆvar(

I

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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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Example

Consider the following survival data: 6, 6, 6, 7+, 8 (in months)

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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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Ties in Survival Times

Let ).exp( jT

j x

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Breslow Approximation

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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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Efron Approximation

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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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Discrete Partial Likelihood

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Discrete Partial Likelihood

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Discrete Partial Likelihood

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