if we can reduce our desire, then all worries that bother us will disappear
DESCRIPTION
If we can reduce our desire, then all worries that bother us will disappear. Survival Analysis. Semiparametric Proportional Hazards Regression (Part II). Inference for the Regression Coefficients. Risk set at time y, R(y), is the set of individuals at risk at time y. - PowerPoint PPT PresentationTRANSCRIPT
Survival Analysis1
If we can reduce our desire, then all worries that bother us will disappear.
Survival Analysis2
Survival Analysis
Semiparametric Proportional Hazards Regression (Part II)
Survival Analysis3
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).
Survival Analysis4
Partial Likelihood
Survival Analysis5
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
Survival Analysis6
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)
Survival Analysis7
Maximum Partial Likelihood Estimate
An estimate for is obtained as the maximiser of PLn(), called the maximum partial likelihood estimate (MPLE).
Survival Analysis8
Score Function
Survival Analysis9
Fisher Information Matrix
Survival Analysis10
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
Survival Analysis11
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?
Survival Analysis12
Example
Consider the following survival data: 6, 6, 6, 7+, 8 (in months)
Survival Analysis13
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.
Survival Analysis14
Ties in Survival Times
Let ).exp( jT
j x
Survival Analysis15
Breslow Approximation
Survival Analysis16
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.
Survival Analysis17
Efron Approximation
Survival Analysis18
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
Survival Analysis19
Discrete Partial Likelihood
Survival Analysis20
Discrete Partial Likelihood
Survival Analysis21
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