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

2. Semiparametric Proportional Hazards Regression (Part II)
Survival Analysis
Semiparametric Proportional Hazards Regression (Part II)
Survival Analysis

3. 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 Analysis

4. Partial Likelihood
Survival Analysis

5. 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 Analysis

6. 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 Analysis

7. Maximum Partial Likelihood Estimate
An estimate for b is obtained as the maximiser of PLn(b), called the maximum partial likelihood estimate (MPLE).
Survival Analysis

8. Score Function
Survival Analysis

9. Fisher Information Matrix
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10. Estimating Covariance Matrix
Let be the MPLE of b, which can be found using the Newton-Rhapson method.
The covariance matrix of is estimated by
Survival Analysis

11. 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 Analysis

12. Example
Consider the following survival data: 6, 6, 6, 7+, 8 (in months)
Survival Analysis

13. 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 Analysis

14. Ties in Survival Times
Let
Survival Analysis

15. Breslow Approximation
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16. 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 Analysis

17. Efron Approximation
Survival Analysis

18. Efron Approximation
Approximation assumes that all possible orderings of tied survival times are equally likely.
Hertz-Picciotto and Rockhill (Biometrics 53, , 1997) presented a simulation study which shows that Efron approximation performed far better than Breslow approximation
Survival Analysis

19. Exact Partial Likelihood
Instead of average of all possible orderings of tied survival times, we sum them together, which is the exact conditional probability (the exact partial likelihood component at the tied time).
e.g.: Subjects 1, 2, 3 tied at time 6;
6 possible orderings called scenarios A1 to A6
The partial likelihood of Ai is the conditional probability of having Ai, denoted as PL(Ai)
The exact conditional probability is therefore:
P(A1 or A2 or …A6)=sum of PL(Ai) from i=1 to 6
Survival Analysis

20. Discrete Partial Likelihood
Survival Analysis

21. Discrete Partial Likelihood
Survival Analysis

22. Discrete Partial Likelihood
The computational burden grows very quickly.
Gail, Lubin and Rubinstein (Biometrika 68, , 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