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
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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

23. Model Selection
In SAS, PROC PHREG, use Efron method to handle ties and conduct backward model selection
Once the final model is selected, refit the final model with Exact method to handle ties
Survival Analysis

24. Estimation of Survival Function
To estimate S(y|X), the baseline survival function S0(y) must be estimated first.
Two estimates:
Breslow estimate
Kalbfleisch-Prentice estimate
Survival Analysis

25. Breslow Estimate
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26. Kalbfleisch-Prentice Estimate
An estimate of h0(y) was derived by Kalbfleisch and Prentice using an approach based on the method of maximum likelihood.
Reference: Kalbfleisc, J.D. and Prentice, R.L. (1973). Marginal likelihoods based on Cox’s regression and life model. Biometrika, 60,
Survival Analysis

27. Example: PBC
Survival Analysis

28. Predicted Curve for a fixed Covariate Value
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29. Estimation of the Median Survival Time
Survival Analysis

30. Assessing the Proportional Hazards Assumption
By empirical score process/simulations
In SAS: add a statement
ASSESS PH/ RESAMPLE;
A p-value will be given to assess the significance level of deviation from the proportional hazards assumption
Survival Analysis

31. Strategies for Non-proportionality
Stratify the covariates with non-proportional effects
No test for the effect of a stratification factor (so only for nuisance covariates)
How to categorize a numerical covariate?
Partition the time axis
Add a time-dependent covariate (advanced)
Use a different model (such as parametric AFT model)
Survival Analysis

32. The End
Good Luck for Finals!!
Survival Analysis