1. survival analysis 1 The greatest blessing in life is in giving and not taking.

2. survival analysis 2 Survival Analysis Nonparametric Estimation of Basic Quantities (Sec. 5.4 & Ch. 6)

3. survival analysis 3 Abbreviated Outline Survival data are summarized through estimates of the survival function and hazard function. Methods for estimating these functions from a sample of right-censored survival data are described. These methods are nonparametric. Non-informative censoring is assumed.

4. Non-informative Censoring The knowledge of a censoring time for an individual provides NO further information about this person’s likelihood of survival at a FUTURE time had the individual continued on the study. survival analysis 4

5. 5 Nonparametric Methods Distribution free: no assumptions about the underlying distribution of the survival times. Less efficient than parametric methods if the survival times follow a theoretical distribution. More efficient when no suitable theoretical distributions are known.

6. survival analysis 6 Nonparametric Methods Estimates obtained by nonparametric methods can be helpful in choosing a theoretical distribution, if the main objective is to find a parametric model for the data.

7. survival analysis 7 Example: 6-MP A case-control study Experimental drug: 6-mercaptopurine (6- MP) for treating acute leukemia 11 American hospitals participated 42 patients with complete or partial remission of leukemia were randomly assigned to either 6-MP or a placebo 21 patients per group Patients were followed until their leukemia relapse or until the end of the study

8. survival analysis 8 Example: 6-MP

9. survival analysis 9 Kaplan-Meier Estimator Also called product-limit estimator The standard estimator of the survival function using right- censoring data

10. survival analysis 10 Kaplan-Meier Estimator Data: n individuals with observed survival times: z1, z2, …, zn. Some of them may be right-censored. There may be > 1 individuals with the same observed survival time. Let r be the number of distinct uncensored survival times among zis.

11. survival analysis 11 Kaplan-Meier Estimator Sort distinct uncensored zis in ascending order: Notation:

12. survival analysis 12 Example: 6-MP Consider the 6-MP group:

13. survival analysis 13 Kaplan-Meier Estimator

14. survival analysis 14 Kaplan-Meier Estimator Let tmax be the largest survival time. For t > tmax,

15. survival analysis 15 Example: 6-MP 6-MP group

16. survival analysis 16 Example: 6-MP Placebo group

17. survival analysis17

18. survival analysis 18 Estimation beyond tmax If tmax is censored, for t > tmax: Efron (1967) suggests Gill (1980) suggests

19. survival analysis 19 Understanding K-M Estimator The K-M estimator was constructed by a reduce-sample approach. The K-M estimator is an extension of the empirical survivor function.

20. survival analysis 20 Standard Error

21. survival analysis 21 Pointwise Confidence Interval Under certain regularity conditions, the K-M estimator is: A mle Consistent Asymptotically normal

22. survival analysis 22 Pointwise Confidence Interval

23. survival analysis 23 Example: 6-MP 95% C. I. for the 6-MP group:

24. survival analysis24

25. survival analysis 25 Potential Problem If is close to 0 or 1, the resulting confidence limits could lie outside [0,1]. A possible solution: complementary log- log transformation

26. survival analysis 26 Complementary Log-log Reference: Collect, Sec. 2.2.3. Comp. log-log transformation: Find C.I. forfirst and then convert it back to.

27. survival analysis 27 Complementary Log-log

28. survival analysis 28 Complementary Log-log By Delta Method:

29. survival analysis 29 Example: 6-MP

30. survival analysis 30 Life-table Estimate Also called actuarial estimate For large data sets Grouping survival times into intervals The process is similar to the formation of a frequency table and a histogram in elementary statistics.

31. survival analysis 31 Life-table Estimate

32. survival analysis 32 Life-table Estimate

33. survival analysis 33 Life-table Estimate Actuarial assumption: The censored survival times in I j are uniformly distributed across I j  The average # of individuals at risk in I j is:

34. survival analysis 34 Life-table Estimate An actuarial estimate of p j is:

35. survival analysis 35 Life-table Estimate

36. survival analysis 36 Life-table estimate

37. survival analysis 37 Estimating the Cumulative Hazard Function

38. survival analysis 38 Estimating the Cumulative Hazard Function Nelson-Aalen estimate:

39. survival analysis 39 K-M Estimate vs. N-A Estimate