The greatest blessing in life is in giving and not taking.

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

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

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.

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. 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. 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. Example: 6-MP

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

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. Kaplan-Meier Estimator • Sort distinct uncensored zis in ascending order: • Notation:

12. Example: 6-MP • Consider the 6-MP group:

13. Kaplan-Meier Estimator

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

15. Example: 6-MP 6-MP group

16. Example: 6-MP Placebo group

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

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. Standard Error

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

22. Pointwise Confidence Interval

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

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. Complementary Log-log • Reference: Collect, Sec. 2.2.3. • Comp. log-log transformation: • Find C.I. for first and then convert it back to .

27. Complementary Log-log

28. Complementary Log-log • By Delta Method:

29. Example: 6-MP

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. Life-table Estimate

32. Life-table Estimate

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

34. Life-table Estimate An actuarial estimate of pj is:

35. Life-table Estimate

36. Life-table estimate

37. Estimating the Cumulative Hazard Function

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

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