1. 6/1/2015survival analysis 1 If you want peace, you must first have peace of mind. To have peace of mind, you must first act according to reason. With reason, you will have peace of mind, and then the whole family will be at peace.

2. 6/1/2015survival analysis 2 Survival Analysis Censoring and Truncation

3. 6/1/2015survival analysis 3 Abbreviated Outline Mechanisms that can lead to incomplete observation of a survival time are discussed.

4. 6/1/2015survival analysis 4 Difficulty of Survival Analysis The possibility that some individuals may not be observed for the full time to failure. Two mechanisms that can lead to incomplete observation of failure time are censoring and truncation.

5. 6/1/2015survival analysis 5 Censoring and Truncation A censored observation arises when the exact failure time is unknown, but can only be determined to lie within a certain interval. A truncated observation is one which is unobservable due to a selection process inherent in the study design.

6. 6/1/2015survival analysis 6 Typical Censoring Mechanisms Right censoring Type I censoring Type II censoring Random censoring Left censoring Double censoring (a data set containing left & right censoring data) Interval censoring

7. 6/1/2015survival analysis 7 Right Censoring Observation begins at the defined time origin and ceases before the event of interest is realized. The survival time is only known to exceed a certain value. Incomplete nature of the observation occurs in the right tail of the time axis.

8. 6/1/2015survival analysis 8 Notation Yi = the survival time of subject i. Y 1, …, Y n are i.i.d. survival times. Ci = the censor time of subject i (or say potential observation duration).

9. 6/1/2015survival analysis 9 Right Censoring The information from subject i can be represented by where Zi = min{ Yi, Ci } and

10. 6/1/2015survival analysis 10 Right Censoring: Type I The censor times, Cis, are fixed. The event of interest is observed only if it occurs prior to some prespecified time. Y1, …,Yn are assumed to be independent of the mechanism generating the fixed censor times.

11. 6/1/2015survival analysis 11 Example: Diet-tumor Study A laboratory investigator is interested in the relationship between diet and the development of tumors. 3 diet groups: low-fat, saturated-fat, unsaturated-fat diets 30 rates per group An identical amount of tumor cells were injected into a foot pad of each rat, and the tumor-free times of the rats were recorded. The study was terminated after 200 days.

12. 6/1/2015survival analysis 12 Example: Diet-tumor Study Tumor-free times (days) for the low-fat group are as follows: 140, 177, 50, 65, 86, 153, 181, 191, 77, 84, 87, 56, 66, 73, 119, 140 and 200+ for the other 14 rats. “+” denotes a censored observation. Q: What are Cis?

13. 6/1/2015survival analysis 13 Example: HIV+ Study Subjects were enrolled from 1/1/1989 to 12/31/1991. The study ended on 12/31/1995. The event of interest is death due to AIDS or AIDS-related complications.

14. 6/1/2015survival analysis 14 Example: HIV+ Study

15. 6/1/2015survival analysis 15 Example: HIV+ Study Study time: calendar time period Patient time: the length of time period that a patient spends in the study

16. 6/1/2015survival analysis16

17. 6/1/2015survival analysis 17 Right Censoring: Type II Arises when n subjects start on study at the same time, with the study terminating once r failures have been observed, where r is some pre- determined integer (r<n). Experiments involving type II censoring are often used in testing of equipment life.

18. 6/1/2015survival analysis 18 Example A life test of aircraft components cannot wait until all components have failed. Others?

19. 6/1/2015survival analysis 19 Right Censoring: Random Arises when other competing events cause subjects to be removed from the study.

20. 6/1/2015survival analysis 20 Right Censoring: Random Some events which cause the subject to be randomly censored, with respect to the event of interest, include Patient withdrawal from a clinical trial Death due to some cause other than the one of interest Migration of human population

21. 6/1/2015survival analysis 21 Right Censoring: Random The censoring times Ci are random variables assumed to be independent of each other and of the survival times Yi, i=1,…,n. Often, the censoring scheme in biomedical studies is a combination of random and type I censoring.

22. 6/1/2015survival analysis 22 Example: Diet-tumor Study

23. 6/1/2015survival analysis 23 Left Censoring Arises when the event of interest has already occurred for the individual before observation time. The survival time is only known to be less than a certain value. Incomplete nature of the observation occurs in the left tail of the time axis.

24. 6/1/2015survival analysis 24 Left Censoring The observed data are where Zi = max{ Yi, Ci } and

25. 6/1/2015survival analysis 25 Left Censoring Left censoring is common when the measurement apparatus has a low resolution threshold.

26. 6/1/2015survival analysis 26 Double Censoring The observed data are where Zi = max{ min{ Yi, ti },li } and (ti: the right censor time) (li: the left censor time)

27. 6/1/2015survival analysis 27 Example: Marijuana Q: When did you first use marijuana? Answer: 1. Exact age 2. I have never used it 3. Cannot recall when the first time was

28. 6/1/2015survival analysis 28 Example: Marijuana

29. 6/1/2015survival analysis 29 Interval Censoring A more general type of censoring occurs when failure is known to occur only within an interval. A generalization of left and right censoring.

30. 6/1/2015survival analysis 30 Example: Cancer Recurrence Survival (failure) time is the time to recurrence of colorectal cancer, following surgical removal of primary tumor. After surgery, patients are examined every 3 months to determine if cancer has recurred.

31. 6/1/2015survival analysis 31 Truncation Truncation is a condition which screens out certain subjects so that the investigator will not be aware of their existence. For truncated data, only subjects who satisfy the condition are observed by the investigator. The condition is usually associated with a truncation time

32. 6/1/2015survival analysis 32 Truncation Time Truncation time for individual i, denoted ti, is the time of the occurrence of the event truncating individual i.

33. 6/1/2015survival analysis 33 Left Truncation Arises when the condition is that the truncation time must occur prior to the event of interest. Only individuals with Yi > ti are observed. Left truncated data are rarely seen in medical research; it is often due to the threshold of an apparatus.

34. Example: astronomical data With a given telescope, we can only detect a very distant stellar object which is brighter than some limiting flux — the object is left-truncate if it lies beyond detection by our telescope – we cannot tell if the object is even there if we cannot see it. 6/1/2015survival analysis 34

35. 6/1/2015survival analysis 35 Example: Channing House Channing House is a retirement center in Palo Alto, CA All the residence were covered by a health care program provided by the center Ages at death of 462 individuals who were in residence during Jan 1964 to July 1975 are recorded Ages at which individuals entered the retirement center are also recorded

36. 6/1/2015survival analysis 36 Example: Channing House What is the time and condition of truncation? The problem can be solved by revising our target population.

37. 6/1/2015survival analysis 37 Right Truncation Arises when only individuals who have experienced the event of interest are included in the sample. That is, ti = the end date of study and only individuals with Yi < ti are observed.

38. 6/1/2015survival analysis 38 Example: AIDS Only those who developed AIDS were asked for their infection dates Data: infection and induction times for 258 adults who were infected with AIDS virus and developed AIDS by 6/30/1986 Time in years infected by AIDS virus (from 4/1/1978) Waiting time to the development of AIDS (from the date of infection) Q: What is the time and condition of truncation?