If you want peace, you must first have peace of mind.

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. survival analysis

2. Survival Analysis Censoring and Truncation survival analysis

3. Abbreviated Outline • Mechanisms that can lead to incomplete observation of a survival time are discussed. survival 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. survival 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. survival analysis

6. Typical Censoring Mechanisms • Right censoring • Type I censoring • Type II censoring • Random censoring • Left censoring • Double censoring • Interval censoring survival analysis

7. Right Censoring • Observation begins at the defined time origin and ceasesbefore 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. survival analysis

8. Notation • Yi = the survival time of subject i. • Y1, …, Yn are i.i.d. survival times. • Ci = the censor time of subject i (or say potential observation duration). survival analysis

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

14. Example: HIV+ Study survival analysis

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

16. survival 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. survival analysis

18. Example • A life test of aircraft components cannot wait until all components have failed. • Others? survival analysis

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

22. Example: Diet-tumor Study survival 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. survival analysis

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

25. Left Censoring • Left censoring is common when the measurement apparatus has a low resolution threshold. survival 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) survival analysis

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

28. Example: Marijuana survival 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. survival 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. survival 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. survival analysis

32. Truncation Time • Truncation time for individual i, denoted ti, is the time of the occurrence of the event truncating individual i. survival analysis

33. Left Truncation • Arises when the condition is that the truncation time must occur prior to the event of interest. (truncation event occurs BEFORE 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. survival analysis

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. survival 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 survival analysis

36. Example: Channing House Truncation event=being alive at 55 years old Truncation condition: truncation event occurs before death The problem can be solved by revising our target population. survival analysis

37. Right Truncation • Arises when only individuals who have experienced the event of interest are included in the sample. That is, truncation event (the end of study) occurs AFTER the event of interest. • That is, ti = time to the end date of study and only individuals with Yi < ti are observed. survival 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) Truncation time is time to the end of study. Only those whose development of AIDS before the end of study are observed. survival analysis