If you want peace you must first have

If you want peace, you must first have peace of mind. To have peace of mind, you must first according to reason. With reason, you will have peace of mind, and then the whole family will be at peace. 1 survival analysis 3/18/2018

Survival Analysis Censoring and Truncation 2 survival analysis 3/18/2018

Abbreviated Outline ¢ 3 Mechanisms that can lead to incomplete observation of a survival time are discussed. survival analysis 3/18/2018

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/no observation of failure time are censoring and truncation. 4 survival analysis 3/18/2018

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. 5 survival analysis 3/18/2018

Typical Censoring Mechanisms ¢ Right censoring Type I censoring l Type II censoring l Random censoring l Left censoring ¢ Double censoring (a data set containing left & right censoring data) ¢ Interval censoring ¢ 6 survival analysis 3/18/2018

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. l Incomplete nature of the observation occurs in the right tail of the time axis. l 7 survival analysis 3/18/2018

Notation ¢ Yi = the survival time of subject i. ¢ Y 1, …, Yn are i. i. d. survival times. ¢ Ci = the censor time of subject i (or say potential observation duration). 8 survival analysis 3/18/2018

Right Censoring ¢ The information from subject i can be represented by where Zi = min{ Yi, Ci } and 9 survival analysis 3/18/2018

Right Censoring: Type I The censor times, Cis, are fixed. ¢ The number of censor cases is random. ¢ The event of interest is observed only if it occurs prior to some prespecified time. ¢ Y 1, …, Yn are assumed to be independent of the mechanism generating the fixed censor times. ¢ 10 survival analysis 3/18/2018

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. 11 survival analysis 3/18/2018

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? 12 survival analysis 3/18/2018

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. ¢ Q: What are Cis? 13 survival analysis 3/18/2018

Example: HIV+ Study 14 survival analysis 3/18/2018

Example: HIV+ Study ¢ Study time: calendar time period ¢ Patient time: the length of time period that a patient spends in the study 15 survival analysis 3/18/2018

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Right Censoring: Type II ¢ ¢ ¢ 17 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

Example A life test of aircraft components cannot wait until all components have failed. ¢ The test was terminated at the 10 th failure out of 13 test items. The 13 observed times (in days) are: 22, 50, 88, 100, 132, 154, 176, 250, 300+, 300+. Identify Cis. ¢ 18 survival analysis 3/18/2018

Right Censoring: Random Arises when other competing events cause subjects to be removed from the study. ¢ The censor times are random and the number of censored cases is also random. ¢ 19 survival analysis 3/18/2018

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 l Death due to some cause other than the one of interest l Migration of human population l 20 survival analysis 3/18/2018

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. 21 survival analysis 3/18/2018

Example: Diet-tumor Study 22 survival analysis 3/18/2018

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. l Incomplete nature of the observation occurs in the left tail of the time axis. l 23 survival analysis 3/18/2018

Left Censoring The observed data are where Zi = max{ Yi, Ci } and 24 survival analysis 3/18/2018

Left Censoring Left censoring is common when the measurement apparatus has a low resolution threshold. ¢ For example, a pollutant apparatus can only detect the pollution if it is higher than 10^-4 ppm (threshold). ¢ But left censoring can happen in other situations too. Example? ¢ 25 survival analysis 3/18/2018

Double Censoring The observed data are where Zi = max{ min{ Yi, ti }, li } and (ti: the right censor time) (li: the left censor time) 26 survival analysis 3/18/2018

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 What are the statuses for each answer? 27 survival analysis 3/18/2018

Example: Marijuana (frequency table) 28 survival analysis 3/18/2018

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. 29 survival analysis 3/18/2018

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. 30 survival analysis 3/18/2018

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 ¢ 31 survival analysis 3/18/2018

Truncation Time ¢ 32 Truncation time for individual i, denoted ti, is the time of the occurrence of the event truncating individual i. survival analysis 3/18/2018

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. ¢ 33 survival analysis 3/18/2018

Example: astronomical data ¢ 34 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 3/18/2018

Example: 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 35 survival analysis 3/18/2018

Example: Channing House What is the time and condition of truncation? The problem can be solved by revising our target population. 36 survival analysis 3/18/2018

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. 37 survival analysis 3/18/2018

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 l Time in years infected by AIDS virus (from 4/1/1978) l Waiting time to the development of AIDS (from the date of infection) Q: What is the time and condition of truncation? 38 survival analysis 3/18/2018