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Censoring and truncation Definitions and types Censoring: time of event is not known precisely Truncation: eligibility or observation of subject or subject-time for study depends on event Use terms left, right, and interval Based on time moving from left to right

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Right censoring: most often dealt with explicitly, several types Fixed censoring, random censoring, other types Use C to denote censoring time Although convention is to use uppercase for random variables, lowercase for fixed quantities, will generally use uppercase for censoring times Will provide justification later Cr / right censoring time

Consider ways right censoring might arise in a study

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Fixed censoring

Simplest: every subject has same censoring time; Type I Arises most naturally in certain types of experiments e.g., animal experiment; start, end experiment at fixed time for all animals Cr is the difference between start and end of experiment for all subjects, is fixed let follow-up time L denote the length of follow-up for a subject L / min(T,Cr) * / I(T<Cr) indicator of whether failure observed (1 = yes; 0 = no) failure-time data usually represented by pair (L,*)

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2 generalizations of fixed censoring progressive type I censoring animals have different fixed-sacrifice times

generalized type I censoring results from staggered entry in many clinical trials, epidemiologic studies, cannot enroll all subjects simultaneously recruitment and subject accrual take place over an interval accrual period may be dictated by availability of subjects, limitations on study resources

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examples: study of kidney graft transplant survival obtaining information on (essentially) all subjects receiving kidney transplants in Delaware Valley, at time of transplant took about 3 years to recruit 750 subjects; recruitment continuing initially planned to end follow-up after 4.5 years from beginning of study Cr varies between 1 and 4.5 years expected to be approximately uniformly distributed over interval study ends at fixed time censoring time: difference interval due to subject availability

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HIP trial: study of screening for breast cancer start time arbitrary in disease process very large pool of potential subjects logistics of study dictate staggered entry subjects recruited from 1963 to 1966 follow-up through mid-1980s potential censoring times range up to 22 years

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In various types of type I censoring, potential censoring time known in advance (will discuss when construct likelihood), even for subjects who fail potential censoring time can be considered baseline covariate X

in many studies, potential censoring time known only approximately in advance

certain failure-time outcomes ascertained only occasionally e.g., kidney transplant study: occurrence of acute rejection ascertained by periodic review of medical records date of medical record review may not be known in advance (although it is possible to make record review refer to predetermined dates)

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Multicenter AIDS Cohort Study (MACS) occurrence of certain events ascertained at regular visits of subjects to study, sometimes through interview may refer to previous time date of visits to study is approximately every 6 months but not known exactly in advance so, analysis which ends at visit 17 will in fact have a potential censoring time which is but approximately known

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Type II censoring: study progresses until the failure of the first r individuals often used in equipment testing

progressive type II censoring when r1 subjects fail, remove n1!r1 subjects wait until r2 additional subjects fail; remove n2!r2 subjects

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Random censoring some other competing event causes removal from study

examples: loss to follow-up; e.g., subjects in MACS move, refuse to participate, come to subsequent visits death from other causes; relevant both when failure-time outcome is causespecific death, other outcomes e.g., HIP study subjects censored when die of non-breast cancer cause relevant both for breast cancer incidence as outcome, breast cancer mortality

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in principle, fundamentally different occurrences loss to follow-up: outcome may actually happen, just not observed death from other causes: outcome never happens, may want to postulate latent failure-time first type poses technical but not conceptual difficulties second type poses both technical and conceptual difficulties what is meaning of term random in phrase random censoring?

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random refers to fact that censoring process/events unplanned/ not under control of investigator, so events occur "randomly"; i.e., censoring events occur as stochastic process does not mean that censoring process is independent of (unobservable) failuretimes (as in terms like missing at random)

will discuss more in unit on competing risks

in some (most) studies, combination of 2 or more types of censoring HIP study, some subjects die of other causes (random censoring); most subjects alive at end of study, staggered entry (generalized type I censoring)

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Left censoring occurs when event occurs before an individual enters study; if it is less than a censoring time study of when first graders learn to read pair of random variables (L,,) L = length of follow-up, , = indicator of left censoring generalization for doubly censored data (right and left censoring): * / censoring indicator (-1 = left censored, 0 = right censored, 1 = failure observed) explain in context of study of 1st graders

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Interval censoring failure-time known to occur in interval (CR,CL); use this notation because, as in right censoring, failure occurs to right of (after) CR, before (to the left of) CL

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Truncation a condition which screens subjects so that the investigator is not aware of their existence left truncation: entry into study depends on some event occurring before event of interest

consider example 3.8 in KM survival study of residents of retirement center. Gives age at entry into center, age at death. Before entry into center, subjects are not known. Thus, left truncated at time of entry. This presumes that the time frame of interest (and possibly for analysis) is chronologic age; in other time frames (e.g., time since entry), not truncated. Thus, truncation is dependent on time frame

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Another example: study of survival of kidney transplants study started in 1998 largely enrolled subjects at the time of their transplant study size limited by the number of transplant recipients per year however, had list of subjects transplanted over previous year enrollment in study depended on consent subjects transplanted before the start of the study left-truncated (also possibly right-censored) reason for enrollment-increase size of study/sample; will need to deal with truncated data correctly natural time scale in study: time since transplant

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Right truncation: entry into study depends on event of interest. Study of time to AIDS from exposure/infection with infected blood (1.19) Only have information on subjects with AIDS, time of their infection.

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Censoring in data other than survival data: provide examples:

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scales, balances; some people heavier than heaviest measurable weight other instruments with limits of detection

studies of kidney function; glomerular filtration rate (GFR) is measure of this when subjects' GFR drops below certain level (about 7), considered to have end-stage renal disease go on dialysis or get transplant can't measure GFR type of left censoring

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