Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 16Page 17Page 18Page 191Censoring and truncationDefinitions and typesCensoring: time of event is not known preciselyTruncation: eligibility or observation of subject or subject-time for studydepends on eventUse terms left, right, and intervalBased on time moving from left to right2Right censoring: most often dealt with explicitly, several typesFixed censoring, random censoring, other typesUse C to denote censoring timeAlthough convention is to use uppercase for random variables, lowercase forfixed quantities, will generally use uppercase for censoring timesWill provide justification laterrC / right censoring timeConsider ways right censoring might arise in a study3Fixed censoringSimplest: every subject has same censoring time; Type IArises most naturally in certain types of experimentse.g., animal experiment; start, end experiment at fixed time for all animalsrC is the difference between start and end of experiment for all subjects, is fixedlet follow-up time L denote the length of follow-up for a subjectrL / min(T,C )r* / I(T<C ) indicator of whether failure observed (1 = yes; 0 = no)failure-time data usually represented by pair (L,*)42 generalizations of fixed censoringprogressive type I censoringanimals have different fixed-sacrifice timesgeneralized type I censoringresults from staggered entryin many clinical trials, epidemiologic studies, cannot enroll all subjectssimultaneouslyrecruitment and subject accrual take place over an intervalaccrual period may be dictated by availability of subjects, limitations on studyresources5examples:study of kidney graft transplant survivalobtaining information on (essentially) all subjects receiving kidney transplantsin Delaware Valley, at time of transplanttook about 3 years to recruit 750 subjects; recruitment continuinginitially planned to end follow-up after 4.5 years from beginning of study rC varies between 1 and 4.5 yearsexpected to be approximately uniformly distributed over intervalstudy ends at fixed timecensoring time: differenceinterval due to subject availability6HIP trial: study of screening for breast cancerstart time arbitrary in disease processvery large pool of potential subjectslogistics of study dictate staggered entrysubjects recruited from 1963 to 1966follow-up through mid-1980spotential censoring times range up to 22 years7In various types of type I censoring, potential censoring time known in advance(will discuss when construct likelihood), even for subjects who failpotential censoring time can be considered baseline covariate Xin many studies, potential censoring time known only approximately in advancecertain failure-time outcomes ascertained only occasionallye.g., kidney transplant study: occurrence of acute rejection ascertained byperiodic review of medical recordsdate of medical record review may not be known in advance (although it ispossible to make record review refer to predetermined dates)8Multicenter AIDS Cohort Study (MACS)occurrence of certain events ascertained at regular visits of subjects to study,sometimes through interviewmay refer to previous timedate of visits to study is approximately every 6 months but not known exactly inadvanceso, analysis which ends at visit 17 will in fact have a potential censoring timewhich is but approximately known9Type II censoring:study progresses until the failure of the first r individualsoften used in equipment testingprogressive type II censoring1 11when r subjects fail, remove n !r subjects222wait until r additional subjects fail; remove n !r subjects10Random censoringsome other competing event causes removal from studyexamples:loss to follow-up; e.g., subjects in MACS move, refuse to participate, come tosubsequent visitsdeath from other causes; relevant both when failure-time outcome is cause-specific death, other outcomese.g., HIP study subjects censored when die of non-breast cancer causerelevant both for breast cancer incidence as outcome, breast cancer mortality11in principle, fundamentally different occurrencesloss to follow-up: outcome may actually happen, just not observeddeath from other causes: outcome never happens, may want to postulate latentfailure-timefirst type poses technical but not conceptual difficultiessecond type poses both technical and conceptual difficultieswhat is meaning of term random in phrase random censoring?12random refers to fact that censoring process/events unplanned/ not under controlof investigator, so events occur “randomly”; i.e., censoring events occur asstochastic processdoes not mean that censoring process is independent of (unobservable) failure-times (as in terms like missing at random)will discuss more in unit on competing risksin some (most) studies, combination of 2 or more types of censoringHIP study, some subjects die of other causes (random censoring); most subjectsalive at end of study, staggered entry (generalized type I censoring)13Left censoringoccurs when event occurs before an individual enters study; if it is less than acensoring timestudy of when first graders learn to readpair of random variables (L,,)L = length of follow-up, , = indicator of left censoringgeneralization for doubly censored data (right and left censoring):* / censoring indicator (-1 = left censored, 0 = right censored, 1 = failureobserved)explain in context of study of 1st graders14Interval censoringRLfailure-time known to occur in interval (C ,C ); use this notation because, as inRLright censoring, failure occurs to right of (after) C , before (to the left of) C15Truncationa condition which screens subjects so that the investigator is not aware of theirexistenceleft truncation: entry into study depends on some event occurring before eventof interestconsider example 3.8 in KMsurvival study of residents of retirement center. Gives age at entry into center,age at death. Before entry into center, subjects are not known. Thus, lefttruncated at time of entry.This presumes that the time frame of interest (and possibly for analysis) ischronologic age; in other time frames (e.g., time since entry), not truncated. Thus, truncation is dependent on time frame16Another example: study of survival of kidney transplantsstudy started in 1998largely enrolled subjects at the time of their transplantstudy size limited by the number of transplant recipients per yearhowever, had list of subjects transplanted over previous yearenrollment in