Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 16Page 17Page 18Page 19Page 20Page 21Page 22Page 231Nonignorable or informative missingness/censoring Often ill-defined conceptDiscussed concept of independent censoringformalizations/expressions for idea:censoring not dependent on future failure timecan make conditional on baseline covariates:alternative version:idea: once one accounts for baseline covariates, failure-time doesn’t predict who is likely to be lost to follow-up2why might this idea not work?1. Haven’t measured enough baseline predictors ofcensoring; assumes that there might be possible tomeasure enough predictorscan’t fully assess ignorability assumption (as before) but for someunmeasured but possibly measurable Uinference about hazards in setting:depends on assumptionsgeneral sorts of assumptions:a. no assumptions/worst-case scenariosb. make specific assumptions about degree ofignorabilitywhat are extreme assumptions about censored subjects?3worst-case scenariosall subjects for whom outcome is unknown take oneextreme: fail right after censoring timedon’t fail at all during follow-upresults in (nonparametric) bounds on survival curves:no assumptions required in nonparametric settingwhat conditions result in more useful/narrowerbounds?answer in terms of degree of censoring, failure4example: asthma study (both groups combined)how would one derive bounds on comparison parameters(risk ratios, risk differences, etc.)?5comparisons of groups and boundsdiscussdoesn’t by self deal with sampling variabilitycan add confidence limits in usual fashion6alternatively, speculate on degree of departures fromindependent/ignorable censoringcan parametrize in terms of censoring process:e.g., or in terms of hazards how like/unlike censored anduncensored subjects aregiven fixed values for these parameters, can then estimatehazards, etc.methods can reduce to standard ones if censoringignorablesensitivity analysis:vary assumptions about degree of ignorability7often have story about reasons for depatures from randomcensoringe.g., people get sick, unable to come for repeat visits,drop out of study/are lost to follow-upif had adequate data on degree of illness at t, wouldexplain censoringexplainable censoringexplain assumption:time-varying covariateshistory of time-varying covariatesis assumption testable?8Adjusting/controlling for dependent censoringPresent one approach: inverse probability of censoringweighting (IPCW)Explain first in terms of discrete-time censoringdistribution, univariate estimationwill generalizeif had information about all subjects at time t, couldestimate by sample average9Structured tree graph to diagram settingsplit at leftcircumference point:change in covariateprocess split at rightcircumference point:censoring/loss tofollow-up (assumedthat subjects lost/notlost comparable)failure shown onsegment not in circle10If no censoring, estimate survival as Kaplan-Meier: Kaplan-Meier biased:why? (Provide intuitive explanation)what approaches already discussed might be useful?11Sicker people drop-out/lost to follow-up moreapproaches to controlling for dependent censoring (list):standardizationweightingboth yield same results hereassumption: ignorable censoring:12standardization:because ofassumption ofignorable censoring,can use rates in sicksubjects who areuncensored torepresent rates insick subjects who arecensored also;eliminate probabilityof censoring fromgraphstandardize survivalrates/risk todistribution ofcovariates inpopulation:S(1) = 0.5*0.5 + 0.8*0.5 = 0.65 (correct)13weighting:weight all subjectsuncensored by endby probability ofbeing uncensored attime 1 given pastcompute survivalusing weights:weights underlined: difference from Kaplan-Meier, withweights 114weighting creates pseudopopulation that would have beenseen without censoringideas extend to settings with progressive censoring overtime15sick subjects more likely to die, be lost to follow-up thanhealthy onesprevious sickness predicts censoring, death as well16weights for S(2):time-varyinguse probability of being uncensored at end of subject’sfollow-upfor subjects who die at time 1, probability of beinguncensored through time 1for subjects followed to end, product of conditionalprobabilities of being uncensored in each intervale.g., for subjects sick in both intervals, weight = 2*2.5 = 51718weights create pseudopopulation; represents what wouldhave been seen in absence of censoring:average over all subjects:19weighting allows calculation as simple weighted sampleaverage of all uncensored subjects20can also compute as product-limit estimate usingconditional survivals over perioduse time-varying weights21Kaplan-Meier:, a substantial overestimate22time-varying weights generalize to generalmodels/methods for hazards (e.g., proportional hazards)at each failure-time, use as weight 1 over the product ofconditional survivals in model treating censoring eventsas failureslet time-varying weight = note that this is not the probability of surviving givenhistory but a pathwise version of thiscan be derived from a subject’s integratedhazard/intensity functionin typical problems, deal with covariate histories of highdimensionwill need to resort to models to estimateissues with weights; as before23method: called Inverse Probability of CensoringWeighting (IPCW)applicable to other missing-data problems (i.e., not justfailure-time/survival