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Bloomberg School BIO 751 - stratification designs

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Principal stratification designsto estimate input data missing due to deathConstantine E. Frangakis1, Donald B. Rubin2,Ming-WenAn1, and Ellen MacKenzie31Department of Biostatistics, Johns Hopkins University2Department of Statistics, Harvard University3Department of Health Policy and Management, Johns Hopkins UniversitySummary. We consider studies of cohorts of individuals after a critical event, such as aninjury, with the following characteristics. First, the studies are designed to measure “input”variables, which describe the period before the critical event, and to characterize the distribu-tion of the input variables in the cohort. Second, the studies are designed to measure “output”variables, primarily mortality after the critical event, and to characterize the predictive (con-ditional) distribution of mortality given the input variables in the cohort. Such studies oftenpossess the complication that the input data are missing for those who die shortly after thecritical event because the data collection takes place after the event. Standard methods ofdealing with the missing inputs, such as imputation or weighting methods based on an as-sumption of ignorable missingness, are known to be generally invalid when the missingness ofinputs is nonignorable, that is, when the distribution of the inputs is different between thosewho die and those who live. To address this issue, we propose a novel design that obtains anduses information on an additional key variable – a treatment or externally controlled variable,which if set at its “effective” level, could have prevented the death of those who died. Weshow that the new design can be used to draw valid inferences for the marginal distribution ofinputs in the entire cohort, and for the conditional distribution of mortality given the inputs,also in the entire cohort, even under nonignorable missingness. The crucial framework that weuse is principal stratification based on the potential outcomes, here mortality under both levelsof treatment. We also show using illustrative preliminary injury data, that our approach canreveal results that are more reasonable than the results of standard methods, in relatively dra-matic ways. Thus, our approach suggests that the routine collection of data on variables thatcould be used as possible treatments in such studies of inputs and mortality should becomecommon.Key Words: Causal inference; Censoring by death; Missing data; Potential Outcomes; Prin-cipal Stratification; Quantum mechanics.1. Introduction.We consider studies that interview cohorts of individuals after a critical event, such as injuryor stroke, with the following two characteristics. First, the studies are designed to measure“input” variables, which describe the period before the critical event, and to characterizethe distribution of the input variables in the cohort. Second, the studies are designed tomeasure “output” variables, primarily mortality after the critical event, and to characterizethe predictive (or conditional) distribution of mortality given the input variables in the cohort.Such studies, however, are often complicated by the fact that the input data are missing forthose who die shortly after the critical event because the data collection takes place after theevent.This problem, input data missing due to death, occurs commonly, for example, in studies ofelders (Cornoni et al., 1993; Reuben, 1995; Cohen, 2002), or victims of injuries (e.g., MacKenzieet al., 2006). The goals we address for such studies are how to estimate the inputs missingdue to death, and how to characterize the predictive (or conditional) distribution of mortalitygiven the input variables in the cohort. Answers to these goals are important because, first,they can be used to better alert the individuals and their physicians about increases in risks,and second, they inform about the pathways of such risks.As a motivating example, consider the National Study on the Costs and Outcomes ofTrauma Centers (NSCOT, MacKenzie et al., 2006). That study used hospital discharge recordsto identify and enroll individuals who received care for injuries. The first follow-up visit wasscheduled at three months. During this visit, patients were interviewed about their pre-injurydisability, as measured by “activities of daily living (ADL)”. It is of interest to evaluate the re-lation that prior disability has to the risk of death following an injury. However, some patientsdied as a result of injury, before this first follow-up visit. Thus, the ADL values are missing forthese patients. If these missing past ADL values have a different distribution than the observed2past ADL values among survivors, standard methods cannot estimate that relation.Another class of examples arises in the evaluation of the effect that a periodic exposure(e.g., to drug) has on the risk of a critical event using a case-crossover design (Maclure, 1991).In its basic form, this design aims to measure, for each one of a group of injury cases, the gaptime between the last exposure and the critical event, and a measure of that person’s typicalfrequency of past exposure. A measure of association between exposure and the critical event isthen defined by comparing the observed gap times to their distribution that would be expectedif the critical event had been unrelated to the exposure process defined by the past frequencies.In this design, even if we know the victims’ most recent exposure to drugs (e.g., by bloodmeasurement), the frequency of past exposure becomes missing for those who die as a result ofsevere injuries, and this missingness is usually ignored (e.g., Vinson et al., 1995). As discussedbelow, such missingness needs to be addressed by new and more appropriate methods. Suchexamples are summarized in Table 1.Table 1 here.Standard methods confronted with missing data from death, as also noted by Zhang andRubin (2003), can be classified into three types. The first type is concerned only with the ob-served data (e.g., cause-specific hazards, dating to Prentice et al. 1978; and partly conditionalon being alive, Kurland and Heagerty, 2005); these methods are not relevant to our problembecause they do not attempt to estimate the missing data. The second type of method assumesignorability (Rubin, 1976) of missing data and essentially replaces them with data matchedfrom fully observed strata, either across time from the same person, or across people for thesame


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