Rejoinder: on the duality of missing data and causal effectsConstantine E. Frangakis, Donald B. Rubin, Ming-Wen An, and Ellen MacKenzieWe thank the editors for the opportunity to have the topic of this paper discussed. We alsothank the discussants for their generally interesting comments.1. Xie and MurphyXie and Murphy (XM) describe our problem using only observed-data representations, andthen discuss some additional practical issues.Physics versus pure empiricism. XM present a reduced form of our problem using only theresultant observed data. Motivated from this representation, XM (and also Robins, Rotnitzky,and Vansteelandt - RRV) suggest that one does not need to invoke potential outcomes andprincipal strata. We disagree: Potential outcomes and principal strata are essential in orderto formulate the problem and goal, to state explicit assumptions (such as ignorable treatmentassignment), and to devise possible designs to address the problem. For instance, with onlynotation for observed data, it is not possible even to define the meaning of a causal effect.That meaning was central in our approach – to regard a value as truly missing (i.e., notobserved but observable), only if there exists, in principle, an intervention that would havecaused it to be observed. The reader can appreciate the need for potential outcomes alsofrom XM’s own writing: when they comment outside of our specific problem, they too invokepotential outcomes and principal strata (see their discussion after question (d) in “practicalconsiderations”, where their Riis defined as the difference of never jointly observable potentialoutcomes: Si(1) − Si(0)).More generally, a representation in terms of potential outcomes and principal strata isrequired if one is to describe the theoretical, physical underlying system of the problem. Manyanalogies regarding such physical versus purely empirical representations can be drawn. Forexample, man went to the moon based on Newton’s theoretical, physical (albeit not quitecorrect) model of nature’s laws. That voyage would not have been possible if Newton had notpersisted in seeking a physical model, but instead had proposed – and if we had accepted asappropriate – some non-differentiable step function (e.g., based on a CART - tree diagram)that would stop after “explaining” empirically only his discrete, few observations.In summary, postulating a theory in terms of its underlying physics has been, and willcontinue to be more beneficial than mere explanation in terms of observed data, because aphysical system is actually more parsimonious and thus more generalizable, and hence morepowerful for predicting other observable events.On practical considerations. A researcher needs to consider the thoughtful questions (a)-(d)that XM raise, and address them based on the ability to obtain data on factors approximatelysatisfying our assumptions. An example is question (c): if the prevention factor z is knownto be effective, why does the decision maker not administer the most effective level of z toall ? The answer involves obstacles external to the decision maker. Taking, for example, thetime to transport an injured patient to the hospital, and adjusting for severity of injury andknowledge that time is important, considerable variation in time can still exist because of otherfactors: how promptly the injury victim was first spotted and reported; how close the nearesthelp was; availablity of fast transport at the time; and traffic and other problems encounteredby the transport mode. This comment also addresses RRV’s point on ethical considerations:variation in such obstacle factors cannot be generally viewed as ethical or not, because theseobstacles are rarely in the control of the ethically charged decision maker for z.Ofcourse,Ziis assigned by the decision maker so as to maximize the anticipated likelihood of survival, butthis likelihood is only conditional on what the decision maker knows, and so after we conditionon that knowledge, we can effectively assume ignorability.Regarding XM’s discussion of more general principal strata, certainly the meaning of thestrata Si(z) depend on the meaning of the prevention factor z, but this is not a complication2of principal strata, but a consequence of meanings changing with problems. Within a problem,though, the meaning of Si(z) does not depend on the assignment mechanism for the actuallevels Zi.XM wonder about the distinction between the covariates we denoted as X and the inputfactor A, stating that “X precedes both A and Z”. This is not generally correct. Somecovariate values are determined prior to both A and Z, such as age or gender, but othercovariate values, and often those used to ensure ignorability, are determined prior to Z butafter A. In our example, X was the severity of injury as judged by the medical personnel afterthe injury occurred, whereas the input variable A was a disability whose value is determinedbefore the injury, but only recorded at the interview after the injury.More important, as we have emphasized in the paper, there is a clear scientific distinctionbetween the critical covariates X and the input A: the covariates X used to ensure ignorabilityneed only be those that were involved in the decision maker’s informed choice to administer ornot the prevention factor (for example, X can often leave out factors causing variation in z suchas the obstacle factors just described). The key fact that makes it easier to record X than A isthis: If the decision maker for z is a person other than the injured victim, we can, in principle,talk to that decision maker (whether or not the victim eventually dies) and ask for the value ofall those variables X that the decision maker used for the assignment of the prevention factor.We cannot do the same for A because, by definition, its accurate measurement depends on thevictim’s ability to be interviewed, which is impossible if the victim dies.2. Ten HaveTen Have commented on the role of an exclusion restriction and the role of covariates, andhas indicated numerous directions for possible fruitful extensions to our methods.On ignorability and exclusion. Ten Have wonders about our assumption of ignorability,3that is, (A, P ) ⊥⊥ Z | X, and its relation to exclusion restrictions typically made in settings ofnoncompliance. Because the factor A is, by design, an input factor that preceeds the preventionfactor z,thevalueofA cannot be changed (for