Phiri, Kelesitse

Tchetgen Tchetgen, Eric

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Harvard UniversityHarvard University Biostatistics Working Paper SeriesYear  Paper Bounds to Evaluate the Pure/natural DirectEffect without Cross-world CounterfactualIndependenceEric Tchetgen Tchetgen∗ Kelesitse Phiri†∗Harvard University, etchetge@hsph.harvard.edu†Harvard University, kphiri@hsph.harvard.eduThis working paper is hosted by The Berkeley Electronic Press (bepress) and may not be commer-cially reproduced without the permission of the copyright holder.http://biostats.bepress.com/harvardbiostat/paper171Copyright c©2014 by the authors.CommentaryBounds to evaluate the pure/natural direct e¤ectwithout cross-world counterfactual independenceEric J. Tchetgen Tchetgen1;2and Kelesitse Phiri2Departments of Biostatistics1 and Epidemiology2; Harvard UniversityCorresponding author: Eric J. Tchetgen Tchetgen, Department of Epidemiology, HarvardSchool of Public Health 677 Huntington Avenue, Boston, MA 02115.1 Hosted by The Berkeley Electronic PressAbstractIdentication of the pure/natural direct e¤ect relies on a cross-world counterfactual independenceassumption which cannot be enforced experimentally and therefore is not strictly subject to scien-tic scrutiny. Here we extend the Robins-Richardson nonparametric bounds for binary exposure,mediator and outcome variables, to handle the situation of exposure-induced confounding. Thenew bounds are a useful alternative to the strong assumptions made by Tchetgen Tchetgen andVanderWeele (Epidemiology, 2014,Mar;25(2):282-91) who point identify the pure direct e¤ect un-der a nonparametric structural equation model with independent errors which implies cross-worldcounterfactual independencies, and make an additional assumption about the exposure-dependentconfounder.2 http://biostats.bepress.com/harvardbiostat/paper171The pure or natural direct e¤ect quanties the e¤ect of an exposure on an outcome that is notmediated by an intermediate variable.1 5 For exposure A, mediator M and outcome Y , let M(a)and Y (a) = Y (a;M(a)) respectively dene the counterfactual mediator and outcome had exposuretaken value a: Likewise, let Y (a;m) dene the counterfactual outcome had exposure and mediatortaken the value a and m; respectively: Finally let Y (a;M(a)) denote the counterfactual outcomehad exposure taken value a and the mediator taken the value it would have under treatment a:The average pure direct e¤ect on the additive scale is then dened for a 6= a :PDE (a; a) = E fY (a;M(a))  Y (a)gThroughout assume that as encoded in the diagram given in Figure 1.a, treatment is randomized,A ?? fY (a;m);M(a)g (1)and also, M is randomized within levels of A,Y (a;m) ??M(a)jA = a (2)Assumptions (1) and (2) are su¢ cient under the 3-node diagram to identify the total averagecausal e¤ect E fY (a)  Y (a)g and the controlled direct e¤ect E fY (a;m)  Y (a;m)g : However,assumptions (1) and (2) taken to be the only counterfactual independencies for each value ofa encoded in the 3-node causal diagram of Figure 1.a, do not su¢ ce to identify counterfactualaverages of the form E fY (a;M(a))g ; and therefore these assumptions alone cannot identifyPDE (a; a).Identication of PDE (a; a) has been controversial even in this simple setting, as it requiresinterpreting the diagram of Figure 1.a to encode additional counterfactual independencies, suchthat in addition to (1) and (2) ; we also have for all a; a :2Y (a;m) ??M(a)jA = a (3)This latter assumption is sometimes described as simply requiring that conditional on exposure,there is no unobserved confounding between the mediator and the outcome, as depicted in Figure1.a.2 4 However, the assumption is much stronger than suggested in Figure 1.a, since it entailsassuming independencies about counterfactuals indexed by distinct treatment interventions, i.e.cross-world counterfactual independencies. Whereas, the traditional statement of no unobservedconfounding (1) and (2) ensures that all counterfactuals involved in the independence statementsare dened for a single treatment value a:6;7 In principle, assumptions (1) and (2) can be madeto hold experimentally, say by randomizing A and subsequently randomizing M conditional on A;while assumption (3) is more audacious and should be made with caution in practice, as it cannotbe experimentally enforced and therefore is not strictly subject to scientic scrutiny.6 The ambigu-ity as to which counterfactual independencies are naturally being assumed in Figure 1.a stems fromthe fact that the counterfactual variables in question (for the mediator and outcome variables) arethemselves not directly represented on the causal diagram. As a remedy to this limiting feature ofthe traditional causal diagram, Richardson and Robins6 have recently developed the single-worldintervention graph (SWIG), a new causal diagram which allows one to represent counterfactualsindexed by a single world intervention directly on the graph. Figure 1.b depicts the SWIG corre-sponding to the causal diagram given in Figure 1.a., for an intervention which sets the treatmentto a xed value ea for all persons in the population. One can then verify using d-separation, thatunlike in Figure 1.a, in the SWIG of Figure 1.b., it is guaranteed that only assumptions (1) and3 Hosted by The Berkeley Electronic Press(2) (for intervention value ea = a), and no cross-world counterfactual independence is encoded inthe graph. Note that to be correctly interpreted, d-separation statements for the SWIG of gure1.b implicitly condition on variables for all split nodes to equal their value under the intervention,i.e. all independence statements are conditional on A = a in Figure 1.b.Upon ruling out cross-world independence assumptions, Robins and Richardson fail to pointidentify PDE (a; a) and have proposed instead reporting nonparametric bounds for PDE(a; a).7For binary Y;A and M; the Robins-Richardson bounds under assumptions (1) and (2) are givenfor PDE (1; 0):L  PDE (1; 0)  U (4)L = max f0;Pr (M = 0jA = 0) + E [Y jA = 1;M = 0]  1g+max f0;Pr (M = 1jA = 0) + E [Y jA = 1;M = 1]  1g   E [Y jA = 0]U = min fPr (M = 0jA = 0) ; E (Y jA = 1;M = 0)g+min fPr (M = 1jA = 0) ; E [Y jA = 1;M = 1]  E [Y jA = 0]gTchetgen Tchetgen and VanderWeele consider identication of PDE(a; a) under the causal dia-gram depicted in Figure 2.a.8 The mediator in this diagram is subject to exposure-induced con-founding which by the recanting witness criterion, renders PDE(a; a) non-identied from theobserved data, even if instead of (3) ; one were to assume the nonparametric structural equa-tions model with independent errors (NPSEM-IE) representation of the diagram in gure 2.a.8Under the NPSEM-IE, we have that conditioning on A and R recovers cross-world counterfactualindependencies for the mediator and outcome,7;8 i.e. for all a and aY (a;m) ??M(a)jA = a;R(a) = r (5)Tchetgen Tchetgen and VanderWeele8 establish that PDE(a; a) becomes identied in the NPSEM-IE for Figure 2.a, provided an additional assumption also holds, either(i) the treatment and confounder R are binary and the e¤ect of treatment on R is monotone atthe individual level, i.e. R(a)  R(a) for a < a; or,(ii) there is no average additive interaction between R and M in their joint e¤ects on Y; i.e.E (Y ja;m; r)  E (Y ja;m; r)  E (Y ja;m; r) + E (Y ja;m; r) = 0Alternative identifying assumptions under the NPSEM-IE were also considered by Robins andRichardson.7 Note that Tchetgen Tchetgen and VanderWeele8 thus continue to make a cross-worldcounterfactual independence assumption now given by (5) : We extend the Robins-Richardsonbounds for PDE(a; a) so that they may be used in the presence of exposure-induced confoundingas depicted in Figure 2.a, upon interpreting the causal diagram strictly as encoding assumptions(1) andY (a;m) ??M(a)jA = a;R = r (6)The SWIG corresponding to this causal diagram is given in Figure 2.b for the intervention ea.In the Appendix we establish that for binary A and M; and a = 1; a = 0; under the SWIG ofFigure 2.b,4 http://biostats.bepress.com/harvardbiostat/paper171L  PDE (1; 0)  U (7)L = max f0;Pr (M (0) = 0) + E [Y (1; 0)]  1g+max f0;Pr (M (0) = 1) + E [Y (1; 1)]  1g   E [Y (a = 0)]U = min fPr (M (0) = 0) ; E [Y (1; 0)]g+min fPr (M (0) = 1) ; E [Y (1; 1)]  E [Y (a = 0)]gTherefore the bounds L and U are identied provided E [Y (a;m)] ; E [Y (a)] and E [M(a)] arethemselves identied. Under the usual no unobserved confounding assumptions (1) and (6), i.e.under the SWIG in gure 2.b for ea = a; we indeed haveE [Y (a;m)] =XrE [Y ja;m; r] f(rja);E [Y (a)] = E [Y ja] ;E [M (a)] = E [M ja] :5 Hosted by The Berkeley Electronic PressAppendixSWIG bounds under (1) and (6): The derivation of L and U is analogous to that of L andU given in Robins and Richardson,6 with a slight generalization to allow for the recanting witnessR. Note thatE fY (1;M(0))g = E fY (1; 1)jM(0) = 1gPr fM(0) = 1g+ E fY (1; 0)jM(0) = 0gPr fM(0) = 0g ;andE fY (1; 1)g = E fY (1; 1)jM(0) = 1gPr fM(0) = 1g+ E fY (1; 1)jM(0) = 0gPr fM(0) = 0g ;which impliesmax f0; (E fY (1; 1)g   Pr fM(0) = 0g) =Pr fM(0) = 1gg  E fY (1; 1)jM(0) = 1gmin f1; E fY (1; 1)g =Pr fM(0) = 1gg  E fY (1; 1)jM(0) = 1glikewiseE fY (1; 0)g = E fY (1; 0)jM(0) = 0gPr fM(0) = 0g+ E fY (1; 0)jM(0) = 1gPr fM(0) = 1gwhich impliesmax f0; (E fY (1; 0)g   Pr fM(0) = 1g) =Pr fM(0) = 0gg  E fY (1; 0)jM(0) = 0gmin f1; E fY (1; 0)g =Pr fM(0) = 0gg  E fY (1; 1)jM(0) = 0gIt follows thatmax f0; (E fY (1; 1)g   Pr fM(0) = 0g)g+max f0; (E fY (1; 0)g   Pr fM(0) = 1g)g  E fY (1;M(0))gmin fPr (M (0) = 1) ; E fY (1; 1)gg+min fPr (M (0) = 0) ; E fY (1; 0)gg  E fY (1;M(0))gproving the result.Acknowledgments: The authors thank James Robins and Thomas Richardson for insightfulcomments and for providing the SWIG gures included in the paper.References[1] Robins, J. M. and S. Greenland (1992). Identiability and exchangeability for direct and indi-rect e¤ects. Epidemiology 3, 143155.[2] Pearl, J. (2001). Direct and indirect e¤ects. In Proceedings of the 17th Annual Conferenceon Uncertainty in Articial Intelligence (UAI-01), San Francisco, CA, pp. 41142. MorganKaufmann.[3] VanderWeele, T.J. and Vansteelandt, S. (2010). Odds ratios for mediation analysis for a di-chotomous outcome - with discussion. American Journal of Epidemiology, 172, 1339-1348.6 http://biostats.bepress.com/harvardbiostat/paper171[4] Imai, K., Keele, L., and Yamamoto, T. (2010). Identication, inference and sensitivity analysisfor causal mediation e¤ects. Statistical Science 25, 5171.[5] Tchetgen Tchetgen EJ and Shpitser I (2012) Semiparametric Theory for Causal MediationAnalysis: e¢ ciency bounds, multiple robustness, and sensitivity analysis).The Annals of Sta-tistics. 40,3: 1816-1845.[6] Richardson TS, and Robins JM (2013). Single World Intervention Graphs (SWIGs): A unica-tion of the counterfactual and graphical approaches to causality. Technical Report 128, Centerfor Statistics and the Social Sciences, University of Washington.[7] Robins, JM, and Richardson TS (2010). Alternative graphical causal models and the identi-cation of direct e¤ects. Causality and psychopathology: Finding the determinants of disordersand their cures (2010): 103-158.[8] Tchetgen Tchetgen EJ and VanderWeele T. On identication of natural direct e¤ectswhen a confounder of the mediator is directly a¤ected by exposure. (2013) Epidemiology.Mar;25(2):282-91.7 Hosted by The Berkeley Electronic Press8 http://biostats.bepress.com/harvardbiostat/paper171

Bounds to Evaluate the Pure/natural Direct Effect without Cross-world Counterfactual Independence

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Bounds to Evaluate the Pure/natural Direct Effect without Cross-world Counterfactual Independence

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