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Bayesian Methods for Inference on the Causal Effects of Mediation

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Title:
Bayesian Methods for Inference on the Causal Effects of Mediation
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1 online resource (112 p.)
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english
Creator:
Kim, Chan Min
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Daniels, Michael Joseph
Committee Members:
Ghosh, Malay
Presnell, Brett Douglas
Perri, Michael G

Subjects

Subjects / Keywords:
bayesian -- causal -- mediation
Statistics -- Dissertations, Academic -- UF
Genre:
Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
This thesis is motivated by the challenges in analyzing effects of an intervention and mediators on a target outcome, which is called mediation analysis. Typically, researchers are interested in decomposing the total effect as the effect of intervention on the outcome through mediators (indirect effect) and the effect passing around mediators(direct effect). Here, we propose a Bayesian approach to conduct inference on direct and indirect effects. The first part of this thesis introduces the potential outcomes framework. In this framework, we illustrate the decomposition of the overall effect into the direct and indirect effect.Further, we distinguish controlled and natural effects; we will focus on natural effects. Also, we briefly review existing approaches for mediation analysis, pointing out advantages and disadvantages. The second part of the thesis focusses on the setting of a single mediator. We propose a fully Bayesian approach to infer natural direct and indirect effects. To identify and then estimate the joint distribution of mediators under two treatment arms, we use a copula model and equate two specific conditional distributions of potential outcomes (up to sensitivity parameters). We also assess sensitivity to our assumptions with sensitivity analysis and informative priors. The third part of the thesis develops a framework to identify and estimate natural direct and indirect effects in the setting of longitudinal mediators and responses. A Bayesian dynamic model is introduced to connect the observed data across different time points. This approach differs from other longitudinal mediation approaches in that we can estimate direct and indirect effects at each time as well as cross-sectional effects. Several conditional independence assumptions (similar to these in Chapter 2) are introduced to identify causal effects at each time. The fourth chapter of thesis explores the setting on multiple mediators. Here, we can decompose the overall effect into the direct effect and the joint indirect effect which is the aggregated effect of all mediators. We further decompose the joint indirect effect into mediator-specific indirect effects and simultaneous effects of pairs, triples,etc. of mediators. For identifiability, we introduce generalized assumptions from the single mediator case with additional sensitivity parameters.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Chan Min Kim.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Daniels, Michael Joseph.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

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Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045730:00001

MISSING IMAGE

Material Information

Title:
Bayesian Methods for Inference on the Causal Effects of Mediation
Physical Description:
1 online resource (112 p.)
Language:
english
Creator:
Kim, Chan Min
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Daniels, Michael Joseph
Committee Members:
Ghosh, Malay
Presnell, Brett Douglas
Perri, Michael G

Subjects

Subjects / Keywords:
bayesian -- causal -- mediation
Statistics -- Dissertations, Academic -- UF
Genre:
Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
This thesis is motivated by the challenges in analyzing effects of an intervention and mediators on a target outcome, which is called mediation analysis. Typically, researchers are interested in decomposing the total effect as the effect of intervention on the outcome through mediators (indirect effect) and the effect passing around mediators(direct effect). Here, we propose a Bayesian approach to conduct inference on direct and indirect effects. The first part of this thesis introduces the potential outcomes framework. In this framework, we illustrate the decomposition of the overall effect into the direct and indirect effect.Further, we distinguish controlled and natural effects; we will focus on natural effects. Also, we briefly review existing approaches for mediation analysis, pointing out advantages and disadvantages. The second part of the thesis focusses on the setting of a single mediator. We propose a fully Bayesian approach to infer natural direct and indirect effects. To identify and then estimate the joint distribution of mediators under two treatment arms, we use a copula model and equate two specific conditional distributions of potential outcomes (up to sensitivity parameters). We also assess sensitivity to our assumptions with sensitivity analysis and informative priors. The third part of the thesis develops a framework to identify and estimate natural direct and indirect effects in the setting of longitudinal mediators and responses. A Bayesian dynamic model is introduced to connect the observed data across different time points. This approach differs from other longitudinal mediation approaches in that we can estimate direct and indirect effects at each time as well as cross-sectional effects. Several conditional independence assumptions (similar to these in Chapter 2) are introduced to identify causal effects at each time. The fourth chapter of thesis explores the setting on multiple mediators. Here, we can decompose the overall effect into the direct effect and the joint indirect effect which is the aggregated effect of all mediators. We further decompose the joint indirect effect into mediator-specific indirect effects and simultaneous effects of pairs, triples,etc. of mediators. For identifiability, we introduce generalized assumptions from the single mediator case with additional sensitivity parameters.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Chan Min Kim.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Daniels, Michael Joseph.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045730:00001


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BAYESIANMETHODSFORINFERENCEONTHECAUSALEFFECTSOFMEDIATIONByCHANMINKIMADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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2013ChanminKim 2

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Tomyfamilyandfriends 3

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ACKNOWLEDGMENTS Itissuchagreatpleasuretothankallforhelpingmesuccessfullycompletethisdissertation.FirstandforemostIexpressmydeepestgratitudetomyadvisor,Dr.MichaelJ.Daniels,whohasadvisedandsupportedmethroughoutmyresearchwithgreatpatienceandinsightfulknowledgewhileencouragingmetoworkinmyownway.Withouthispersistentguidance,thisdissertationwouldnotbepossible.IwouldliketothanktoDr.MalayGhosh,Dr.BrettPresnellandDr.MichaelPerriforservingonmycommitteeandgivingmelotsofhelpfulcommentstoimprovethequalityofthisdissertation.Also,IwouldlikeacknowledgethecommentsfromDr.JasonRoyandDr.JosephHogan.Mylifeasadoctoralstudentforthelast5yearshasbeenterricbecauseofmanyfriendsfromSouthKoreaandthoseImetatUniversityofFlorida.Particularly,Iappreciatemylongtimefriends,TaehyungKwon,JonghwanHwang,JoowonLee,JoonhoShin,JoonwooSohnandSukwonWangfortheirspecialfavorsfromKoreaaswellasmyclassmates,AlexanderKirpich,LeiHuang,KeLi,MinzhaoLiuandmanyothersfortheirhelpandsupport.Ithanktomyparents,YounghoKimandInsookHwang,whoseimmeasurablelove,supportandpridemadethisentirestudypossibleandwellworththeeffort.Ialsowishtothankmyparents-in-law,MyungchulChoiandJongdukKim,fortheirconstantencouragementandsupportwithsincerelove.Otherfamilymembers,SoohyungLee,SookhyunKimandYuminChoiwerealsosourcesofmyenergyforbrushingmyway.Lastly,Ioweaprofounddebtofgratitudetomywife,SooyounChoi,whohaspatientlyenduredaseeminglyendlessgraduatecareerwithallherheart. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTIONTOMEDIATIONANALYSIS ................... 12 1.1BasicFramework ................................ 12 1.2LiteratureReviewfortheSingleMediatorCase ............... 14 1.3LiteratureReviewforLongitudinalCausalEffects .............. 17 1.4LiteratureReviewforMultipleMediators ................... 19 2BAYESIANINFERENCEFORTHECAUSALEFFECTOFMEDIATION .... 21 2.1Introduction ................................... 21 2.2InferenceonCausalEffects .......................... 25 2.2.1Notation ................................. 25 2.2.2Assumptions .............................. 26 2.2.3AlternativeAssumptionsRequiredforNon-ParametricIdentication 30 2.2.4IdenticationofJointDistributionsforComputationofDirectandIndirectEffects ............................. 30 2.2.5ModelsandEstimation ......................... 32 2.3PosteriorComputations ............................ 33 2.4SensitivityAnalysisandElicitation ...................... 33 2.5SimulationStudytoAssessSensitivitytoViolationsofAssumption3 ... 34 2.6TOURS:WeightManagementTrial ...................... 36 2.6.1DescriptionofTOURS ......................... 36 2.6.2Models .................................. 36 2.6.3ElicitationofSensitivityParameters .................. 37 2.6.4Results ................................. 37 2.6.5ComparisonwithBaronandKennyTypeEstimators ........ 38 2.7Discussion ................................... 39 3BAYESIANAPPROACHFORLONGITUDINALMEDIATIONANALYSIS .... 41 3.1IntroductiontoLongitudinalMediationAnalysis ............... 41 3.2InferenceonCausalEffectsofLongitudinalMediators ........... 44 3.2.1CausalEffectsofMediators ...................... 44 3.2.2BayesianDynamicModel ....................... 45 3.2.3AssumptionsforLongitudinalMediationAnalysis .......... 46 5

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3.2.4IdenticationofJointDistributionsforComputationofDirectandIndirectEffectsatEachTime ..................... 49 3.2.4.1Obervationmodel ...................... 51 3.2.4.2Evolutionmodel ....................... 52 3.3PosteriorComputationsofLongitudinalMediationAnalysis ........ 53 3.4SensitivityAnalysis ............................... 54 3.5AnalysisofMediationinCTQII ........................ 55 3.5.1DescriptionofCTQII .......................... 55 3.5.2PriorsandSensitivityParameters ................... 56 3.5.3ResultsofLongitudinalMediationAnalysis .............. 56 3.6DiscussiononLongitudinalMediationAnalysis ............... 65 4BAYESIANINFERENCEFORTHECAUSALEFFECTSINTHESETTINGOFMULTIPLEMEDIATORS ............................ 66 4.1IntroductiontotheCausalEffectsofMultipleMediators .......... 66 4.2NewFrameworkforMultipleMediators .................... 68 4.2.1NotationandDenition ......................... 68 4.2.2ProblemsoftheRegressionApproachforMultipleMediators ... 71 4.3InferenceontheCausalEffectsofMultipleMediators ............ 73 4.3.1AssumptionsforMultipleMediators .................. 73 4.3.2IdenticationandModelSpecication ................. 76 4.3.3SpecicationofModels ......................... 77 4.4PosteriorComputationsintheCausalEffectsofMultipleMediators .... 78 4.5MediationintheSTRIDETrial ......................... 79 4.5.1DescriptionofSTRIDE ......................... 79 4.5.2PriorsandSensitivityParameters ................... 80 4.5.3ResultsofMultipleMediators ..................... 81 4.5.4ComparisonwiththeRegressionApproach ............. 83 4.6DiscussionontheCausalEffectsofMultipleMediators ........... 83 5CONCLUSION .................................... 85 5.1SummaryofContributions ........................... 85 5.2FutureResearch ................................ 86 APPENDIX ACOMPARISONOFVARIANCES .......................... 88 A.1VariancewithoutAssumption5 ........................ 88 A.2VariancewithAssumption5 .......................... 89 A.3ComparisonofVarianceswithandwithoutAssumption5 ......... 89 BIMPLEMENTATIONOFDIRICHLETPROCESSPRIORSINBUGS ....... 91 CDETAILSOFPOSTERIORCOMPUTATION .................... 92 6

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DORDINALCATEGORICALMEDIATOR ...................... 94 D.1Assumption3foranOrdinalMediator .................... 94 D.2SpecicationofModels ............................ 95 D.3SensitivityAnalysis ............................... 95 ECOMPARISONOFVARIANCESINLONGITUDINALMEDIATION ....... 97 E.1VariancewithoutAssumption4inLongitudinalMediation ......... 97 E.2VariancewithAssumption4inLongitudinalMediation ........... 98 E.3ComparisonofVarianceswithandwithoutAssumption4inLongitudinalMediation .................................... 98 FMONTECARLOINTEGRATIONINLONGITUDINALMEDIATION ....... 100 GELICITATIONOFTHESENSITIVITYPARAMETER ............... 101 HCOMPARISONOFVARIANCESINMULTIPLEMEDIATORS .......... 102 H.1VariancewithoutAssumption4inMultipleMediators ............ 102 H.2VariancewithAssumption4inMultipleMediators .............. 103 H.3ComparisonofVariancesinMultipleMediators ............... 103 IMONTECARLOINTEGRATIONINMULTIPLEMEDIATORS .......... 105 REFERENCES ....................................... 106 BIOGRAPHICALSKETCH ................................ 112 7

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LISTOFTABLES Table page 2-1SimulationstoassesssensitivityofestimateofNIEtoviolationsinAssumption3 ............................................ 35 2-2Posteriormeansand95%C.I.'sforpriorsUnif[0,1],Unif[50,100],Unif[1,1.3]. ................................... 38 2-3Posteriormeansand95%C.I.'sfor2f50,75,100g,2f1.0,1.15,1.3g. ... 40 3-1EstimatesandstandarddeviationsoftheNIEandNDEfor=0.3. ........ 59 3-2EstimatesandstandarddeviationsoftheNIEandNDEfor=0.5. ........ 60 3-3EstimatesandstandarddeviationsoftheNIEandNDEfor=0.7. ........ 61 4-1Bayesianestimatesand95%credibleintervalsofTE,JNIEandNDEforeachcase .......................................... 82 4-2Bayesianestimatesand95%credibleintervalsofNIE'sforeachcase ..... 82 4-3Bayesianestimatesand95%credibleintervalsofSNIE'sforeachcase .... 83 4-4Estimatesand95%C.I.'sofTE,JNIEandNDEfromtheregressionapproach 83 4-5Estimatesand95%C.I.'sofNIE'sfromtheregressionapproach ........ 83 8

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LISTOFFIGURES Figure page 3-1Thelongitudinalmediationmodelwitht=1,2,3. ................. 45 3-2Cessationratesoftheexerciseinterventionandthecontrol. ........... 57 3-3NIE,NDEandTEforweek4,5,6,7,8when=0.3. ............... 62 3-4NIE,NDEandTEforweek4,5,6,7,8when=0.5. ............... 63 3-5NIE,NDEandTEforweek4,5,6,7,8when=0.7. ............... 64 4-1PartitioningoftheJNIEfor3mediators ....................... 70 4-2Possiblerangesof1and2forthepositivedenitecovariancematrixw/andw/o1>2 ...................................... 81 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyBAYESIANMETHODSFORINFERENCEONTHECAUSALEFFECTSOFMEDIATIONByChanminKimAugust2013Chair:MichaelJ.DanielsMajor:StatisticsThisthesisismotivatedbythechallengesinanalyzingeffectsofaninterventionandmediatorsonatargetoutcome,whichiscalledmediationanalysis.Typically,researchersareinterestedindecomposingthetotaleffectastheeffectofinterventionontheoutcomethroughmediators(indirecteffect)andtheeffectpassingaroundmediators(directeffect).Here,weproposeaBayesianapproachtoconductinferenceondirectandindirecteffects.Therstpartofthisthesisintroducesthepotentialoutcomesframework.Inthisframework,weillustratethedecompositionoftheoveralleffectintothedirectandindirecteffect.Further,wedistinguishcontrolledandnaturaleffects;wewillfocusonnaturaleffects.Also,webrieyreviewexistingapproachesformediationanalysis,pointingoutadvantagesanddisadvantages.Thesecondpartofthethesisfocussesonthesettingofasinglemediator.WeproposeafullyBayesianapproachtoinfernaturaldirectandindirecteffects.Toidentifyandthenestimatethejointdistributionofmediatorsundertwotreatmentarms,weuseacopulamodelandequatetwospecicconditionaldistributionsofpotentialoutcomes(uptosensitivityparameters).Wealsoassesssensitivitytoourassumptionswithsensitivityanalysisandinformativepriors.Thethirdpartofthethesisdevelopsaframeworktoidentifyandestimatenaturaldirectandindirecteffectsinthesettingoflongitudinalmediatorsandresponses.A 10

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Bayesiandynamicmodelisintroducedtoconnecttheobserveddataacrossdifferenttimepoints.Thisapproachdiffersfromotherlongitudinalmediationapproachesinthatwecanestimatedirectandindirecteffectsateachtimeaswellascross-sectionaleffects.Severalconditionalindependenceassumptions(similartotheseinChapter2)areintroducedtoidentifycausaleffectsateachtime.Thefourthchapterofthesisexploresthesettingonmultiplemediators.Here,wecandecomposetheoveralleffectintothedirecteffectandthejointindirecteffectwhichistheaggregatedeffectofallmediators.Wefurtherdecomposethejointindirecteffectintomediator-specicindirecteffectsandsimultaneouseffectsofpairs,triples,etc.ofmediators.Foridentiability,weintroducegeneralizedassumptionsfromthesinglemediatorcasewithadditionalsensitivityparameters. 11

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CHAPTER1INTRODUCTIONTOMEDIATIONANALYSIS 1.1BasicFrameworkInferencesforcausaleffectsareofinteresttosocialandbehavioralresearchers.Inparticular,theyareofteninterestedinsituationswhereintermediatevariables,calledmediators(Kraemeretal.,2002),arepresentonthecausalpathwayandtrytomeasuretheeffectofthetreatmentonoutcomesthroughmediatorsaswellasthetreatmenteffectonoutcomesthatpassaroundthemediators.TheprocessofhowtreatmentsaffectoutcomeshasoftenbeenaBlackBox(Imaietal.,2011)fordecadesandseveralapproacheshavebeenproposedbyresearcherstolookinsidethebox.Sincemostoftheirapproachesarebasedonstrongassumptionsorrestrictedtocertainsituations,therearenosetrulesforidentifyingcausaleffectsofinterest.Westartoutbydeningtheeffectofaninterventioninthesettingofasinglemediator.Thegraphbelowexplicitlyillustratestheidea,Z)166(!Y&%MHere,thedirecteffectofinterventionZonoutcomeYisthehorizontalarrowatthetop.TheindirecteffectofinterventionZonYisthearrowpassingthroughmediatorMonitspathway.Then,thetotaleffectofexposureZonoutcomeYissimplythesumoftheabovetwoeffects.Thisnotionofdeningcausaleffectsisusuallyaccompaniedbythepotentialoutcomeframework,socalledRubin'scausalmodel(RCM)(Rubin,1974;Holland,1986).ForarandomizedinterventionZ2f0,1g,therandomvariableMzdenotesavalueofthemediatorforanindividualassignedtointerventionZ=zandthepairofrandomvariables(M0,M1)denotespotentialvaluesofamediatorunderinterventionz=0,1.Similarly,letYz,Mz0denoteapotentialoutcomefrominterventionZ=zandamediatorvalueinducedbyZ=z0.Onlyoneoftheoutcomes(Y0,M0,Y1,M1) 12

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canbeobservedforanindividual.WecanneverobserveY0,M1orY1,M0.Then,foranindividual,thetotaleffect(TE)isdenedas TE=Y1,M1)]TJ /F4 11.955 Tf 11.96 0 Td[(Y0,M0=(Y1,M1)]TJ /F4 11.955 Tf 11.95 0 Td[(Y1,M0)+(Y1,M0)]TJ /F4 11.955 Tf 11.96 0 Td[(Y0,M0) (1) =IE+DEwheretherstparenthesesof( 1 )quantiestheindirecteffect(IE)whichcapturestheeffectofthemediatoronoutcomewhilethetreatmenteffectiscontrolledatz=1.Thesecondcapturesthedirecteffect(DE)oftreatmentonoutcomegiventhemediatorisxedatthenaturalvalueunderz=0.(Pearl,2001).Inthisequation,theunobservablequantityY1,M0isacounterfactualvariable(Greenlandetal.,1999)thatwouldresultifanindividualreceivingtreatmentz=1hadtheirmediatorxedatthevaluethatwouldresultfromnotreatment.MostofdifcultiesofinferenceoncausaleffectsstemfromhowtohandleY1,M0properly.Manyresearchershaveusedthestructuralequationmodels(BaronandKenny,1986)toestimatedirectandindirecteffects.However,forthisframeworktoresultinthecausaleffectofmediators,strongandpotentiallyunrealisticassumptionsareneeded(Imaietal.,2010).MoredetailsonthisareinSection 1.2 .WeobviouslycannotidentifythetotaleffectattheindividuallevelsinceonlyeitherY1,M1orY0,M0isobservedforanindividual.Thatiswhyresearcherstrytoestimatetheaveragecausaleffect,E(Y1,M1)]TJ /F4 11.955 Tf 12.61 0 Td[(Y0,M0),ofapopulationlevel,themeandifferenceofoutcomesbetweentwotreatments.Theeffectsgivenin( 1 )arenaturaleffects(Pearl,2001).ThesedeneYz,Mz0asapotentialoutcomeaffectedbytheinterventionzandamediatorissettohaveitsnaturalvalueunderz0.Thesearedistinguishedfromcontrolledeffects(Pearl,2001).Controlledeffectsmeasuretheeffectsofthepre-determinedvalueofmediators.Thatis,togetacontrolledeffect,xavalueofthemediatoratacertainlevelm2MandthenY1,m)]TJ /F4 11.955 Tf 11.15 0 Td[(Y0,m 13

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measuresadirecteffect.InthisdissertationwewilldevelopnonparametricBayesianmethodstoinfernaturaldirectandindirecteffects.ThesecanbeaccomplishedbymakingappropriateassumptionsthatmakeitpossibletoidentifythejointdistributionofmediatorsandtheposteriordistributionofY1,M0.Sinceapairofpotentialvaluesofmediators(M0,M1)canneverbeobservedatthesametime,weborrowtheideaoftheGaussiancopulamodel(Nelsen,1999)tospecifythejointdistributionofmediatorswhichallowedforexiblespecicationofthemarginaldistributionsofmediators.AnotherkeyaspectofourapproachisthatweequatethedistributionofthecounterfactualvariableY1,M0totheobservabledistributionofY1,M1uptosensitivityparameters.Here,westratifythepopulationintothoseforwhomthetreatmenthasalargeeffectandsmalleffectonthemediatorsformakingtheassumptionlessrestrictive.Inthisdissertation,weconsiderthreemediatorsettings:asinglemediator,longitudinalmediators,andmultiplemediators.InChapter2,wedevelopamethodtohandleasinglemediator.Thissituationhasbeenwidelystudiedbyseveralresearcherswhicharediscussedinthenextsection.InChapter3,weextendthepreviousideatoestimatecausaleffectsinthesettingoflongitudinalmediators.Forthelongitudinalmediationanalysis,weincorporateaBayesiandynamicmodel(WestandHarrison,1997)foridentifyingthejointdistributionofnaturaldirectandindirecteffectsatdifferenttimepoints.InChapter4,weadoptthisapproachtothecaseofmultiplemediatorswithintroducingfewnewterminologiessuchasthejointindirecteffect,mediator-speciceffectsandsimultaneouseffects.Inthefollowingtwosubsections,wereviewthecrucialcontributionstotheliteratureforeachcase. 1.2LiteratureReviewfortheSingleMediatorCaseFirst,weconsidertheproblemofhowtomeasurecausaleffectsofmediationinthepresenceofsinglemediator.Asamatteroffact,therehavebeenseveralstudiesmadeonhowatreatmenthasaneffectonaresponseviaamediatingvariable.The 14

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mostcommonlycitedmethodisfromBaronandKenny(1986).Thismethodusesasetofthreeregressionmodelsgivenas Y=1+1Z+1M=2+2Z+2 (1) Y=3+3Z+M+3Theindirecteffectcanbeestimatedby2(MacKinnonandDwyer,1993)andthedirecteffectby3.ThereasonforthepopularityoftheBaronandKennyapproachliesinitssimplicitytoimplementandinterpret.Itisoftenreferredtoasthestructuralequationmodel(SEM)approach.Therehavebeenseveralmethodsproposedforassessingwhetheramediatedeffectisstatisticallysignicant.Thecausalstepsapproach,outlinedbyBaronandKenny(1986)andJuddandKenny(1981),hasbeendevelopedmostlybyMacKinnonandothers(MacKinnonetal.,2002).Thecausalstepsapproachtests1)ifZsignicantlyaccountsforvariabilityofY,2)ifZsignicantlyaccountsforvariabilityofMand3)ifMalsoaccountsforvariabilityofYaftercontrollingforZin( 1 )inordertoprovetheexistenceofmediatedeffects.However,duetoitslowpowertodetecttheeffectofamediator(FritzandMacKinnon,2007),Sobel(theproduct-of-coefcientsapproach,1982)andothers(PreacherandHayes,2004)haveproposedatestoftheproductterm2assumingthesamplingdistributionof2isanormaldistribution.However,thisassumptionisvalidonlyinlargesamples.Toovercomethelackofnormality,bootstraptestshavebeenproposedfortestingmediationeffects(BollenandStine,1990;PreacherandHayes,2008).FortheSEMbasedmodels,thecausaleffectofthemediatorcanonlybeidentiedunderasetofassumptions,namelysequentialignorabilityandnointeraction(Imaietal,2010).Theformerstatesthatposttreatmentvariables(i.e.,potentialmediator)canberandomizedwithineachtreatmentlevel(possiblygivenbaselinecovariates). 15

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Differentversionsofsequentialignorabilityassumptionshavebeensuggestedbymanyresearchers(Pearl,2001;Robins,2003;Petersenetal.,2006;HafemanandVanderWeele,2011,Imaietal.,2010).Thelatter,nointeractionassumption,correspondstohavingnointeractionterm(ZM)inthelastequationof( 1 ).Thisistypicallytoostrong.Forcontrolledeffects,itcorrespondstoE(Y1m)]TJ /F4 11.955 Tf 11.96 0 Td[(Y0m)notdependingonm.TorelaxtheseassumptionsrequiredforSEM,severalotherapproacheshavebeenproposed.Principalstratication(PS)(FrangakisandRubin,2002),dividessubjectsintostrataintermsofthepotentialvaluesofmediators(orcovariates)andmeasuresthecausaleffectswithineachstratum(principaleffects).ThePSmodelisanalternativeforSEMsinceitallowsconfoundingonthecausalpathwayanddoesnotrequirethesequentialignorabilityassumption.Gallopetal.(2009)denedprincipalstratumbasedonthepotentialvaluesofmediators.Then,theyassumeaparametricmodelwithineachstratumanddrawcausaleffects.Forexample,whentreatmentandmediatorarebinary,(1)M0=M1=0;(2)M0=0,M1=1;(3)M0=1,M1=0and(4)M0=M1=1arepossiblestrataandcausaleffectsareinferredforeachstratum.Ellittetal.(2010)proposedaPSapproachinaBayesiancontext.TheproblemwiththePSapproachis,sincecausaleffectsareestimatedwithineachstratum,itisnotclearhowtodrawthepopulationdirectandindirecteffectfromthismodel.Rubinhasalsocastdoubtonitsutilityformeasuringindirectanddirecteffects(Rubin,2004).VanderWeele(2008)discoveredsomerelationshipsbetweenprincipalstraticationanddenitionsofdirectandindirecteffectsinmediationanalysis.AnotherdrawbackofPSliesinthecaseofcontinuousmediators,asunclearhowtoformprincipalstratabasedonpairsofpotentialmediator(M0,M1).Anotherapproachisinstrumentalvariables(IV)(Angristetal.,1996;ImbensandRubins,1997).ThebasicideaofIVistoincorporateanextravariable(aninstrumentalvariable)inthestructuralequationmodelsothatwecancontrolconfoundingonthe 16

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causalpathway.Albert(2008)andSobel(2008)proposedanIVapproachformediationanalysis.However,intheIVapproach,theyassumean`exclusionrestriction'whichstatesthattreatmentsaffectpotentialoutcomesonlythroughmediators.Thatis,fortreatmentz,z02Zandmediatorm2M,potentialoutcomesY(z,m)=Y(z0,m).Accordingly,theIVapproachgenericallydoesnotallowfordirecteffects.Semiparametricapproachesusingthestructuralmeanmodel(SMM)(Fischer-LappandGoetghebeur,1999;TenHaveetal.,2004;VansteelandtandGoetghebeur,2007)havebeenproposedtoestimatecausaleffects.SMMdoesnotassumesequentialignorabilityofthemediator,butitrequiresanexclusionrestrictionwhichblocksthedirecteffect.TenHaveetal.(2007,2012)suggestedamodiedversionwhichmakesano-interactionassumptioninstead.Marginalstructuralmodel(MSM)(Robinsetal.,2000;Hernanetal.,2001)hasalsobeenproposed.Theyuseinverse-probabilityweighting(IPW)todealwithmissingcounterfactualsandtohandleconfounding.ThismodeldiffersfromSEMsinceitmakesuseofIPWforadjustingforconfoundersinsteadofaddingmorevariablestotheregression.However,adrawbackisthatestimatorsarenotstablewhenmediatorsarecontinuous,asanindividualunlikelyvalueofamediatorcanhaveahugeinuenceontheestimates(Vansteelandt,2009).Recently,therehasbeensomeapproachesthatconsidercounterfactualvariablesasmissingdataandimputethosevaluesbasedonobservedquantities(BondarenkomandRaghunathan,2010;Zhang,2012).WedescribeourapproachinChapter2. 1.3LiteratureReviewforLongitudinalCausalEffectsIncontrasttothesinglemediatorcase,therearealimitednumberofliteratureonthecausaleffectsoflongitudinalmediators.ColeandMaxwell(2003)andMaxwelletal.(2011)proposedanautoregressivemodeltoassessindirecteffectsanddirecteffects.ForeachinterventionZt,mediatorMtandresponseYt,theypre-speciedastructuralmodelandcorrelationsbetweenZt,MtandYtateachtimeandoveralltimeperiods. 17

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Then,theydenedthemediatedeffectasthesumofallpossiblepathsfromZ1tothenalresponseYTthroughanyofMt's.AndthedirecteffectisdenedasthedirectpathfromZ1toYT.However,Inthisframework,causaleffectsofinteresthighlydependonthemodelingstructuresandpredeterminedcorrelations.AnotherSEMbasedmodelisthelatent-growthmodel(LGM)(MuchenandCurran,1997;Cheongetal.,2003).Inthisapproach,mediatorsandoutcomesaremeasuredandmodeledastwodifferentparallelprocesseswithlatentfactors.Afterconstructingtwoseparategrowthmodels,theysetstructuralequationmodelsoflatentfactorsandmeasurecausaleffectsviatheproductoftheircoefcients.However,thisapproachmeasuresthedirectandtheindirecteffectsnotateverytimebutonlyattheendofstudy.OtherapproachesbasedonSEMareoutlinedinMacKinnonetal.(2007).Generally,longitudinalSEMmodelsaretooparametric.Linetal.(2008,2009)examinedtime-varyingmediation(ofcompliance).Theyusedhierarchicallatentclassstructuresthatcharacterizesubjectcompliancebehavioralpatternsovertime.Afterthat,theyformedprincipalstrataofcomplianceandidentiedeachcomplianceclasswithalatentvariable.Byutilizinglevelsofamediatorasprincipalstrata,theycandrawprincipalstratadirectandindirecteffectswhichareoutlinedinthecross-sectionalcasebyRubin(2004).AndbyVanderWeele(2008),thisprincipalstratamediationanalysiscanhaveasimplerelationshipwithpopulationdirectandindirecteffects.However,itishardtoadoptthisapproachtoacontinuousmediatorcasesincetheprincipalstratacannotbedenedintheusualsense.Also,VanderWeele(2011)pointedoutthattheindirecteffectcannotbeidentiedintheprincipalstrataapproachingeneral.InvanderLaanandPetersen(2004),treatmentalsomayvarywithtimeandintermediatevariablesandoutcomesaremeasuredateachtimeperiod.Theytamarginalstructuralmodel(MSM)forthemarginalmeanofpotentialoutcomeE(Yz,Mz0)wherezdenotestreatmenthistoriesandMz0denoteshistoryofmediatorsinduced 18

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bythemultivariatetreatmentregimez0.Then,thedirecteffectoftreatmentregimezisE(Yz,M0)]TJ /F4 11.955 Tf 12.79 0 Td[(Y0,M0)andtheindirecteffectofzisdenedasE(Yz,Mz)]TJ /F4 11.955 Tf 12.79 0 Td[(Yz,M0).Foridentication,theymakeanewconditionalindependenceassumptionwhichstatesthat,withineachstratumofbaselinecovariates,thedirecteffectoftreatmentataxedlevelofthemediatordoesnotdependonasubject'scounterfactuallevelofthemediatorundercontrol.Inourapproach,wetrytoavoidstrongassumptionsandproposeamethodtoinferdirectandindirecteffectsateachtime.ByusingaBayesiandynamicmodel,wecanupdatecausaleffectsacrosstimeperiods.FurtherdetailsaregiveninChapter3. 1.4LiteratureReviewforMultipleMediatorsForthemultiplemediatorscase,thereareevensmallernumberofliteraturethantheprevioustwocases.ThemethodbasedonBaronandKenny(1986)hasbeenmostcommonlyusedinthemultiplemediatorscasetoo.MacKinnon(2008)extendsBaronandKennymethodstraightforwardly.Foreachadditionalmediatorintroducedinthemodel,weneedtotacorrespondingregressionmodeloftheadditionalmediator.Forinstance,iftwomediatorsarenowtestedinourcausalmodel,weneedtworegressionmodelsofmediatorsM1=1+1Z+1M2=2+2Z+2Y=3+3Z+1M1+2M2+3insteadofoneregressionmodelasinthesinglemediatorcase 1 .Then,theindirecteffectsofM1andM2aredenedas11and22,respectively.Also,thedirecteffectis3andthetotaleffectisthesumofalleffects.However,asinthesinglemediatorcase,thisextendedmethodalsorequiressequentialignorabilitywhichcannotbejustiedinpractice. 19

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PreacherandHayes(2008)reviewedfewmethodsbasedontheBaronandKennymethod.TheypointedoutthatsincetheProduct-of-Coefcientapproachinthesettingofmultiplemediatorvariablesrequiresthemultivariatedeltamethodtoderivethestandarddeviationoftheindirecteffect,itcanbeonlyvalidundertheassumptionofmultivariatenormalityorthesituationofverylargesamples(Sobel,1986).Torelaxthisnormalityassumption,theyrecommendedtousebootstrapping.However,evenwithrelaxingthenormalityassumption,wehavethesameidenticationissuesaddressedbefore.Recently,ImaiandYamamoto(2013)developedamethodinthepresenceofcausallydependentmultiplemediatorsunlikethepreviousregressionapproacheswhichassumedcausalindependenceamongmultiplemediators.Intheirarticle,theyassumemediatorsaresequentiallyrelated.Thatis,onemediatoriscausallyaffectedbytheothersequentially.However,wearemoreinterestedinthesituationthatmultiplemediatorsaremeasuredatthesametime(notsequentially)withpossibleoverlappingeffectsamongmediators.DetailsabouttheframeworkwillbediscussedinChapter4. 20

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CHAPTER2BAYESIANINFERENCEFORTHECAUSALEFFECTOFMEDIATION 2.1IntroductionBehavioralscientistsandotherappliedresearchersareofteninterestedinboththecausaleffectofaninterventiondirectly,andonthecausaleffectoftheinterventionontheoutcomethroughitseffectonotherprocesses,calledmediators(Kraemeretal.2002).Forexample,interventionssuchascognitivebehavioraltherapy(CBT)typicallyinuenceoneormoreprocesses,suchasselfefcacyormotivation,whichinturnleadstoachangeinbehavior,suchasreducedconsumptionofalcoholorlossofweight.Thegraphbelowillustratesthebasicideainthesettingofasinglemediator,M:Z)166(!Y&%MInthisgraph,thedirecteffectofexposureZonoutcomeYisthehorizontalarrowatthetop.TheindirecteffectofZonYpassingthroughmediatorMiscapturedbythearrowsthatowfromZtoMtoY.Thestatisticalchallengeisquantifyingthedirectandindirecteffects.Thisissimilarinstructuretothesurrogateendpointsproblem(JoffeandGreene,2009;WolfsonandGilbert,2010;Lietal.,2011).Weformalizetheaboveasfollows.First,letZ2f0,1gdenoterandomizedintervention.Denethepair(M0,M1)asthepotentialvaluesofamediatorvariableunderinterventionz=0,1,withMobs=ZM1+(1)]TJ /F4 11.955 Tf 12.69 0 Td[(Z)M0observed.EachsubjectcouldbethoughtofashavingapotentialoutcomeYz,Mzforeverycombinationofzandm.TwowaystocharacterizetheeffectofZthatpassesaroundM(directeffect)havebeenproposed(RobinsandGreenland,1992;Pearl,2001).Ineachcase,comparisonsaremadebetweenpotentialoutcomeswithaconstantmediatorbutdifferenttreatments.ThenaturaldirecteffectisdenedbyNDE=E(Y1,M0)]TJ /F4 11.955 Tf 12.07 0 Td[(Y0,M0).ThisquantiestheeffectoftheinterventionZobtainedbysettingMtoits`natural'valueM0;i.e.,itsrealization 21

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intheabsenceoftheintervention.Notethatherethevalueofthemediatorwillnotbeconstantacrosssubjects,butrathersettoeachsubject'svalueofMintheabsenceoftreatment.Alternatively,onecandenethecontrolleddirecteffectoftreatmentbyE(Y1m)]TJ /F4 11.955 Tf 12.93 0 Td[(Y0m),forallm.Here,thedirecteffectoftreatmentinvolvessettingMtoaparticularvalueforthewholepopulationandvaryingthetreatment.Inmanytrialsofabehavioralintervention,thepotentialmediatorisabehavior,symptom,orperceptionofanindividual.Forexample,inatrialdesignedtoexaminetheeffectoftherapyfordepressiononsmokingcessationmighttrials,depressivesymptomscouldbeviewedasamediator.Manybehavioraltrialsalsoexaminemeasuresofmotivationorexpectationofsuccessfulbehaviorchangeaspotentialmediators.Becausethesevariablescannotbedirectlymanipulatedbytheexperimenter,theuseofcontrolledeffectscanbedifculttojustify.Theuseofnaturaldirectandindirecteffectsinbehavioralinterventiontrialsisconceptuallyeasiertojustify,particularlywhentheinterventionbeingadministeredhasmultiplecomponentsdesignedtoinuencespecicmediators(orpathstowardchangeinthetargetedbehavior).ThenaturalindirecteffectisdenedasNIE=E(Y1,M1)]TJ /F4 11.955 Tf 10.49 0 Td[(Y1,M0),ortheeffectofchangingfromM0toM1,hadeveryonereceivedtheintervention.WecanthendenethetotalcausaleffectofZonYasTE=NDE+NIE=E(Y1,M1))]TJ /F4 11.955 Tf 12 0 Td[(E(Y0,M0).Referringtothegureabove,thiscapturestheaggregateeffectofZthatpassesthroughandaroundM.Tointerpretthemeaningofnaturaldirectandindirecteffects,andparticularlytointerpretthemeaningofY1,M0,weusetheweightmanagementtrial(describedattheendofthissection)asanexample.Supposetheinterventionhasacomponentthatistargetedtohelppeopletrackfoodintake.Thenthedirecteffectistheeffectoftheinterventionifthecomponentoftreatmentthatisaffectingfoodintakemonitoringweresomehowtoberemoved.Thisimpliesthatthepathfromtheinterventiontofoodintakemonitoringwillbeblocked,butallothercomponentsofthetreatmentwillbe 22

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implementedandcanpotentiallyaffectweightlossthroughpathsthatdonotinvolvefoodintakemonitoring.Inpractice,mediationanalysisisoftenbasedonsolvinglinearsystemsofequations(MacKinnon,2008).Forexample,BaronandKenny(1986)usedthefollowingthreeregressionmodels:Y=1+1Z+"1M=2+2Z+"2Y=3+3Z+M+"3,although,giventhesecondtworegressions,therstisredundant(Imaietal.,2010).Here,theproposedTEis3+2,theNDEeffectis3andtheNIEis2.Thecontrolleddirecteffectoftreatmentisalso3.However,causalinterpretationsoftheseparametersdependonsequentialignorabilityandnointeractionassumptions(Imaietal.,2010);moredetailontheformercanbefoundinSection 2.2.3 .Thenointeractionassumptionisparticularlystrongforcontrolledeffects,asitrequiresthat,forexample,E(Y1m)]TJ /F4 11.955 Tf 11.98 0 Td[(Y0m)doesnotdependonm.Inadditiontotherandomizationandnointeractionassumptions,themodelalsorequirescorrectspecicationofthelinearsystem.ABayesianversionoftheregressionapproachcanbefoundinYuanandMacKinnon(2009).Newsemiparametricmethodshaverecentlybeenproposedforestimatingmediationeffects.TenHaveetal.(2007)proposedestimatingmediationeffectsusingmodelsthatmakeassumptionsaboutstructuralinteractions,ratherthansequentialignorability.VanderWeele(2009)proposedusingtwomarginalstructuralmodels(Robins,1999)toestimatenaturaldirectandindirecteffects.However,thesemethodscanbeproblematicforcontinuousmediatorsduetounstableweights(Vansteelandt,2009).Parametriclikelihood-basedorBayesianmethodsformediationhaveprimarily 23

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beenproposedinaprincipalstratication(PS)framework(FrangakisandRubin,2002),inwhichcausaleffectsaredenedwithinstratadeterminedbypost-randomizationoutcomes.SeeGallopetal.(2009)andElliott,Raghunathan,andLi(2010)forexamples.Inthemediationcontext,thePSapproachhasbeenusedtodenetreatmenteffectsconditionalonM0andM1,andhencefocusesonlatentsubpopulationsdenedbypairsfM0,M1g.Forabinarymediator,thedirecteffectofZisdenedasE(Y1)]TJ /F4 11.955 Tf 12.29 0 Td[(Y0jM1=M0),orthecausaleffectofZamongpeoplewhosevalueofMwouldnotbeaffectedbyZ.WhenMiscontinuousratherthanbinary,thePSapproachwillgenerallyrequireadditional,untestablemodelingassumptionsbecausestratadenedbyM0=M1willbesparseorevenemptyinnitesamples.BecausePS-basedinferencesapplytolatentsubpopulations,directcomparisonsbetweenPSandothermethodsisnotstraightforward;however,VanderWeele(2008)andJoffeandGreene(2009)providedetaileddiscussionanddescribelinkagesbetweenPS-basedinferencesandbothcontrolledandnaturaldirectandindirecteffects.Ourapproachisdistinctfromothermediationapproachesintheliteratureinseveralways.WetakeafullyBayesianapproachtoinferringnaturaldirectandindirecteffects.Becausewewillfocusonnaturaleffects,wecanfocusonasubsetofthepotentialoutcomesYz,Mz:fY1,M1,Y1,M0,Y0,M0g,withYobs=ZY1,M1+(1)]TJ /F4 11.955 Tf 12.25 0 Td[(Z)Y0,M0observed.Forexample,Y1,M1istheoutcomethatwouldbeobservedifwesetZ=1andM=M1.Inthisframework,wedonotrequirethatYzmbedenedforallvaluesofm;itisonlynecessarytodeneYzmfortherealizationsofM0andM1.WemodelthemarginaldistributionsofM0andM1non-parametrically,andthenspecifyacopulamodeltoobtaintheirjointdistribution.Inthisway,wecanallowexibilityofthemarginaldistributionsofthemediators.Weavoidmakingsomeofthestrongassumptionsthatarerequiredforsomeofthealternativemethodsdescribedabove.Instead,ourmodelisidentiedifthreesensitivityparametersarespecied.Althoughourapplicationhasabinaryoutcomeand 24

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continuousmediator,ourgeneralapproachcouldbeusedforothertypesofoutcomesandmediators.Weillustratethemethodologyusingdatafromaweightmanagementtrial,TOURS(Perrietal.,2008).Subjectscompletedastandardsixmonthlifestylemodicationprogramandthenwererandomizedtoeitherextendedcareortoaneducationcontrolgroup.Adherencetobehavioralweight-managementstrategies,asmeasuredbythenumberofdayswithself-monitoringrecordsforfoodintake,istheproposedmediatorofweightchange.Theoutcomewasa(binary)measureofweightchange(describedinSection 2.6 ).Weestimateboththedirecteffectoftheweightmanagementprogramsontheweightchangeoutcome,aswellastheindirecteffectoftheprogramsontheoutcomethroughtheeffectonadherencetofoodintakeself-monitoring.InSection 2.2 ,wediscussinferenceonthecausaleffectofmediationbyrstintroducingsomenotation,thenstatingourassumptions,andnallyshowingthatourassumptionsaresufcienttoidentifythenaturaldirectandindirecteffects.WeprovidedetailsonposteriorcomputationsinSection 2.3 .Section 2.4 outlinesourapproachtoelicitationforthesensitivityparametersandsubsequentsensitivityanalysis.SimulationstoassesssensitivitytoviolationsofassumptionscanbefoundinSection 2.5 .Section 2.6 containsouranalysisoftheTOURStrial.Finally,wewrapupanddiscussextensionsinSection 2.7 2.2InferenceonCausalEffects 2.2.1NotationLetfz,Mz0(y)denotethedistributionofYz,Mz0,for(z,z0)2f0,1g2.Similarly,wedenotetheconditionaldistribution[Yz,Mz0jMz0=mz0]byfz,Mz0(yjmz0).LetD=(M1)]TJ /F4 11.955 Tf 11.66 0 Td[(M0).Theconditionaldistribution[Yz,Mz0jMz=mz,M0z=mz0,D=d]isdenotedbyfz,Mz0(yjmz,mz0,d).Thejointconditionaldistribution[Y1,M1,Y1,M0,Y0,M0jM0=m0,M1=m1]isdenotedbyf(1,M1),(1,M0),(0,M0)(y11,y10,y00jm0,m1).Othermultivariatedistributionsaredenedusingsimilarnotationbelow. 25

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2.2.2AssumptionsRecalltheobserveddataisMobs=ZM1+(1)]TJ /F4 11.955 Tf 9.62 0 Td[(Z)M0andYobs=ZY1,M1+(1)]TJ /F4 11.955 Tf 9.62 0 Td[(Z)Y0,M0.Theobserveddataarenotsufcienttoidentifytheconditionaldistributionf(1,M1),(1,M0),(0,M0)(y11,y10,y00jm1,m0)andthejointdistribution,fM0,M1(m0,m1)whicharenecessarytoidentifythejointposteriordistributionofNIEandNDEwithoutassumptions.Thus,wemakethefollowingassumptions.Assumption1.(Randomizationassumption) f(z0,M),Mz(yz0,m,mzjz)=f(z0,M),Mz(yz0,m,mz).(2)Thisassumptionwillholdinourapplicationsincethetreatmentwasrandomized.Assumption2stratiesthepopulationintothoseforwhomthetreatmenthasalargeandsmalleffectonthemediator.Assumption2a.Foraxedzandforsome,fz,Mz0(yz,Mz0=1jMz0=m,jdj<)=fz,Mz(yz,Mz=1jMz=m,jdj<).Notethatweassumebinaryresponsesthroughtout.TherandomvariableDquantiesthetreatmenteffectonthemediator.Aconsequenceoftheassumptionisthat,forexample,f1,M1(y1,M1=1jM0=m0,M1=m1,jdj<)=f1,M0(y1,M0=1jM0=m1,M1=m0,jdj<).Itmeansthat,amongpeopleforwhomthetreatmenteffectonthemediatorissmall(asquantiedby),thedistributionoftheoutcomeissamewhetherthatmediatorvaluewasinducedbyZ=1orZ=0.Itdoesnotimplyanexclusionrestriction.Thatis,wearenotassuming[Y1,M0jM0=m]=[Y0,M0jM0=m]. 26

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Assumption2b.ThenextassumptionisforthesubgroupofsubjectsforwhomZhasagreaterthaneffectonM.Forthisgroup,foraxedz,,and,weassumefz,Mz0(yz,Mz0=1jMz0=m,jdj)=sgn(d)fz,Mz(yz,Mz=1jMz=m,jdj).wherethesensitivityparameterisarelativeriskwiththefollowingrestriction:2(0,1=fz,Mz(yz,Mz=1jMz=m,jdj))forsgn(d)+or2(fz,Mz(yz,Mz=1jMz=m,jdj),1)forsgn(d))]TJ /F5 11.955 Tf 21.91 0 Td[(.Notewedifferentiatem1>m0+fromm0>m1+throughthesgn(d)intheaboveexpression.WediscusselicitationofandinSection 2.4 .NotethatwithAssumption2,weimplicitlyassumeadiscontinousrelationship(astepfunctionat)betweentheconditionalprobabilitiesandthetreatmenteffectonthemediator,D.Therearenotgoodalternativestothis,e.g.,asmoothfunctionofD,sincethisisnotidentiablefromthedata(andwouldinvolveadditionalsensitivityparameters).Weviewthestepfunctionassumptionasareasonablealternative.Byconsideringaseveralcombinationsofand,weshouldbeabletocapturemanyplausiblescenarios.Thekeyisdifferentiatingthepopulationintothosewheretheinterventionhasalargeversussmalleffectonthemediator.Assumption3.Thefollowingconditionalindependenceholds:fM0z(m0zjmz,yz,mz)=fM0z(m0zjmz).Thisassumptionssaysthatthepotentialvalueofthemediatorundertreatmentz0isindependentofthepotentialoutcomeundertreatmentzconditionalonthepotentialvalueofthemediatorundertreatmentz;forexample,M1?Y0,M0jM0.Thisassumptionalsoimpliesfz,Mz(yz,Mzjmz,mz0)=fz,Mz(yz,Mzjmz). 27

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Thatis,thepotentialoutcomesYz,Mzareindependentofthemediatorundertheothertreatment,mz0conditionalonthemediatorassociatedwiththepotentialoutcome,mz;forexample,Y1,M1?M0jM1.Thusthisassumptionsaysthatnoadditionalinformationisprovidedaboutthepotentialoutcomes,Yz,Mzfromthemediatorundertheothertreatment,Mz0afterweconditiononthemediatorundertreatmentz.NoteitclearlydoesnotimplyY1,M1?M1jM0.Thisassumptionisnotrequired,butconsiderablysimpliescomputations.Byusingthisassumption,weonlyneedtoestimatetheconditionaldistributionoftheobservableoutcomes.WeexaminesensitivitytothisassumptionviasimulationsinSection 2.5 .Assumption4.WeassumethejointdistributionofthemediatorfollowsaGaussiancopulamodel(Nelsen,1999),FM0,M1(m0,m1)=2)]TJ /F6 7.97 Tf 6.59 0 Td[(11fFM0(m0)g,)]TJ /F6 7.97 Tf 6.58 0 Td[(11fFM1(m1)g,where1istheunivariatestandardnormalCDFand2isthebivariatenormalCDFwithmean(0,0)T,variance(1,1)Tandcorrelation2()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1).Thejointdistributionofthecontinuousmediatorscanbeidentieduptoasensitivityparameterbyrstspecifyingthetwomarginaldistributions.Thereisnoinformationinthedataaboutbecauseitrepresentstheassociationbetweentwovariablesthatareneverobservedsimultaneously.Wewillthereforetreatasknownandvaryitaspartofasensitivityanalysis.Thespecialcase=1impliesequipercentileequatingofthemediators(i.e.,theranksofM0andM1arethesame).InSection 2.2.5 ,wediscussBayesiannonparametricestimationofthemarginaldistributionswhichareidentiedfromMobsasoutlinedinSection 2.2.4 .ThechoiceoftheGaussiancopulahereisforseveralreasons:1)itallowscompleteexibilityinthemarginals(whichwemodelinSection 2.2.5 usinganonparametricBayesianapproach)and2)itisparsimoniousintermsofsensitivityparameters(hereonlyonesensitivityparameter,). 28

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Assumption5.(Conditionalindependencebetweenpotentialoutcomes)f(1,M1),(1,M0),(0,M0)(y11,y10,y00jm0,m1)=f1,M1(y11jm0,m1)f1,M0(y10jm0,m1)f0,M0(y00jm0,m1).NotethatAssumption5isnotnecessarytoestimateE[NIEjdata]andE[NDEjdata];forthese,wejustneedthemarginalposteriordistributionsforthepotentialoutcomes.However,itisnecessarytoestimateotherfeaturesoftheposteriordistributionofNIEandNDE.Inparticular,theposteriormeanoftheNIEandNDEisnotaffectedbythisassumption;howevertheposteriorvarianceis.Infact,thisassumptionprovidesanupperboundonthevarianceoftheNIEassumingdeviationsonlyinvolvingpositivedependencebetweenthepotentialoutcomes.Inparticular,thedifference(whichwedenoteasA)betweenthevarianceoftheNIEunderAssumption5andunderthecasethatAssumption5doesnothold(withthestrongestpossibleconditionaldependencebetweentheoutcomes)isA2Zs.d.(Y10)s.d.(Y11)f(m0,m1)dm0dm1,where=p expfsgn(d)logI(jdj)gP)]TJ /F1 11.955 Tf 11.96 0 Td[(expfsgn(d)logI(jdj)gP s.d.(Y10)s.d.(Y11)andP=f1,M1(y11=1jm1)f1,M1(y11=1jm0).Forfurtherdetailsonthisandtheentirederivation,seeAppendix A .Thisassumptionstatesthatthecorrelationbetweenthepotentialoutcomesiscompletelyexplainedbythetwovaluesforthepotentialmediator;implicitly,itisassumingtherearenoothermediators.Wecanweakenthisassumption,butnotwithoutaddingadditionalsensitivityparameters.Inthedataexample,weprovideinformationonthechangestotheposteriorvarianceunderviolationsofAssumption5.Another 29

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optiontoweakenthisassumptionwouldbetohaveitholdonlyconditionalonbaselinecovariates;wediscussthisextensioninSection 2.7 .Weemphasizethatnoneoftheseassumptionsare`checkable'fromtheobserveddata. 2.2.3AlternativeAssumptionsRequiredforNon-ParametricIdenticationTheaverageNIEandNDEcanbeidentiednon-parametricallywithanalternativesetofassumptions(Imaietal.,2010;Robins,1999).Inparticular,Imaietal.(2010)showedthatnon-parametricidenticationrequiredthetreatmentassignmentignorability( 2 )andignorabilityofthemediator(i.e.,sequentialingnorability),fz0,M(yz0,mjm,z)=fz0,M(yz0,mjz)forz,z0=0,1.Inaddition,apositivityassumptionisrequiredfortreatmentandthemediator:P(Z=z)>0andfMz(mjZ=z)>0forallm,z.Theaboveassumptionsaretypicallymadeconditionalonpre-treatmentcovariates.Asensitivityanalysiscanbeusedtoquantifyeffectsofunmeasuredconfounding(Imaietal.,2010a;Imaietal.,2010b;VanderWeele,2010).Wedonotmakethesequentialignorabilityassumption.Asstatedearlier,thisistypicallynotareasonableassumptionformediatorsinbehavioraltrials.Forexample,ourAssumption2ballowsforadependencebetweenM0,M1andthepotentialoutcomesthatisnotassumedtovanishafterconditioningonZ(unlikewithsequentialignorability).However,werequireadditionalassumptionsaboutthejointdistributionof(M0,M1)becauseweneedtoidentifytheposteriordistributionsofNDEandNIE,notjustthemeans. 2.2.4IdenticationofJointDistributionsforComputationofDirectandIndirectEffectsInthefollowing,wewilldemonstratethatAssumptions1-5aresufcienttoidentifythejointdistributionofNIEandNDE.Westatethisformallyinthefollowingtheorem. 30

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Wealsonotethatbyrandomizationofthetreatment,( 2 ),thedistributionsfMz(mz),fMzjYz(mzjyz,Mz)andfz,Mz(yz,Mz)areestimablefrom(Yobs,Mobs).Theorem.ThejointposteriordistributionofNIEandNDEisidentiedunderAssumptions1-5.Proof:Considerthefollowingfactorizationofthejointdistributionofthetwopotentialoutcomes(oneofwhichisobserved),whichwewilldenoteasB, f(0,M0),(1,M1),M0,M1(y00,y11,m0,m1)=f(M0,Y0),(M1,Y1)(m0,m1jy00,y11)f(0,M0),(1,M1)(y00,y11). (2) WecanfurtherfactorBas B=f(0,M0),(1,M1)(y00,y11jm0,m1)fM0,M1(m0,m1)=f0,M0(y00jm0,m1)f1,M1(y11jm0,m1)fM0,M1(m0,m1)(A5)=fM0,M1(m0,m1jy00)f0,M0(y00) fM0,M1(m0,m1)fM0,M1(m0,m1jy11)f1,M1(y11) fM0,M1(m0,m1)fM0,M1(m0,m1)=fM1(m1jm0)fM0jY0(m0jy00)f0,M0(y00) fM0,M1(m0,m1)fM0(m0jm1)fM1jY1(m1jy11)f1,M1(y11)(A2)where`A'correspondsto`Assumption'intheabove.Eachcomponentinthelastequalityisidentiedbyrandomization(Assumption1)and/orAssumption4.Toobtaintheposteriordistributionofindirecteffects,weneedf(1,M1),(1,M0)(y11,y10)=Zf(1,M1),(1,M0)(y11,y10jm0,m1)fM0,M1(m0,m1)dm0dm1.ThesecondtermintheintegrandisafunctionoftheestimablequantitiesinthelastequalityofB.UsingAssumption5,thersttermintheintegrandcanbefactoredasf(1,M1),(1,M0)(y11,y10jm0,m1)=f1,M1(y11jm0,m1)f1,M0(y10jm0,m1).ByAssumption3,thersttermisequaltof1,M1(y11jm0,m1)=f1,M1(y11jm1)whichcanbeestimatedusingtheobserveddataviarandomization(andafunctionofcomponents 31

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inthelastequalityofB).Also,weobservethepairs(Y1,M1,M1).Thesecondterm,f1,M0(y10jm0,m1)isidentiedbyAssumptions2and3.FromAssumption2,weidentifyf1,M0(y10jm0,m1)usingf1,M1(y11jm0,m1)andthesensitivityparameters,(,),whichisnowidentiedbyf1,M1(y11jm0)viaAssumption3.Similarly,toobtaintheposteriordistributionofdirecteffects,weneedf(1,M0),(0,M0)(y10,y00)=Zf(1,M0),(0,M0)(y10,y00jm0,m1)fM0,M1(m0,m1)dm0dm1.Therstterm,f(1,M0),(0,M0)(y10,y00jm0,m1)canbefactoredviaAssumption5f(1,M0),(0,M0)(y10,y00jm0,m1)=f1,M0(y10jm0,m1)f0,M0(y00jm0,m1).TheidenticationofthersttermwasoutlinedintheidenticationoftheNIE.Forthesecondterm,f0,M0(y00jm0,m1)=f0,M0(y00jm0)byAssumption3,whichisestimablefromtheobserveddataandrandomization(sincefunctionofquantitiesinthelastequalityofB). 2.2.5ModelsandEstimationThemodelsrequiredforinferenceintheprevioussectioncanbespeciednonparametricallyandestimatedusingtheobserveddata.Inparticular,weneedthefollowingnonparametricmodels:Yz,MzBer(z,Mz):z=0,1.WespecifyDirichletprocesspriorsforthedistributionsFMz,y(mzjYz,Mz=y)fory=0,1;z=0,1.WealsoplaceindependentUnif(0,1)priorsonz,Mz.TherelevantposteriorcanbesampledinWinBUGS(seeAppendix B ).Notethattheidentiedquantitiesintheprevioussubsection,fz,Mz(yjm)canbeestimatedquiteeasilyusingthemodels;thisisclearifwerewritefz,Mz(yjm)asfMz,y(mjy)fYz,Mz(y)=fMz(m). 32

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2.3PosteriorComputationsWeconstructanalgorithmtosamplefromtheposteriordistributionofthedirectandindirecteffects.Weproceedusingthefollowingsteps. 1. Fixthesensitivityparameters,(,,). 2. Sample[FM1,1,FM1,0,FM0,1,FM0,0,1,M1,0,M0]fromtheposteriordistributionp(FM1,1,FM1,0,FM0,1,FM0,0,1,M1,0,M0jmobs,yobs)wheremobs=fMzi,i=1,...,ngandyobs=fYzi,Mzi,i=1,...,ngusingWinBUGS. 3. Foreachsample(FM1,1,FM1,0,FM0,1,FM0,0,1,M1,0,M0),computeNDEandNIE. 4. RepeatSteps2-3Qtimes.Ifweplaceaprioronthesensitivityparameters,Step1isreplacedbysamplingthepriorandStep4becomesrepeatSteps1-3Qtimes.DetailscanbefoundinAppendix C 2.4SensitivityAnalysisandElicitationAssumptions2and4containthreesensitivityparameters,(,,).Wediscussageneralstrategytoelicitarangeforeachsensitivityparameter.Assumption2.Tohelpunderstandthersttwosensitivityparameters,weassume,wlog,thatthetreatmenthasanon-negative(non-decreasing)effectonthemediatorandusingAssumption2,wehavethefollowingexpression fz,Mz0(yz,Mz0=1jmz,mz0,d) fz,Mz(yz,Mz=1jmz,mz0,d)=exp(log)=.(2)Inthefollowing,wechooseZ=1(wlog)andassume(m1)]TJ /F4 11.955 Tf 11.99 0 Td[(m0)>.Inaddition,wecansimplifytheexpressionin( 2 ),whichwillfacilitateelicitation,asfollows,f1,M1(y1,M1=1jM1=m1,M0=m0,d)=f1,M1(y1,M1=1jM1=m1,M0=m0,d<)=f1,M0(y1,M0=1jM1=m0,M0=m1,d<).TherstequalitycomesfromAssumption3;thesecondfromAssumption2a.Sowecanrewrite( 2 )as f1,M0(y1,M0=1jm,d) f1,M0(y1,M0=1jm,d<)=.(2) 33

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wheremisthevalueofthemediatorunderthecontrolarm.Thenumeratorcorrespondstom1>(m0+)(assumingalargervalueforthemediatorisbetter).Ifweassumethetreatmenthasalargereffectonothermediators(notmeasured)orotherrelevantmechanisms,thenwemightexpecttheprobabilityinthenumeratortobelargerthanthedenominatorcorrespondingtoalargerdirecteffect.Weuseexpression( 2 )foreliciting.Toelicitlikelyvaluesfor,weconsiderhowbigdshouldbeforthefollowingratiotobenotequaltoone,f1,M0(y1,M0=1jm0,d=0[m1=m0]) f1,M0(y1,M0=1jm0,d=[m1=m0+]).Assumption4.TheparameterinAssumption4correspondstotherankcorrelationbetweenthemediatorvaluesunderthetreatmentandcontrolarms,with=1correspondstoaperfectcorrelationand=0correspondingtoindependence.Weusethesetwobenchmarkstoelicitavalue.Aconservativeapproachwouldbejusttoconsideranyvaluein[0,1)(assumingtherelationshipwaspositive).Weelicitarangeofvaluesforeachsensitivityparameter. 2.5SimulationStudytoAssessSensitivitytoViolationsofAssumption3WeexplicitlysuggestapproachesforsensitivityanalysiswithsensitivityparametersforAssumptions2and4.ForAssumption5,wederivedanalyticresultsthatdemonstrateitsimpact(onlyontheposteriorvariance).Inthebelow,weassess,viasimulations,sensitivitytoviolationsofAssumption3.Forthesimulation,similartothedataexample,weassumeY1,M1Ber(0.71).Weconsiderthefollowing(simple)violationsofassumption3.Firstofall,weassumelogit(M0)jlogit(M1),Y1,M1N(,2)where=0+1logit(M1)+2Y1,M1andthelogittransformationisontheinterval[0,350].Basedonthedata(andsetting=.3inAssumption4),weobtain0=)]TJ /F5 11.955 Tf 9.3 0 Td[(1.5241and1=0.1842.WeconsiderdeviationsfromAssumption3(2=0)intermsofthefollowingvaluesfor2,f1.28,2.56,5.12gwhicharehalf,fullandtwiceofs.d.ofm0afterthelogittransformation.Forthesimulation, 34

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Table2-1. SimulationstoassesssensitivityofestimateofNIE(standarddeviationsinparentheses)toviolationsinAssumption3:n=60 Case=50=75=100 10Assumption30.0244(0.058)0.0304(0.055)0.0295(0.059)Truth0.0244(0.058)0.0304(0.055)0.0295(0.059)11.28Assumption30.0052(0.059)0.0066(0.056)0.0079(0.061)Truth0.0021(0.058)0.0036(0.055)0.0041(0.058)12.56Assumption3-0.0251(0.051)-0.0174(0.056)-0.0165(0.048)Truth-0.0034(0.049)0.0073(0.052)0.0068(0.046)15.12Assumption3-0.0587(0.044)-0.0720(0.049)-0.0625(0.049)Truth0.0045(0.038)0.0012(0.038)0.0007(0.039)1.150Assumption30.0214(0.053)0.0296(0.053)0.0286(0.062)Truth0.0214(0.053)0.0296(0.053)0.0286(0.062)1.151.28Assumption3-0.0046(0.057)0.0026(0.058)0.0057(0.049)Truth-0.0075(0.057)0.0000(0.056)0.0032(0.047)1.152.56Assumption3-0.0150(0.062)-0.0152(0.055)-0.0164(0.054)Truth0.0043(0.058)0.0039(0.055)0.0034(0.050)1.155.12Assumption3-0.0602(0.048)-0.0482(0.052)-0.0758(0.060)Truth-0.0007(0.032)0.0073(0.039)-0.0109(0.045)1.130Assumption30.0200(0.056)0.0303(0.061)0.0172(0.055)Truth0.0200(0.056)0.0303(0.061)0.0172(0.055)1.131.28Assumption30.0005(0.056)-0.0015(0.053)-0.0055(0.058)Truth-0.0014(0.056)-0.0049(0.053)-0.0076(0.058)1.132.56Assumption3-0.0127(0.054)-0.0116(0.055)-0.0152(0.063)Truth0.0045(0.052)0.0056(0.051)0.0022(0.058)1.135.12Assumption3-0.0408(0.053)-0.0545(0.052)-0.0515(0.052)Truth0.0147(0.037)0.0021(0.041)0.0066(0.043)20Assumption30.0151(0.053)0.0108(0.063)0.0213(0.055)Truth0.0151(0.053)0.0108(0.063)0.0213(0.055)21.28Assumption3-0.0053(0.052)0.0109(0.055)0.0127(0.059)Truth-0.0068(0.054)0.0107(0.053)0.0105(0.057)22.56Assumption30.0167(0.058)0.0004(0.058)0.0006(0.061)Truth0.0232(0.052)0.0099(0.053)0.0126(0.057)25.12Assumption3-0.0049(0.064)-0.0181(0.056)-0.0201(0.056)Truth0.0309(0.049)0.0229(0.040)0.0208(0.044) wealsoconsidervaryingthesensitivityparametersfromAssumption2asfollows:2f1,1.15,1.3,2gand2f50,75,100g.Foreachscenario,wecomputetheNIEassumingAssumption3holdsandcompareittothetrueNIEwhenAssumption3doesnothold.TheresultsareinTable 2-1 35

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TheposteriormeanandstandarddeviationoftheNIEarenotverysensitivetosmalltomediumsizeviolationsofAssumption3withtheestimatesnotdifferingbymuchmorethan.02.However,forthelargeviolation(2standarddeviationchange),theestimatescandifferbyasmuchas.05to.08.Therearenoconsistentpatternsofbias,includingbiastowardthenull. 2.6TOURS:WeightManagementTrial 2.6.1DescriptionofTOURSThiswasarandomizedtrialtocomparetheeffectivenessofextendedcareprogramsdesignedtopromotesuccessfullongtermweightmanagement.Participantscompletedastandardsixmonthlifestylemodicationprogramandthenwererandomlyassignedtotelephonecounseling,face-to-facecounselingoraneducationcontrolgroup(Perrietal.,2008).ThiscompletedtrialisreferredtoasTOURS.Averyimportantquestioninthistrial,andobesityresearchingeneralareidentifyingmediatorsofweightchange.Inthistrial,differentmeasuresofadherencetobehavioralweight-managementstrategieswererecorded.Here,wefocusonthe(continuous)mediator,thenumberofdayswithself-monitoringrecordsforfoodintake(whichtakesvalues0to350)duringtheweightmanagementphaseofthetrial,6to18months.Amongthosethatlostatleast5%oftheirweightby6months,wedenethe(binary)outcomeofinteresttobewhetherornottheymaintainedthelossofatleast5%from6to18months.Intheanalysisoftheoriginaltrial,thetelephoneandface-to-facetreatmentarmsresultedinsimilarweightmaintenancethatwasconsiderablylargerthantheeducationcontrolarm.Here,weassesstheNIEandNDEofthemediatorfortheface-to-face(FTF)vs.educationcontrol(EC)arms.Thesamplesizesforthetwotreatmentarmswere63and62,respectively. 2.6.2ModelsWeassumethefollowingpriorfortheconditionaldistributionofthemediatorsgiventhebinaryresponse(y2f0,1g),FMz,y(mzjYz,Mz=y)DP(Kz,Wz 36

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Beta[0,350](1z,1z)+(1)]TJ /F4 11.955 Tf 12.66 0 Td[(Wz)Beta[0,350](2z,2z)),wherethebasemeasureisamixtureofBetadistributionsontheinterval[0,350]andKZistheprecisionparameter.Weplacethefollowingpriorsonthehyperparameters,KzDiscUnif[1,20]andizUnif(0,70)andizUnif(0,70)fori=1,2andWzUnif(0,1):z=1,2. 2.6.3ElicitationofSensitivityParametersThecombinedexpertiseoftheauthorsinweightmanagementtrialsandcausalinferencewereutilizedthedeterminereasonablevaluesforthesensitivityparameters.Assumption2.Regardingthesensitivityparameter,itwasthoughtthatadifferenceofatleastonedayperweekinllingoutthefoodintakerecordscouldbeinterpretedasclinicallyimportantandsignicant;wediscussthisissuefurtherinthediscussionsection.Asaresult,weconsidervaluesof2(50,100);roughlycorrespondingtoadifferenceof1to2daysperweek.Inaddition,intermsoftheratioin( 2 ),theimpactofthetreatmentonthemediatorbeingmorethan50dayscouldreectapositiveimpactonotherfactorsinnatetotheindividualuptoarelativeriskofabout1.3.Thus,weconsideredvalues2(1.0,1.3).Assumption4.Forassumption4,thecorrelationbetweenm0andm1wasthoughttobepositive.So,wefollowedtheconservativeapproachfromSection2.4andconsider2[0,1).Fortheanalysis,wealsoconsiderindependentuniformpriorsovertheseranges. 2.6.4ResultsForsamplingfromtheposteriordistributionofthemodelsfortheobserveddatainSection2.6.2,weran10000iterationsanddiscardedtherst5000asburn-in.Weranmultiplechainsandtraceplotsindicatedconvergence.Thetotaleffectofface-to-face(FTF)versusmail(EC)correspondedtoamarginallysignicantriskdifferenceof.085()]TJ /F5 11.955 Tf 9.3 0 Td[(.070,.25)suggestingtheefcacyoftheFTFtreatment(Table 2-3 ).Forallcombinationsofthesensitivityparametersconsidered,theconclusionswerequiterobustcorrespondingtoalargeNDErangingfromabout 37

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.077to.089,withcredibleintervalsthatcoveredzero(seeTable 2-3 ).TheNIEwasalwaysmuchsmallerinmagnitude,lessthan.01inabsolutevaluewithcredibleintervalscenteredclosetozero.Theresultswereleastsensitivetothecorrelationbetweenmediators(seeAssumption4)andtheNDEdecreased(slightly)asepsilonincreasedbutincreasedastheRR,increased.Whenweassumedindependentuniformpriorsonthesensitivityparameters(basedontheirrangeselicitedinSection 2.6.3 ),wedrewsimilarconclusions(Table 2-2 ). Table2-2. Posteriormeansand95%C.I.'sforpriorsUnif[0,1],Unif[50,100],Unif[1,1.3]. NDENIETE 0.081(-0.073,0.25)0.003(-0.086,0.12)0.085(-0.070,0.25) Thus,basedonouranalysis,therewassomeevidencefortheefcacyoftheFTFtreatment,butminimalevidencethattheeffectoftheFTFtreatmentwasmediatedbythenumberofself-monitoringrecordscompletedoverthe12monthmanagementportionofthetrial.Themaximalinuence(ontheposteriorvariance)foraviolationofAssumption5isA.39. 2.6.5ComparisonwithBaronandKennyTypeEstimatorsForcomparison,wealsoestimatedthedirectandindirecteffectsusingtheBaronandKennyapproachundertheassumptionsofsequentialignorabilityandnointeraction.WeusetheRfunctionmediate(Imaietal.,2010)andlinearmodelsasoutlinedintheBaronandKennyapproachinSection 2.1 .Thenaturaldirecteffectwasestimatedtobe.031()]TJ /F5 11.955 Tf 9.29 0 Td[(.12,18)ofsimilarmagnitudetothenaturalindirecteffect.054()]TJ /F5 11.955 Tf 9.3 0 Td[(.000,.12),aquitedifferentconclusionfromtheanalysisabove.However,theassumptionsunderlyingtheBaronandKennyapproachareunlikelytobereasonableinour(behavioralscience)applicationandthus,weprefertheanalysis(inSection 2.6.4 )undertheassumptionsproposedinSection 2.2 .Notethatthesequentialignorabilityassumptionisoften 38

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weakenedbyincludingbaselinecovariatesandconductingsensitivityanalysis(Imaietal.,2010;vanderWeele,2010),whichwedidnotdohere. 2.7DiscussionWehaveproposedaBayesianapproachtothecausaleffectofmediationthatinvolvesthreesensitivityparametersandnoparametricmodelsfortheobserveddata.Strategiestoelicitthesensitivityparameterswereprovided.SimulationstudiessuggestedthatestimationoftheNIEisnotverysensitivetosmalltomediumsizeviolationsofAssumptions3andAssumption5providesanupperboundontheposteriorvarianceoftheNIE.FortheTOURStrials,theeffectoftheface-to-facecounselingtreatmentvs.theeducationcontrolwasmarginallysignicant.However,basedonouranalysis,thepotentialmediator,thenumberofself-monitoringfoodrecordscompletedwasnotamediatorofthisrelationship.Weproposethisasageneralapproachtoassessmediationthatallowseasytointerpretsensitivityparametersandrealisticassumptionsforbehavioraltrials.Thereareseveralextensionstothecurrentmodelingapproach.First,wemightincorporatebaselinecovariatestoweakensomeofourassumptionsandpotentiallygainefciencyinestimationofthenaturalindirecteffects;wearecurrentlyworkingonthisextension.Second,wecoulddevelopamoredetailedframeworkforelicitingaprior(notjusttherange)forthesensitivityparameters.Third,extendingthecurrentframework(bothdeningcausaleffectsandmodels)tothesettingofmultiplemediatorsisanopenquestion.Fourth,wemightconsideralternativestoAssumption2;inaddition,wecangeneralizeAssumption2byreplacingtherelativeriskformulationwithanoddsratio(exponentialtilt)formulationthatwouldbeappropriateforbothabinaryandacontinuousresponse.TherearealsonumerousinterestingextensionsbasedontheTOURSdata.Twelvesubjects(7.4%)droppedoutbefore18months.Wehavenotincludedthemintheanalysis.Futureanalyseswillincludethesesubjectsunderspecicassumptionsabout 39

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thedropout.Inaddition,wehavedenedthemediatorhereasthetotalnumberofdayswithself-monitoringrecordsoffoodintakeoverthe12monthperiod.However,thismaybetoocoarseasummary.Futureworkwillexaminetherecordcompletionprocess,basicallya350-dimensionalvectorof0and1's(thatsumuptoourmediator)astheremaybea(clinical)distinctionbetweenllingoutnorecordsperweekversusoneperweekasopposedtotwoperweekvsthreeperweek(thatbothcorrespondtoadifferenceof50daysofrecords).WeareworkingonmakingthemethodsavailableasanRpackage. Table2-3. Posteriormeansand95%C.I.'sfor2f50,75,100g,2f1.0,1.15,1.3g. NDENIETE 05010.077(-0.078,0.25)0.007(-0.088,0.12)0.085(-0.070,0.25)1.150.083(-0.073,0.26)0.001(-0.100,0.11)0.085(-0.070,0.25)1.300.089(-0.085,0.26)-0.003(-0.100,0.10)0.085(-0.070,0.25)07510.078(-0.070,0.25)0.006(-0.086,0.11)0.085(-0.070,0.25)1.150.082(-0.083,0.25)0.002(-0.095,0.11)0.085(-0.070,0.25)1.300.086(-0.073,0.26)-0.001(-0.100,0.10)0.085(-0.070,0.25)010010.078(-0.075,0.25)0.007(-0.090,0.11)0.085(-0.070,0.25)1.150.081(-0.077,0.25)0.004(-0.091,0.11)0.085(-0.070,0.25)1.300.086(-0.072,0.26)-0.001(-0.100,0.10)0.085(-0.070,0.25)0.35010.078(-0.073,0.25)0.007(-0.086,0.12)0.085(-0.070,0.25)1.150.082(-0.074,0.25)0.003(-0.100,0.11)0.085(-0.070,0.25)1.300.087(-0.078,0.26)-0.002(-0.100,0.10)0.085(-0.070,0.25)0.37510.077(-0.076,0.25)0.007(-0.095,0.12)0.085(-0.070,0.25)1.150.081(-0.076,0.25)0.003(-0.091,0.11)0.085(-0.070,0.25)1.300.086(-0.079,0.26)-0.001(-0.100,0.10)0.085(-0.070,0.25)0.310010.078(-0.076,0.26)0.006(-0.092,0.11)0.085(-0.070,0.25)1.150.080(-0.076,0.26)0.004(-0.092,0.11)0.085(-0.070,0.25)1.300.085(-0.079,0.26)0.001(-0.100,0.10)0.085(-0.070,0.25)0.75010.077(-0.073,0.25)0.007(-0.092,0.13)0.085(-0.070,0.25)1.150.082(-0.079,0.25)0.002(-0.100,0.11)0.085(-0.070,0.25)1.300.088(-0.085,0.26)-0.003(-0.100,0.09)0.085(-0.070,0.25)0.77510.077(-0.066,0.25)0.007(-0.088,0.12)0.085(-0.070,0.25)1.150.082(-0.087,0.26)0.003(-0.097,0.10)0.085(-0.070,0.25)1.300.086(-0.087,0.26)-0.001(-0.097,0.10)0.085(-0.070,0.25)0.710010.078(-0.069,0.25)0.007(-0.091,0.12)0.085(-0.070,0.25)1.150.080(-0.076,0.25)0.004(-0.088,0.11)0.085(-0.070,0.25)1.300.084(-0.084,0.26)0.001(-0.100,0.10)0.085(-0.070,0.25) 40

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CHAPTER3BAYESIANAPPROACHFORLONGITUDINALMEDIATIONANALYSIS 3.1IntroductiontoLongitudinalMediationAnalysisInmanyclinicalandbehavioralstudies,thereisinterestintheevolvingeffectofaninterventiononanoutcomeandamediatingvariable.Beforewediscussthelongitudinalcase(whereboththeoutcomeandmediatoraremeasuredrepeatedlyovertime),however,itisusefultobeginwithnon-longitudinalmediation.Bynon-longitudinalmediationwemeanthesituationwherethereisaninitialinterventionassignment,followedsometimelaterbythemeasurementofamediator(orintermediatevariable),followedlaterbythemeasurementoftheoutcome.Eachoftheseismeasuredatjustonetime.Wenextintroducesomenotationtoformalizethediscussion.LetZ2f0,1gdenoterandomizedintervention(e.g.,1fortreatmentand0control).Denethepair(M0,M1)asthepotentialvalueofmediatorunderinterventionz=0,1,respectively.Then,apotentialoutcomeYz,Mz0isapotentialvalueoftheoutcomeforeverycombinationofinterventionzandapotentialmediatorMz0.ThesetofpotentialoutcomesYz,Mz0forz,z02f0,1gcannotbeobservedatthesametimeforeachindividualandsomeofthose(Y1,M0andY0,M1)cannotbeobserved;thisisthefunda-mentalproblemofcausalinference(Holland,1986).Continuingwiththenon-longitudinalsetting,wenextdenedirectandindirecteffectsintermsofpotentialoutcomes.Tostart,wedistinguishbetweencontrolledandnaturaleffects(Pearl,2001).Theformermeasurescausaleffectsatpre-determinedvalueofmediators.ThecontrolleddirecteffectisdenedasCDE=E(Y1,m)]TJ /F4 11.955 Tf 12.55 0 Td[(Y0,m)andthecontrolledindirecteffectisCIE=E(Y1,m0)]TJ /F4 11.955 Tf 12.45 0 Td[(Y1,m0)forpre-determinedvaluesmandm0.ThenaturaldirecteffectisdenedbyNDE=E(Y1,M0)]TJ /F4 11.955 Tf 12.96 0 Td[(Y0,M0)wherethemediatorisheldxedatthevalueitwouldtakeinits`natural'(treatment-free)state.Inotherwords,thetermY1,M0canbethoughtofastheoutcomeundertheinterventionz=1withthepathfromtheinterventiontothemediatorbeingblocked. 41

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TheNDEmeasuresthedirecteffectofintervention.ThenaturalindirecteffectisdenedasNIE=E(Y1,M1)]TJ /F4 11.955 Tf 12.88 0 Td[(Y1,M0),ortheeffectofthemediatorchangingfromM0toM1wheneveryonereceivestheinterventionz=1.ThetotalcausaleffectofZonYasTE=NDE+NIE=E(Y1,M1))]TJ /F4 11.955 Tf 12.14 0 Td[(E(Y0,M0).ThiscapturestheaggregateeffectofZthatpassesthroughandaroundthemediatorM.Inthefollowinggure,NIEstandsforthedirectarrowfromZtoYandNDEstandsforthearrowfromZtoMandMtoY.Z)166(!Y&%MOverthepastdecade,therehasbeensomeworkonlongitudinalmediation(wherethemediatorandoutcomeareobservedrepeatedlyovertime).Structuralequationmodel(SEM)(BaronandKenny,1986)basedapproachesareprevalentinthebehavioralsciences.ColeandMaxwell(2003)andMaxwelletal.(2011)proposeanautoregressivemodeltoassessindirectanddirecteffects.Foreachtime-varyinginterventionZt,mediatorMtandresponseYt,theypre-specifyastructuralmodelandcorrelationsbetweenZt,MtandYtateachtimeandoveralltimeperiods.Then,theydenetheoverallindirecteffectassumofallpossiblepathsfromtheinitialinterventiontothenalresponseYTthroughanyofMt's.Forexample,int=1,2,3,theoverallindirecteffectissumofmeasuresfrom1)Z1!M2!M3!Y4;2)Z1!Z2!M3!Y4;3)Z1!M2!Y3!Y4undertheassumptionthatthereisnopathfromMt)]TJ /F6 7.97 Tf 6.59 0 Td[(1toZtorfromYt)]TJ /F6 7.97 Tf 6.58 0 Td[(1toMt.ThedirecteffectcanbemeasuredbasedonthedirectpathbetweenZ1andY4.Accordingly,thenumberofpathwaystobesummedincreasesasthenumberofobservationtimestincreasesandthecausaleffectinthisframeworkhighlydependsontheparametricmodelstructureandpredeterminedcorrelations,whichcouldleadtobiasedestimatesifthemodelingassumptionsareincorrect.Alsointhetime-varyinginterventionsetting,vanderLaanandPetersen(2004)proposeanmarginalstructuralmodelapproachofestimatingdirectandindirecteffectsforstudies 42

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containingatime-dependentinterventionandcorrespondingtime-dependentcovariatesofinterest(mediators).However,thisdiffersfromoursettingofaninitialinterventionandlongitudinaloutcomes.AnSEMbasedmodelunderaninitialintervention(non-timevarying)isthelatent-growthmodel(LGM)orparallel-processmodel(MuchenandCurran,1997;Cheongetal.,2003).Inthisapproach,mediatorsandoutcomesaremodeledastwodifferentparallelprocesseswithlatentfactors.Theyspecifystructuralequationmodelsforthelatentfactorsandmeasurecausaleffectsviatheproductoftheircoefcients.However,thisapproachmeasuresthedirectandindirecteffectonlyattheendofstudy.OtherapproachesbasedonSEMareoutlinedinMacKinnonetal.(2007).UnlikeSEMbasedmodelswhichareveryparametric,ourframeworkwillbeconsiderablylessparametric.Linetal.(2008,2009)examinetime-varyingmediation(wherethemediatorwascompliance).Theyusehierarchicallatentclassstructuresthatcharacterizesubjectcompliancebehavioralpatternsovertime.Then,theyformcomplianceprincipalstrataandidentifyeachcomplianceclasswithalatentvariable.Byusingamediatortoformprincipalstrata,theycandrawprincipalstratadirectandindirecteffectswhicharedescribedinthecross-sectionalcasebyRubin(2004)andmoregeneralcaseinVanderWeele(2008).However,theprincipalstrataapproachbecomesmoredifcultinthecaseofacontinuousmediatorintermsofdeningtheprincipalstrata.Also,Pearl(2011)arguesthatusingprincipalstratabasedonmediatorsmayprohibitonefromregardingamediatorasacausebutmayforceonetouseamediatorjustasstatisticalconditioningandVanderWeele(2011)pointsoutthattheindirecteffectintheprincipalstrataapproachcannotbeidentiedingeneral.Thischapterextendsourpreviousworkforacross-sectionalmediator(Danielselat.2012)usingaBayesiandynamicmodel(WestandHarrison,1989)tolinkoutcomesandmediatorsmeasuredovertime.Oneofadvantagesofourmethodiswecanstill 43

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makeinferenceonthecross-sectionalcausaleffects.OurapproachismotivatedbydatafromtheCommittoQuit(CTQ)IItrial(Marcusetal.,2005).CTQIIisaclinicaltrialtostudyeffectofmoderate-intensityexerciseonsmokingcessation.Thistrialenrolled217womenandtestedtheeffectonsmokingcessationofmoderate-intensityexercisevs.equivalentstaffcontacttimecare.CTQIIrecordedquitstatusweeklyover8weeks.Foreachweek,participantswererequiredtoattendonesupervisedexercisesessionforthemoderate-intensityexercisegrouporattendonelecturewithlmsandhandoutsforthecontactconditiongroup.Thestudyrecordedquitstatusateachweekaswellasweight.Aquestionofinterestiswhetherweightchangesmediatethetreatmenteffectonsmokingcessation.Thepotentialmediatorintheanalysisisdenedasthedifferencebetweenbaselineweightandweightmeasuredateachweek.InSection 3.2 ,wedenethecausaleffectsofinterestinthelongitudinalsettingandspecifytheobservationmodelandassumptionssufcientforidentication.Section 3.3 outlinesthestepsforposteriorcomputation.Weusethreesensitivityparametersinourassumptions.InSection 3.4 ,weoutlineelicitationofvaluesforthesensitivityparameters.InSection 3.5 ,weapplyourmethodtoassessmediationinCTQII. 3.2InferenceonCausalEffectsofLongitudinalMediators 3.2.1CausalEffectsofMediatorsForlongitudinalcausaleffects,wedenethepotentialmediatorMztasthevalueofthemediatorattimet=1,,TifthesubjectwasassignedtotreatmentgroupZ=z.FullhistoriesofmediatorsaredenedasM1=(M11,,M1T)andM0=(M01,,M0T).Onlyonesetoflongitudinalpotentialmediatorsisobserved,M=ZM1+(1)]TJ /F4 11.955 Tf 12.15 0 Td[(Z)M0.InFigure 3-1 ,wedepictthegraphicalmodelforT=3.Here,outcomesareonthecausalpathwaywhileeachofthem(e.g.,Yt)isaffectedbytheprecedingmediator(Mt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)whichisalsoonthecausalpathwayofmediators.Also,mediatorattimetisaffectedbytheoutcomeattimet.Additionally,eachoutcomeandmediatorhavedirectpathsfromthetreatmentZ.Notethatwecanmeasurethecausaleffectofmediationstartingfrom 44

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Figure3-1. Mediationmodelwitht=1,2,3.Theleftguredepictsallpossiblepathwaysinthismodel.Theleftguredepictsthesituationwhenwearemeasuringeffectsattime3.Thedirectandindirecteffectcanbemeasuredbythepath(a)andthepath(b)!(c),respectively,afterupdatinginformationfromthepast(dashedarrows). t=2inthisframework.ThepotentialoutcomeYz,Mzt)]TJ /F13 5.978 Tf 5.76 0 Td[(1tdenotestheoutcomethatwouldbeobservedattimetifZ=zandMzt)]TJ /F6 7.97 Tf 6.58 0 Td[(1=mzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Onlyonesetofpotentialoutcomescanbeobserved,Yt=ZY1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1t+(1)]TJ /F4 11.955 Tf 12.72 0 Td[(Z)Y0,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1t.Similarly,anobservedmediatorattimetisMt=ZM1t+(1)]TJ /F4 11.955 Tf 12.78 0 Td[(Z)M0t.Wecanthendenenaturaldirectandindirecteffectsateverytimeinthexedtimepointcase.Thekeydifferencehereisthatthenaturaldirectandindirecteffectswillbedenedateverytimet=2,,T.Thisispossibleafteradjustingforpastdata,sinceinFigure 3-1 theinitialtreatmenthasdirectpathstoallmediatorsandoutcomes.Forexample,thenaturaldirecteffectattimetisNDEt=E(Y1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1t)]TJ /F4 11.955 Tf 12.02 0 Td[(Y0,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1t)whereY1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1timpliestheoutcomeattimetundertreatmentstatus1whilethedirectpathfromthetreatmenttothemediatorattimet)]TJ /F5 11.955 Tf 12.47 0 Td[(1beingblocked.Similarly,thenaturalindirecteffectisNIEt=E(Y1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1t)]TJ /F4 11.955 Tf 12.19 0 Td[(Y1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1t)andthetotaleffectisTEt=NDEt+NIEt=E(Y1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1t)]TJ /F4 11.955 Tf 11.58 0 Td[(Y0,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1t).Werstdescribethemodelfortheobservedlongitudinaldata,thenassumptionstoidentifythecausaleffectsofinterest. 3.2.2BayesianDynamicModelTomodeltheobserveddataovertime,weuseaBayesiandynamicmodel(West 45

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andHarrison,1989).DeneVt=(Yt,Mt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)tobetheobservedoutcomeattimetandmediatorattimet)]TJ /F5 11.955 Tf 12.8 0 Td[(1andassumeamodelpt(Vtjt)parameterizedbyt.Weassumethatconditionalonapastobservation,Vt)]TJ /F6 7.97 Tf 6.59 0 Td[(1=(Yt)]TJ /F6 7.97 Tf 6.58 0 Td[(1,Mt)]TJ /F6 7.97 Tf 6.58 0 Td[(2),andavectorofstateparametersattimet,t,Vt=(Yt,Mt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)isindependentofVs)]TJ /F6 7.97 Tf 6.59 0 Td[(1andsforallvaluesofs
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valueofthemediatorwouldhavebeenobservedundercontrolz=0attimet)]TJ /F5 11.955 Tf 12.1 0 Td[(1.Alsoletyzz0tdenoteyz,Mz0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1tfornotationalsimplicity.Assumption1a.Foraxedateachtimet,f1,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y10tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=m,jDt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j<,t)=f1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y11tjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m,jDt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j<,t).Here,therandomvariableDt)]TJ /F6 7.97 Tf 6.58 0 Td[(1quantiesthetreatmenteffectonthemediator.Thisassumptionstatesthatamongsubjectforwhomtreatmentwouldhaveminimalimpactonthemediator(asquantiedby),thedistributionsoftheoutcomearethesamewhetherthemediatorvaluewasinducedbyz=1orz=0.IntheCTQIIexample,amongpeopleforwhomexercise(z=1)wouldhavelittleimpactonweightchangesfromthebaselineateachtimet)]TJ /F5 11.955 Tf 12.72 0 Td[(1,theconditionaldistributionsofcessationstatusarethesamewhetherweightchangewasinducedundertheexerciseintervention(z=1)orthecontrol(z=0).Assumption1b.ThesecondpartoftheassumptionisforthesubgroupofsubjectsforwhomtheinterventionhasagreaterthaneffectonMt)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i.e.,jDt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j).Forthisgroup,forxedand,weassumef1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y10tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=m,jDt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j,t)/e( ()y11t)f1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11tjM1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=m,jDt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j,t),where ()=logsgn(Dt)]TJ /F13 5.978 Tf 5.76 0 Td[(1).IntheCTQIIexample,thisassumptionstatesthatamongsubjectsforwhomexercise(z=1)hasagreaterthaneffectonthedifferenceofweightchangesattimet)]TJ /F5 11.955 Tf 12.67 0 Td[(1,theconditionaldistributionofcessationstatusoftheexerciseinterventionafterblockingtheeffectofexerciseonweightchanges(themediatorvalueremainsatitsvalueunderthecontrol,z=0)isproportionaltotheconditionaldistributionofcessationstatusundertheexerciseinterventionnotblockingthepathofexerciseonweightchange(themediatorvalueunderexercise,z=1)throughanexponentialtiltrelationship.Assumption1differentiatesthepopulationintothoseforwhichthetreatmenthasa 47

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largeeffectonthemediatorversusthoseforwhichthetreatmenthasasmalleffectonthemediator.WediscusselicitationofsensitivityparametersandinSection 3.4 .Assumption2.Foraxedzateachtimet,fz,Mzt)]TJ /F13 5.978 Tf 5.75 0 Td[(1(yzztjmzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m(1)]TJ /F9 7.97 Tf 6.58 0 Td[(z)t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)=fz,Mzt)]TJ /F13 5.978 Tf 5.76 0 Td[(1(yzztjmzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t).ThisimpliesthatthepotentialoutcomesYz,Mzt)]TJ /F13 5.978 Tf 5.76 0 Td[(1tareindependentofthemediatorundertheothertreatment,m(1)]TJ /F9 7.97 Tf 6.59 0 Td[(z)t)]TJ /F6 7.97 Tf 6.59 0 Td[(1conditionalonthemediatorassociatedwiththepotentialoutcomes,mzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1andthevectorofstateparameterst.InCTQIIstudy,thisassumptionstatesthatcessationstatusattimetfromtheexerciseintervention(orthecontrol)isindependentofpotentialweightchangesattimet)]TJ /F5 11.955 Tf 13.27 0 Td[(1fromthecontrol(ortheexerciseintervention)giventhevalueofweightchangesattimet)]TJ /F5 11.955 Tf 12.2 0 Td[(1fromtheexerciseintervention(orthecontrol).SinceM1tandM0tcannotbeobservedatthesametime,weneedanassumptionaboutthejointdistributionofthetwomediators.Assumption3.Foracontinuousmediator,weassumethejointdistributionofmediatorsfollowsaGaussiancopulamodel(Nelsen,1999)suchthatFM0t,M1t(m0t,m1tjt)=2[)]TJ /F6 7.97 Tf 6.59 0 Td[(11fFM0t(m0tjt+1)g,)]TJ /F6 7.97 Tf 6.59 0 Td[(11fFM1t(m1tjt+1)g],where1istheunivariatestandardnormalCDFand2isthebivariatenormalCDFwithmean(0,0)T,variance(1,1)Tandcorrelation=cor[)]TJ /F6 7.97 Tf 6.59 0 Td[(11fFM1t(M1tjt+1)g,)]TJ /F6 7.97 Tf 6.58 0 Td[(11fFM0t(M0tjt+1)g].Thereisnoinformationinthedataaboutsensitivityparametersincethetwomediatorsareneverobservedatthesametime.WeconsidersensitivityanalysisinSection 3.4 .Foranordinalmediator,weprovideamodiedassumptionaswellasdetailsonmodelspecicationandposteriorcomputationinAppendix D .Assumption4.(Conditionalindependence)Potentialoutcomesateachtimetareindependentofeachotherconditionalonallpotentialmediatorsattimet)]TJ /F5 11.955 Tf 12.5 0 Td[(1andthe 48

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vectorofstateparameters,t,f(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),(1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),(0,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1)(y11t,y10t,y00tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)=f1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y11tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)f1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y10tjm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)f0,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y00tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t).NotethatAssumption4isnotnecessarytoestimateE[NIEtjdata]andE[NDEtjdata];forthese,wejustneedthemarginalposteriordistributionsforthepotentialoutcomes.However,itisnecessarytoestimateotherfeaturesoftheposteriordistributionofNIEtandNDEt.Underanassumptionofpositivedependencebetweenthepotentialoutcomesconditionalonmediators,thedifferencebetweenthevarianceofnaturalindirecteffectsunderthecasethatAssumption4doesnotholdattimet(Vart)andthevarianceunderAssumption4(Varwt),denotedasDifft=Varwt)]TJ /F1 11.955 Tf 11.95 0 Td[(Vart,is0Difft2Z(p Q1)]TJ /F4 11.955 Tf 11.95 0 Td[(Q2)fM0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)p(tjVt)dm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dtwhereQ1=Rexpfsgn(dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)logI(jdt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j)ytg(yt)2f1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(ytjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)dytR(yt)2f1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(ytjm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)dytandQ2=Rexpfsgn(dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)logI(jdt)]TJ /F6 7.97 Tf 6.58 0 Td[(1j)ytgytf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(ytjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)dytRytf1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(ytjm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)dyt.Therefore,Assumption4providesanupperboundonthevarianceofNIEt.TheproofisgiveninAppendix E 3.2.4IdenticationofJointDistributionsforComputationofDirectandIndirectEffectsatEachTimeInthefollowing,weshowthatAssumptions1-4alongwiththemodelin( 3 )-( 3 )aresufcienttoidentifythejointdistributionofNIEtandNDEt.Westatethisformallyinthefollowingtheorem.Wealsopointoutthatbyrandomizationofthetreatment,thedistributionsfMzt)]TJ /F13 5.978 Tf 5.75 0 Td[(1(mzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)andfz,Mzt(yzzt)areestimablefrom Vt=(Vt,Vt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,,V1).Theorem1.ThejointposteriordistributionofNIEtandNDEtforeacht(t=1,,T)isidentiedunderAssumptions1-4andthemodelin( 3 )-( 3 ).Proof:ToobtainthejointposteriordistributionofNIEtandNDEt,weneedtoidentifythe 49

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followingcomponent, f(0,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),(1,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1),(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1),M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y00t,y10t,y11t,m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)=f(0,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1),(1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1)(y00t,y10t,y11tjm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1jt)=f0,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y00tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)f1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)f(1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1)(y10tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)(A4)=f0,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y00tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)f1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y11tjm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)expfsgn(dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)logI(jdt)]TJ /F6 7.97 Tf 6.58 0 Td[(1j)y10tgf1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(Y11t=y10tjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1jt).(A1,2) (3) Here,`A'indicatesassumption.Allcomponentsin( 3 )areidentiablebytheobserveddataviarandomizationexceptthelasttermwhichcanbeidentiedbyAssumption3.ByintegratingoutM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1andt,weobtainthejointdistributionofpotentialoutcomes,Y11t,Y10tandY00t, f(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1),(1,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1),(0,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1)(y11t,y10t,y00t)=Zf(0,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1),(1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y00t,y10t,y11t,m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)p(t)dm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1dt=Zf0,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y00tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)f1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y11tjm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)expfsgn(dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)logI(jdt)]TJ /F6 7.97 Tf 6.58 0 Td[(1j)y10tgf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(Y11t=y10tjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)p(t)dm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dt.wherep(t)istheposteriordistributionoftfromtheBayesiandynamicmodelinSection 3.2.2 InSection 3.3 ,wewillgivedetailsonderivationoftheposteriordistributionoft.ThejointposteriordistributionofNDEtandNIEtisafunctionofthejointdistributionabove.Theparametersoftheobservationandevolutionmodelsrequiredforinferenceareestimatedusingtheobserveddata.Weonlyneedtomodeltwodistributionsf(z,Mzt)]TJ /F13 5.978 Tf 5.75 0 Td[(1),(z,Mzt)]TJ /F13 5.978 Tf 5.75 0 Td[(2)(yzzt,yzzt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)andthefMzt)]TJ /F13 5.978 Tf 5.75 0 Td[(1,Mzt)]TJ /F13 5.978 Tf 5.75 0 Td[(2(mzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,mzt)]TJ /F6 7.97 Tf 6.59 0 Td[(2jyzzt,yzzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)forz=0,1andt=2,,Twhicharethejointdistributionoftheobservedresponsesattimestand 50

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t)]TJ /F5 11.955 Tf 12 0 Td[(1andthejointdistributionofobservedmediatorsattimest)]TJ /F5 11.955 Tf 12.01 0 Td[(1andt)]TJ /F5 11.955 Tf 12.01 0 Td[(2conditionalonthetwoobservedoutcomesattimestandt)]TJ /F5 11.955 Tf 12.17 0 Td[(1,respectively.Fort=2,weneedtomodeldistributionsf(z,Mz1),Yz1(yzz2,yz1)andfMz1(mz1,mz0,jyzz2,yz1)insteadwhereyz1denotesthepotentialoutcomeatt=1withouttheeffectofthemediatorandmz0denotesthemediatoratbaseline.Wespecifythemodelingstrategybasedonacontinuousmediator.InAppendix D ,thecaseofanordinalcategoricalmediatorisdiscussed.Here,weassumetheoutcomeisdichotomous(asinourdataexample).Wetrytominimizeparametricassumptioninourmodelspecication. 3.2.4.1ObervationmodelTheobservationmodelfortheresponsecanbespeciedas)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(Yzzt,Yzzt)]TJ /F6 7.97 Tf 6.58 0 Td[(1Mult(z1t,z2t,z3t)fort=2,,T.Forthedistributionoftheobservedmediatorsconditionalontheobservedresponseattimestandt)]TJ /F5 11.955 Tf 12.66 0 Td[(1,aconjugatenormal-normalMDPmodel(MacEachernandMuller,1998)isspeciedforeachz=0,1.Weproposeaparticularformulationbelowthatweuseforthedataanalysis.Wefactorizethisdistributionastwoconditionaldistributions,fMzt)]TJ /F13 5.978 Tf 5.76 0 Td[(2(mzt)]TJ /F6 7.97 Tf 6.59 0 Td[(2jyzzt,yzzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)andfMzt)]TJ /F13 5.978 Tf 5.76 0 Td[(1(mzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1jmzt)]TJ /F6 7.97 Tf 6.59 0 Td[(2,yzzt,yzzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)withfollowingspecicationsMzt)]TJ /F6 7.97 Tf 6.59 0 Td[(2,ijYzzt,i=yt,Yzzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,i=yt)]TJ /F6 7.97 Tf 6.59 0 Td[(1N(z0t,i+z1tyt+z2tyt)]TJ /F6 7.97 Tf 6.58 0 Td[(1,1=zt,i),i=1,,N,z0t,i,zt,iG,i=1,,N,GDP(,G0),G0=N(bzt,1=zt)G zt zt,zt,andMzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,ijMzt)]TJ /F6 7.97 Tf 6.59 0 Td[(2,i=mt)]TJ /F6 7.97 Tf 6.59 0 Td[(2,Yzzt,i=yt,Yzzt)]TJ /F6 7.97 Tf 6.58 0 Td[(1,i=yt)]TJ /F6 7.97 Tf 6.59 0 Td[(1N(z0t,i+z1tmt)]TJ /F6 7.97 Tf 6.58 0 Td[(2+z2tyt+z3tyt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,1=zt,i),i=1,,N,z0t,i,zt,iH,i=1,,N, 51

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HDP(,H0),H0=N(azt,1=zt)G zt zt,zt,whereG(a,b)denotesagammadistributionwithshapeparametera>0andscaleparameterb>0(i.e.,meanisab)andzt,i,zt,zt,iandztdenoteprecisionparametersofnormaldistributions.TheprecisionparametersofDP,and,aregivenUnif(0,10).Notethatweonlymixedovertheinterceptparametersandprecisionsinthetwoconditionaldistributions,z0t,i,z0t,i,zt,iandzt,i.Parameters,z1t,z2t,z1t,z2t,z3t,areestimateusingonlythedataattimet.Wespecifyuniformpriors,Unif(1,5),onscaleparametersztandztinthegammapriors.Here,thevectorofstateparametersist=f(z1t,z2t,z3t,bzt,azt,zt,zt, zt, zt):z=0,1g. 3.2.4.2EvolutionmodelTheevolutionmodelfortheresponseneedstoupdateparameters,zt=(z1t,z2t,z3t),viaztjzt)]TJ /F6 7.97 Tf 7.59 0 Td[(1Dir(ztzt)]TJ /F6 7.97 Tf 7.59 0 Td[(1),wherethehyperparametertisgivenauniformshrinkageprior(Strawderman,1971;Daniels,1999)tocontrolthe`smoothness'oftheevolutionoftheparametersztandisderivedfromauniformprioronnzt=(nzt+zt),wherenztisthenumberofobservationsattimetforz=0,1.Forthemediator,weupdateparametersofthebasemeasuresoftheDP,bzt,azt,zt,zt, zt, ztasfollows:bztjbzt)]TJ /F6 7.97 Tf 6.58 0 Td[(1N(bzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,!z),aztjazt)]TJ /F6 7.97 Tf 6.59 0 Td[(1N(azt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,!z),ztjzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1G1 zzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,z,ztjzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1G1 zzt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,z,and ztj zt)]TJ /F6 7.97 Tf 6.59 0 Td[(1G(1 cz zt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,cz), ztj zt)]TJ /F6 7.97 Tf 6.59 0 Td[(1G(1 cz zt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,cz). 52

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Wespecifyuniformpriors,Unif(1,5)onthescaleparametersz,z,cz,czinthegammahyperpriors.Thevarianceparameters!zand!zaregivenuniformshrinkagepriors (zct)2 = (zct)2+!z 2and (zct)2 = (zct)2+!z 2wherezctandzctareharmonicmeansofzt,iandzt,i,respectively. 3.3PosteriorComputationsofLongitudinalMediationAnalysisRecallthattisthevectorofstateparametersattimet.Historicalinformationisembeddedintheposteriorp(t)]TJ /F6 7.97 Tf 7.59 0 Td[(1j Vt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)attimet)]TJ /F5 11.955 Tf 12.38 0 Td[(1priortoevolutionthrough( 3 ).Theevolutionstepisp(tj Vt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)=Zpe(tjt)]TJ /F6 7.97 Tf 7.6 0 Td[(1)p(t)]TJ /F6 7.97 Tf 7.59 0 Td[(1j Vt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)dt)]TJ /F6 7.97 Tf 7.59 0 Td[(1andafterobservingcurrentinformationVt,theposteriorfortimetisobtainedviatheupdatingstep p(tj Vt)/p(tj Vt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)pt(Vtj Vt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)=p(tj Vt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)pt(VtjVt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t). (3) Theequalityin( 3 )isamodelingassumption(whichcanbecheckedfromtheobserveddata).SamplingfromtheposteriordistributioncanbedoneinWinBUGS.TocomputeNIEtandNDEtateachtimet,wesampleNsetsofstateparameterstfromtheposteriordistributionp(tj Vt)attimet.Then,foreachsetoftandxedvaluesforthesensitivityparameters,,,wedothefollowingthreesteps. 1. GenerateKsetsof(v1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1)wherev1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=(y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(2)bysamplingfromf(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(2),M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(2(y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(2jM1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)andfM0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1jt). 2. Computef1,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(ytjt)viaMonteCarlointegrationusingtheKsetsof(v1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1)asfollows, f1,M0t(ytjt)=PKi=1expflogsgn(dt)]TJ /F13 5.978 Tf 5.76 0 Td[(1,i)ytI(jdt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,ij)gf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(ytjm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,i,v1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,i,t) KC(,), (3) 53

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wherethenormalizingconstantC(,)forxedvaluesandisdenedasC(,)=PKi=1C0f1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(1jm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,i,v1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,i,t)+f1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(0jm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,i,v1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,i,t) K,whereC0=expfsgn(dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,i)logI(jdt)]TJ /F6 7.97 Tf 6.58 0 Td[(1,ij)g.Adetailedderivationof( 3 )isinAppendix F .Tocomputef1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y11tjm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,v1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t),werstnotethatf(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1),M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,V1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11t,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,v1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)=fM1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1jy11t,y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(2,t)fM1t)]TJ /F13 5.978 Tf 5.76 0 Td[(2(m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(2jy11t,y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)f(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1),(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(2)(y11t,y11t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t).TheformsofthealltermsabovearegiveninSection 3.2.4.1 .Then,f1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,v1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)=f(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,V1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11t,m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(2,t) fM1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,V1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(2,t),wherefM1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,V1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,y11t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(2,t)=f(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,V1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(1,m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(2,t)+f(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,V1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(0,m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(2,t). 3. Computethedirectandindirecteffects,NIE(t)=E(Y11tjt))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y10tjt)=f1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11t=1jt))]TJ /F4 11.955 Tf 11.96 0 Td[(f1,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y10t=1jt),NDE(t)=E(Y10tjt))]TJ /F4 11.955 Tf 11.96 0 Td[(E(Y00tjt)=f1,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y10t=1jt))]TJ /F4 11.955 Tf 11.95 0 Td[(f0,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y00t=1jt).wheref1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11t=1jt)=f(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(2)(y11t=1,y11t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=1jt)+f(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(2)(y11t=1,y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=0jt)andf0,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y00t=1jt)=f(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1),(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(2)(y00t=1,y00t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=1jt)+f(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1),(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(2)(y00t=1,y00t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=0jt).Fromthisprocedure,weobtainasampleofsizeNfromtheposteriorofnaturaldirecteffectandindirecteffect,NIEtandNDEt.Ifweplacepriorsonthesensitivityparameters,and,wedotheabovethreestepsNtimesforeachsampleoft,,and. 3.4SensitivityAnalysisAssumptions1-4containthreesensitivityparameters,(,,).Here,wediscussstrategiestoelicitvaluesforeach.TofacilitateelicitationofinAssumption1,wespecify=0andassumeM1t)]TJ /F4 11.955 Tf 9.72 0 Td[(M0t>withprobability1.Then,wecanexpressasthefollowingoddsratio(derivationisin 54

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Appendix G ), = f1,M0t)]TJ /F23 4.068 Tf 5.4 0 Td[(1(yt=1jt) = f1,M0t)]TJ /F23 4.068 Tf 5.39 0 Td[(1(yt=0jt) f1,M1t)]TJ /F23 4.068 Tf 5.4 0 Td[(1(yt=1jt) = f1,M1t)]TJ /F23 4.068 Tf 5.39 0 Td[(1(yt=0jt) (3) Thisistheratiooftheoddsofanoutcomeoccurringontheinterventionwiththemediatorsetatitsnaturalvaluetotheoddsofanoutcomeontheinterventionwiththemediatorsetatitsvalueundertheintervention.Ifweassumethemediatorhasapositiveeffectontheprobabilityofanoutcome,thenweexpecttheoddsinthedenominatortobelargerthantheoddsinthenumeratorin( 3 ).Weuseexpression( 3 )foreliciting.Toelicitpossiblevaluesof,weneedtodeterminehowlargeDtshouldbeforthefollowingratio,R,tobenotequaltoone, R=f1,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(Yt=yjm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,Dt)]TJ /F6 7.97 Tf 6.58 0 Td[(1=0[m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1],t) f1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(Yt=yjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,Dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1=[m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1+],t), (3) whichisthedefaultelicitationstrategyinDanielsetal.(2012).InAssumption3,thesensitivityparameteristhecorrelationparameterbetween)]TJ /F6 7.97 Tf 6.58 0 Td[(11fFM0t(M0t)gand)]TJ /F6 7.97 Tf 6.58 0 Td[(11fFM1t(M1t)g.Ifweassumethecorrelationispositive,adefaultapproachmightbetospecify2[0,1). 3.5AnalysisofMediationinCTQII 3.5.1DescriptionofCTQIICTQIIwasarandomizedclinicaltrialtostudytheeffectofmoderate-intensityexerciseonsmokingcessation.Thistrialenrolled217smokingwomenandtestedtheeffectonsmokingcessationofmoderate-intensityexerciseintervention(109)vs.equivalentstaffcontacttimecontrol(108).Over8weeks,participantswererequiredtoattendonesupervisedexercisesessionforthemoderate-intensityexercisegrouporattendonelecturewithlmsandhandoutsforthecontactconditiongroup(control).Thestudyrecordsquitstatusateachweekaswellasparticipants'weightchanges.Here,weareinterestedinhowweightchangesofanindividualmediatethe 55

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interventioneffectovertime.Wedenethemediatoratweakttobethedifferencebetweenbaselineweightandweightmeasuredatweekt.Ateachweek,wedenethebinaryoutcometobethequitstatusofanindividual.Intheanalysisoftheoriginaltrial,cessationratesatweek1and2werealmost0(0and0.01,respectively)sincethetargetquitweekwasweek3.Hence,weassessthelongitudinalcausaleffectsfromweek4toweek8.Additionally,alargenumberofsubjectsmissedsomesessionsordroppedoutbeforetheendofthestudy.Weassumethatthismissingnessisignorablewhichisimplicitlyaddressedinourdynamicmodel. 3.5.2PriorsandSensitivityParametersThepriorsfortheMDPmodelfortheobserveddata(Section 3.2.4.1 )arespeciedasfollows.ForthescaleparametersztandztintheGammadistributionsofbasemeasures,wespecifyUnif(1,5).Forthescaleparametersz,z,cz,czinthegammadistributionsfortheevolutionmodel,weusethesamepriors.Fortheregressionparameters,z1t,z2t,z1t,z2t,z3t,wespecifyvaguenormalpriors,N(0,100).Thedifferenceintheaverageweightchangesbetweenthetwogroupsamongthosecontinuouslyabstinentatweek8was0.7poundswithstandarddeviation(S.D.)of1.65.Weusedthisinformationtodetermine.Inparticular,weconsiderdifferencesmorethanaverage(0.7)asthecasethatthemediatorunderinterventionz=1havinganimpactonthedenominatorofRinsection 3.4 .Thatis,differenceslargerthan0.7arevaluesofsuchthatRisnotequalto1.Weconsiderisvaryingfrom0.7(average)to2.3(average+S.D.).Weassumedtheimpactofthetreatmentonthemediatorbeingmorethan12poundscouldreectanegativeimpactuptoanoddsratioin( 3 )ofabout0.5.Hence,weconsideredvalues2(0.5,1).ForAssumption4,weassumethecorrelationbetweenM0tandM1twaspositive,2[0,1). 3.5.3ResultsofLongitudinalMediationAnalysisFortheMCMCalgorithmtoobtaintheposteriorforeacht,weran15000iterationsanddiscardedtherst5000asburn-in.FortheMCcomputationsinSection3,weset 56

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N=100(bytakingevery10thposteriorsamplefromthelast1000samplesinordertominimizeautocorrelation)andK=500sincetheeffectivesamplesizeoftheMCintegrationtoensuretheestimateiswithindistance0.025fromthetruevaluewithprobability0.95isapproximately380.Figure 3-2 showstheposteriormeansofthecessationratesfromthetwogroups(exerciseinterventionandcontrol)with95%credibleintervals.Inbothgroups,the Figure3-2. Cessationratesoftheexerciseintervention(1t)andthecontrol(0t)fromtheposteriordistributionateachtimetwith95%credibleintervals. cessationratestendtoincreaseuntilweek7anddropatweek8.Thetotaleffectofmoderate-intensityexerciseversusequalstaffcontacttimecorrespondstoanegativeriskdifferenceduringeachweek(rangingfrom-0.12to-0.03).Theseresults(posteriormeanandSD)aredisplayedinTables 3-1 3-3 .Becausethetotaleffectisinvarianttothevaluesofthesensitivityparameters,onemaysimplyfocusattentiononthethirdrowofTable 3-1 .Thus,theinterventiondidnotincreasecessationoverall.Infact,otherthan 57

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att=8,thereisevidenceoftheexercisearmhavingpoorercessationrates.TheposteriormeansandSDsfortheNIEandNDEaredisplayedinTables 3-1 3-3 forvariouscombinationsofthesensitivityparameters.Inaddition,posteriormeansfortheTE,NIE,NDEaredisplayedinFigures 3.5.3 3.5.3 .Formostcombinationsofsensitivityparametersconsidered,NIEshaveapositiveeffectwhen0.75andNDEshavenegativeeffectsforallvaluesoverallperiods.Overall,theNIE(NDE)decreased(increased)asavalueofdecreased.Inparticular,theresultweremoresensitive(butveryslightly)towhen=0.7andleastsensitivetothecorrelationparameter=0.3.Forsensitivityparameter,NIE(NDE)increased(decreased)asincreasedfrom0.7to2.3.Theinterpretationoftheseparametersisasfollows.TheNDEattimetistheaveragedifferenceincessationratesattimetcomparingpotentialoutcomesunderz=1andz=0,ifeveryone'sweightchangeatt)]TJ /F5 11.955 Tf 11.96 0 Td[(1wassettowhatitwouldhavebeenintheabsenceoftheintervention(z=0).Thus,negativevaluesofNDEsuggestthattheeffectoftheexercisetreatmentonsmokingthatisnotthroughitseffectonweightchangeisnegative(i.e.decreasestheprobabilityofquittingrelativetothecontrolarm).TheNIEattimetisthecessationrateattifeveryonewasintheexercisearmandhadtheirweightchangeatt)]TJ /F5 11.955 Tf 12.69 0 Td[(1settowhatitwouldhavebeenifassignedexercisetreatment,comparedwiththecessationrateifzwassettoexercisetreatmentbutweightchangewassettowhatitwouldhavebeenunderthecontrolcondition.PosteriordrawsofNIEtendedtobepositiveinmostscenarios,suggestingthattheeffectthatexercisetreatmenthasonweightchangetendstomakesmokinglesslikely.Fromthelongitudinalperspective,TEwasincreasingandNIEwasslightlydecreasingfromthebeginning.Theeffectofthemoderateexercisetreatmentonthesmokingcessationwasmediatedpositivelybythemediatorattimet(baselineweight-weightattimet)andthismediationeffectwasslightlydecreasingoverweeks.Thereessentiallyappearedtobeadiminishingimpactoftheintervention(effectstrendingtoward0)overtime. 58

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Table3-1. Estimatesandstandarddeviations(inparentheses)ofthenaturalindirecteffects(NIE)andthenaturaldirecteffects(NDE)atweek4,5,6,7and8.Sensitivityparametersconsideredare=0.5,0.75,1and=0.7,1.5,2.3when=0.3. EffectsWeek4Week5Week6Week7Week8 0.71NIE0.0770.0470.0530.0180.046(0.018)(0.024)(0.019)(0.030)(0.020)NDE-0.194-0.141-0.178-0.094-0.084(0.037)(0.048)(0.043)(0.050)(0.045)0.70.75NIE0.0420.0180.019-0.0130.021(0.016)(0.024)(0.023)(0.027)(0.023)NDE-0.158-0.112-0.144-0.063-0.059(0.037)(0.047)(0.044)(0.047)(0.050)0.70.50NIE-0.024-0.046-0.040-0.073-0.033(0.023)(0.029)(0.027)(0.026)(0.031)NDE-0.092-0.048-0.085-0.003-0.005(0.042)(0.051)(0.047)(0.048)(0.055)1.51NIE0.0780.0460.0530.0160.047(0.016)(0.019)(0.018)(0.030)(0.018)NDE-0.194-0.141-0.178-0.092-0.085(0.037)(0.044)(0.041)(0.049)(0.046)1.50.75NIE0.0460.0190.025-0.0110.024(0.018)(0.025)(0.021)(0.028)(0.022)NDE-0.162-0.114-0.150-0.065-0.062(0.040)(0.049)(0.044)(0.048)(0.047)1.50.50NIE-0.019-0.039-0.035-0.067-0.028(0.022)(0.033)(0.025)(0.026)(0.029)NDE-0.097-0.056-0.090-0.009-0.010(0.043)(0.056)(0.046)(0.048)(0.053)2.31NIE0.0780.0460.0520.0160.050(0.018)(0.021)(0.019)(0.027)(0.021)NDE-0.195-0.140-0.177-0.092-0.088(0.039)(0.046)(0.043)(0.045)(0.046)2.30.75NIE0.0470.0210.026-0.0050.029(0.018)(0.023)(0.020)(0.027)(0.021)NDE-0.164-0.116-0.151-0.071-0.067(0.039)(0.049)(0.042)(0.048)(0.048)2.30.50NIE-0.010-0.032-0.027-0.057-0.021(0.023)(0.029)(0.025)(0.027)(0.029)NDE-0.107-0.062-0.098-0.019-0.017(0.044)(0.053)(0.046)(0.049)(0.054) 59

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Table3-2. Estimatesandstandarddeviations(inparentheses)ofthenaturalindirecteffects(NIE)andthenaturaldirecteffects(NDE)atweek4,5,6,7and8.Sensitivityparametersconsideredare=0.5,0.75,1and=0.7,1.5,2.3when=0.5. EffectsWeek4Week5Week6Week7Week8 0.71NIE0.0770.0480.0540.0150.045(0.017)(0.020)(0.017)(0.031)(0.019)NDE-0.193-0.142-0.178-0.091-0.083(0.038)(0.045)(0.041)(0.049)(0.046)0.70.75NIE0.0370.0150.021-0.0160.023(0.018)(0.026)(0.022)(0.026)(0.024)NDE-0.154-0.109-0.146-0.060-0.061(0.040)(0.049)(0.046)(0.048)(0.050)0.70.50NIE-0.033-0.053-0.050-0.078-0.039(0.022)(0.031)(0.025)(0.027)(0.031)NDE-0.083-0.042-0.0750.0030.001(0.041)(0.052)(0.047)(0.047)(0.055)1.51NIE0.0760.0460.0530.0190.046(0.016)(0.020)(0.019)(0.027)(0.018)NDE-0.192-0.140-0.178-0.094-0.084(0.038)(0.045)(0.042)(0.048)(0.046)1.50.75NIE0.0400.0160.023-0.0120.022(0.018)(0.022)(0.021)(0.027)(0.023)NDE-0.157-0.111-0.148-0.063-0.060.(0.039)(0.048)(0.044)(0.048)(0.050)1.50.50NIE-0.026-0.045-0.038-0.074-0.034(0.023)(0.028)(0.027)(0.026)(0.029)NDE-0.090-0.049-0.086-0.002-0.004(0.043)(0.052)(0.049)(0.049)(0.053)2.31NIE0.0770.0480.0530.0160.045(0.016)(0.019)(0.017)(0.030)(0.019)NDE-0.194-0.142-0.178-0.091-0.083(0.037)(0.044)(0.040)(0.051)(0.046)2.30.75NIE0.0450.0200.023-0.0080.027(0.019)(0.023)(0.021)(0.026)(0.022)NDE-0.161-0.115-0.148-0.068-0.065(0.041)(0.048)(0.044)(0.048)(0.046)2.30.50NIE-0.016-0.037-0.031-0.066-0.021(0.023)(0.032)(0.028)(0.025)(0.029)NDE-0.101-0.058-0.093-0.010-0.017(0.043)(0.054)(0.050)(0.048)(0.053) 60

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Table3-3. Estimatesandstandarddeviations(inparentheses)ofthenaturalindirecteffects(NIE)andthenaturaldirecteffects(NDE)atweek4,5,6,7and8.Sensitivityparametersconsideredare=0.5,0.75,1and=0.7,1.5,2.3when=0.7. EffectsWeek4Week5Week6Week7Week8 0.71NIE0.0770.0480.0540.0150.045(0.017)(0.020)(0.017)(0.031)(0.019)NDE-0.193-0.142-0.178-0.091-0.083(0.038)(0.045)(0.041)(0.049)(0.046)0.70.75NIE0.0320.0110.016-0.0220.020(0.019)(0.026)(0.023)(0.026)(0.025)NDE-0.148-0.105-0.140-0.053-0.058(0.041)(0.050)(0.047)(0.048)(0.051)0.70.50NIE-0.046-0.060-0.060-0.091-0.044(0.022)(0.032)(0.026)(0.026)(0.034)NDE-0.071-0.034-0.0650.0150.006(0.041)(0.054)(0.048)(0.047)(0.057)1.51NIE0.0760.0460.0530.0190.046(0.016)(0.020)(0.019)(0.027)(0.018)NDE-0.192-0.140-0.178-0.094-0.084(0.038)(0.045)(0.042)(0.048)(0.046)1.50.75NIE0.0360.0130.019-0.0180.021(0.018)(0.023)(0.022)(0.027)(0.024)NDE-0.152-0.107-0.144-0.058-0.059(0.039)(0.048)(0.044)(0.048)(0.051)1.50.50NIE-0.035-0.050-0.046-0.084-0.037(0.024)(0.030)(0.029)(0.026)(0.031)NDE-0.081-0.045-0.079-0.009-0.001(0.043)(0.054)(0.050)(0.049)(0.054)2.31NIE0.0770.0480.0530.0160.045(0.016)(0.019)(0.017)(0.030)(0.019)NDE-0.194-0.142-0.178-0.091-0.083(0.037)(0.044)(0.040)(0.051)(0.046)2.30.75NIE0.0430.0180.023-0.0130.025(0.018)(0.025)(0.021)(0.027)(0.023)NDE-0.159-0.112-0.148-0.063-0.063(0.039)(0.048)(0.043)(0.048)(0.047)2.30.50NIE-0.020-0.032-0.035-0.071-0.023(0.024)(0.032)(0.030)(0.027)(0.033)NDE-0.097-0.062-0.090-0.005-0.015(0.043)(0.054)(0.051)(0.048)(0.058) 61

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Figure3-3. Thenaturalindirecteffect(NIE),directeffect(NDE)andtotaleffect(TE)forweek4,5,6,7,8withcombinationsofsensitivityparameters=0.5,0.75,1and=0.7,1.5.2.3when=0.3. 62

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Figure3-4. Thenaturalindirecteffect(NIE),directeffect(NDE)andtotaleffect(TE)forweek4,5,6,7,8withcombinationsofsensitivityparameters=0.5,0.75,1and=0.7,1.5.2.3when=0.5. 63

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Figure3-5. Thenaturalindirecteffect(NIE),directeffect(NDE)andtotaleffect(TE)forweek4,5,6,7,8withcombinationsofsensitivityparameters=0.5,0.75,1and=0.7,1.5.2.3when=0.7. 64

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3.6DiscussiononLongitudinalMediationAnalysisWehaveproposedacausalframeworkforlongitudinalmediationwhichassessestime-varyingdirectandindirecteffectsofmediators.Weattemptedtominimizeparametricassumptionsfortheobserveddata.Severalassumptionsnecessarytoidentifytheeffectsofinterestarespecieduptosensitivityparameters.Alimitationwithourapproachisthatelicitatingaplausiblerangeforthesensitivityparameters,especiallyand,ischallenging.Weproposedmethodstohelpwithpriorelicitation,butopinionsaboutwhatconstitutesaplausiblerangewouldprobablydiffer.FortheCTQII,theeffectofmoderateexercisetreatmentvs.thestaffcontactcontrolwasmarginallysignicant(inanegativeway)overalltimeperiods.However,thelongitudinalmediator,weightchangefrombaselinehadasignicantpositiveimpactoncessation.Thissuggeststhatmoderateintensityexercisemightmakepeoplemorelikelytosmoke(negativedirecteffect),buttothedegreethatitaffectsweightchangeitdecreasestheprobabilityofsmoking(positiveindirecteffect).Aprevioustrialofmoreintenseexerciseshowedapositiveeffectoncessation(Marcusetal.,1999).OnecouldspeculatethatmoreintenseexercisehadalargerpositiveNIEthandidthemoderateexerciseprogram,leadingtoapositivetotaleffect.However,longitudinalweightswerenotrecordedinthattrial.Nevertheless,thisapplicationdemonstratesthewaysinwhichestimatingNDEandNIEovertimecanleadtoabetterunderstandingoftheinterventioneffectsandmechanisms.Futureworkwillincorporatetime-varyingcovariates.IntheCTQII,weassumeignorablilityfordropoutsandintermittentmissingness.Wecandevelopaframeworktoaddressnonignorablemissingnessbyusingpatternmixturemodels(DanielsandHogan,2008)intheBayesiandynamicmodelsetting. 65

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CHAPTER4BAYESIANINFERENCEFORTHECAUSALEFFECTSINTHESETTINGOFMULTIPLEMEDIATORS 4.1IntroductiontotheCausalEffectsofMultipleMediatorsTheimportanceofassessingtheprocessbywhichabehavioralinterventionaffectsanoutcomethroughintermediatevariables(mediators)cannotbeoverstated.Unliketrialsofapharmacologicagent,theprocessormethodbywhichabehavioral`treatment'orinterventionworkscannotberepresentedbyaconcretephysicalmodel(suchasrateofuptakeofadrug).Asidefromtheusualgoalofndingoutwhetheraninterventionaffectsatargetedoutcome,amajorobjectiveofmostbehavioralinterventiontrialsistounderstandwhetherandhowintermediatebehavioralprocessesthatarecausallylinkedtothetargetoutcomewerechangedasaresultofapplyingtheintervention.Forexample,Marcusetal.(2003,2005)conductedaseriesoftrialstostudytheeffectofanexerciseinterventiononsmokingcessationinwomen.Theunderlyingbehavioralmodeltheorizedthatexercisewouldincreasesmokingcessationbyaffectingintermediatevariablesrelatedtoriskofrelapse;theseincludedweightgain,concernaboutweightgain,andselfefcacyrelatedtobehaviorchange.Understandinghowaninterventionworksthroughintermediatevariablesservestoconrmorupdateabehavioralmodel,andservestoguidedevelopmentofmoreeffectivelytargetedinterventions(Kraemeretal.,2002).Untilrecently,thepredominantclassofstatisticalmethodsusedbybehavioralscientistswasbasedonthetwo-stageregressionapproachofBaronandKenny(1986).Thesemethodstendtoappealtointuitionandtheyareeasytoimplement,butgenerallyarenotwellsuitedtoestimatingcausaleffectsofanintermediatevariable,exceptunderstrongsetsofassumptions(e.g.,thesequentialignorabilityconditionthattheintermediatevariableormediatorhasbeenrandomlyallocated;seeImaietal.,2010a,b).Researchersinstatistics,epidemiologyandcomputersciencehavedevelopedformalframeworksforrepresentingdirectandindirecteffectsinacausalframework 66

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(RobinsandGreenland,1992;Pearl,2001;MealliandRubin,2003).Wearenowseeingtheemergenceofanalyticmethodsfordrawingcausalinferencesaboutdirectandindirecteffects(FrangakisandRubin,2002;VanderWeele,2008),whichincludesmethodsbasedonstructuralmodeling(TenHaveetal.,2007;VanderWeele,2009),weightedestimatingequations(TchetgenTchetgenandShpitser,2011),imputation(Vansteelandtetal.,2012;Zhang,2012)andBayesianinference(Danielsetal.,2012).Acausalmodelofmediationcanbeformulatedintermsofeithernaturaleffectsorcontrolledeffects(GreenlandandRobins,1992;Pearl,2001;MealliandRubin,2003).Theessentialdifferencebetweentheseframeworksiswhetheroneconceptualizesthemediatorasavariablethatcouldpotentiallybeassignedaxedvalueoverthepopulation(CDE),orwhetherthisassumptionisnotplausible(NDE).Inbehavioralinterventionstudies,theintermediatevariablesfrequentlyarerelatedtounderlyingindividualtraits(e.g.concernaboutweightgain);characterizingmediationintermsofCDEwillgenerallynotbearealisticapproachinthesesettings.Despitesomeverypromisingadvancesinthisarea,thereremainseveralunresolvedmethodologicalissues.Mostofthemethodscitedabovedealonlywiththecaseofasinglemediator;inpractice,severalmediatorsaremeasured,sometimesatthesametimeandsometimesseriallyovertime.Whereasthesinglemediatorcasehasbeenwidelystudiedwiththeabovemethods,causalinterpretationsinthepresenceofmultiplemediatorshasreceivedconsiderablylessattentionandmostinvolveextensionsoftheregressionapproach(MacKinnon,2008).PreacherandHayes(2008)reviewedseveralmethodsofestimatingmultiplemediationeffectsbasedontheextensionoftheregressionapproach.TheypointedoutthatsincetheusualProduct-of-Coefcientapproachinthesettingofmultiplemediatorvariablesrequiresthemultivariatedeltamethodtoderivethestandarddeviationoftheindirecteffect,itcanbeonlyvalidundertheassumptionofmultivariatenormalityorthesituationofverylargesamples(Sobel,1986).Intheirreview,theyrecommendedto 67

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usebootstrapping,toestimatecausaleffectsofinterestandcorrespondingsamplingdistributions.Eventhoughthisnonparametricmethodrelaxedthenormalityassumption,themainstructurethemethodisappliedtoistheextensionofthestandardregressionmodelwhichassumeadditivityofeachindirecteffect.Recently,ImaiandYamamoto(2013)proposedadifferentapproachallowingthepresenceofcausallydependentmultiplemediators,whichassumesmediatorsaresequentiallyrelatedtoeachother.Inthisarticle,wearemoreinterestedinthesituationthatmultiplemediatorshaveeffectsonoutcomeatthesametime(notsequentially)withpossibleoverlappingeffectsamongmediators.Inaddition,aswithallcausalinferencemethods,validityoftheanalysesdependsonuntestableassumptionsaboutthedistributionofunobservablepotentialoutcomes.Methodsforassessingsensitivitytodeparturesfromuntestableassumptionsarenotwelldevelopedbutareessentialtocomprehensivemediationanalysis.Bayesianapproachesprovideanaturalframeworkforcharacterizinguncertaintyabouttheseassumptionsthroughnon-degeneratepriorsonsensitivityparametersencodingthoseassumptions.Inthischapter,wedeneaframeworktoassesscausaleffectsofmultiplemediatorsandintroduceaBayesianapproachtoestimatethoseeffects.InSection 4.2 ,wedenethecausaleffectsofinterestinthepresenceofmultiplemediatorsandreviewtheregressionapproach.InSection 4.3 ,weproposeaBayesianapproachanddetailedposteriorcomputations.InSection 4.5 ,theBayesianapproachisappliedtoSTRIDEdata(Marcusetal.,2007)whichcontainsmultiplepsychosocialmediators. 4.2NewFrameworkforMultipleMediators 4.2.1NotationandDenitionLetZdenoteassignmenttobehavioralintervention;forclarityweassumetwolevelsofZ,Z=1fortheinterventionandZ=0forthecontrol.LetXdenoteavectorofbaselinecovariateinformation;bothZandXarerecordedattimet=0.Amediatorof 68

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interestismeasuredatsometimepointt1>0,andthetargetedoutcomeismeasuredattimet2>t1.WeadoptthepotentialoutcomesnotationofPearl(2000,2001).Foreachindividual,thereisasethavingtwopotentialoutcomesforeachofKcandidatemediators,fMkz:z=0,1;k=1,...,Kg,allmeasuredatsametime,t1.TherandomvariableMkzistherealizationifthek-thmediatorunderassignmenttoarmZ=z.Eachindividualhasasetof2K+1potentialtargetoutcomesfYz,M1z1M2z2MKzKgwherefz,z1,...,zKg2f0,1g(K+1),whichdenotestherealizationofthetargetedoutcomeunderrandomizationtointerventionlevelzandrealizedmediatorvaluesM1z1,,MKzK.TheobserveddataarerelatedtothepotentialoutcomesviaMk=ZMk1+(1)]TJ /F4 11.955 Tf 11.24 0 Td[(Z)Mk0andY=ZY1,M11M21MK1+(1)]TJ /F4 11.955 Tf 13.11 0 Td[(Z)Y1,M10M20MK0.Fornotationalsimplicity,letMz=fM1z,M2z,,MKzgwhichdenotesasetofKcandidatemediatorsallunderinterventionZ=z.Then,thepotentialoutcomeYz,M1zM2zMKzcanbere-writtenasYz,Mzforz=0,1.Thegraphbelowillustratesthedirecteffect(thehorizontalarrow)andtheindirecteffects(thearrowfromZtoMkandfromMktoY)ofanintervention,Z2f0,1g,onaresponse,Y,withmultiplemediators,Mk's.Z)334()223()222()222()222()335(!Y&...%M1,,MKThepairs(Mk0,Mk1)arethesetofpotentialvaluesofthek-thmediatorunderZ2f0,1g.ThepotentialoutcomeYzm1mKcanbedenedforeverypossiblecombinationofvaluesofZandM1,,MK.ThenaturaldirecteffectisdenedasNDE=E(Y1,M0)]TJ /F4 11.955 Tf 12.7 0 Td[(Y0,M0).Andthejointnaturalindirecteffectofallmediators(JNIE)isderivedbysubtractingthenaturaldirecteffectfromthetotaleffect,JNIE=TE)]TJ /F1 11.955 Tf 11.95 0 Td[(NDE=E(Y1,M1)]TJ /F4 11.955 Tf 11.96 0 Td[(Y1,M0).Withoutlossofgenerality,wewillintroduceapartitioningoftheJNIEusingthecaseof3mediators.DeneMkzasthek-thmediatorunderinterventionz=0,1. 69

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Figure4-1. PartitioningoftheJNIEfor3mediators ThepotentialoutcomeYz,M1z0,M2z00,M3z000istheoutcomefromtreatmentz2f0,1gwhilecorrespondingmediatorsareinducedbyinterventionz0,z00,z0002f0,1g.TheJNIEcanbedecomposedintomediator-specicNIEandthejointeffectsofallpossiblepairs,triples,etc.ofthesetofcandidatemediators(Figure4-1).Themediator-specicNIEforthekthmediatorisdenedasNIEk=E(Y1,M11Mk)]TJ /F13 5.978 Tf 5.75 0 Td[(11Mk1Mk+11MK1)]TJ /F4 11.955 Tf 11.95 0 Td[(Y1,M11Mk)]TJ /F13 5.978 Tf 5.76 0 Td[(11Mk0Mk+11MK1).Forexample,inthecaseofthreemediators,themediator-specicNIEsformediator1,2and3areNIE1=E(Y1,M1)]TJ /F4 11.955 Tf 11.95 0 Td[(Y1,M10M21M31),NIE2=E(Y1,M1)]TJ /F4 11.955 Tf 11.95 0 Td[(Y1,M11M20M31)andNIE3=E(Y1,M1)]TJ /F4 11.955 Tf 11.95 0 Td[(Y1,M11M21M30).InasimilarfashionwecandeneSNIEjkasthesimultaneousNIEattributabletothejointeffectsofmediatorsjandkandSNIEjklasthesimultaneousNIEattributabletothejointeffectofmediatorsj,kandl.Intermsofthecaseofthreemediators,simultaneousnaturalindirecteffectsbetweenmediator1and2,between1and3andbetween2and3areSNIE12=E(Y1,M1)]TJ /F4 11.955 Tf 11.96 0 Td[(Y1,M10M21M31)]TJ /F4 11.955 Tf 11.96 0 Td[(Y1,M11M20M31+Y1,M10M20M31), 70

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SNIE13=E(Y1,M1)]TJ /F4 11.955 Tf 11.96 0 Td[(Y1,M10M21M31)]TJ /F4 11.955 Tf 11.96 0 Td[(Y1,M11M21M30+Y1,M10M21M30)andSNIE23=E(Y1,M1)]TJ /F4 11.955 Tf 11.96 0 Td[(Y1,M11M20M31)]TJ /F4 11.955 Tf 11.96 0 Td[(Y1,M11M21M30+Y1,M11M20M30).Asimultaneousnaturalindirecteffectbetweenthreemediators1,2and3isSNIE123=E(Y1,M1)]TJ /F4 11.955 Tf 11.95 0 Td[(Y1,M10M21M31)]TJ /F4 11.955 Tf 11.96 0 Td[(Y1,M11M20M31)]TJ /F4 11.955 Tf 11.96 0 Td[(Y1,M11M21M30+Y1,M10M20M31+Y1,M10M21M30+Y1,M11M20M30)]TJ /F4 11.955 Tf 11.96 0 Td[(Y1,M0).Usingthesecomponents,wecancomputevariousnaturalindirecteffects.IfonewantstoestimateacombinedNIEoftwomediatorsinthepresenceofthreemediators,theindividualindirecteffects,NIEk,canbeaddedandthensimultaneousindirecteffectofthosetwomediators,SNIEjk,subtracted.Forinstance,estimatingthecombinedeffectofmediator1and2giventhepresenceofmediator3isE(Y1,M1)]TJ /F4 11.955 Tf 12.39 0 Td[(Y1,M10M20M31)=NIE1+NIE2)]TJ /F1 11.955 Tf 11.95 0 Td[(SNIE12.Fork=1,2,,K,theJNIEcanbeformallydecomposedas,JNIE=KXk=1NIEk)]TJ /F14 11.955 Tf 17.04 11.35 Td[(X1jKSNIEjk+)]TJ /F5 11.955 Tf 28.56 0 Td[(()]TJ /F5 11.955 Tf 9.3 0 Td[(1)KSNIE12K. 4.2.2ProblemsoftheRegressionApproachforMultipleMediatorsMacKinnon(2008)extendedthesinglemediatorregressionmodelapproachtomultiplemediators.Foreachadditionalmediator,thisapproachrequiresonemoreregressionmodelofthismediatoronthetreatmentvariable,Z.Inthepresenceoftwomediators(M1,M2),itisnecessarytotthreelinearregressionmodels, M1=1+1ZM2=2+2Z (4) Y=3+3Z+1M1+2M2.Here,theproposedNDEis3,theNIEoftherstmediator(M1)is11,theNIEofthe 71

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secondmediator(M2)is22andtheTEisthesumofalleffects,TE=3+11+22.TheJNIEinthisframeworkisthesumoftwoindividualNIE's,JNIE=11+22.AdditivityoftheNIE'sisimplicitlyassumedinthisapproach, JNIE=nXk=1NIEk. (4) Underanassumptionofmultivariatenormalityofmediatorsoralargesamplesize,standarderrors(S.E.'s)ofNIE1andNIE2canbeestimatedusingthemultivariatedeltamethod.PreacherandHayes(2008)advocateestimatingS.E.'sinusingthebootstrap.AcleardrawbackofthestandardregressionmodelapproachistheassumptionofadditivityoftheNIE's.Inthepresenceofoverlappingeffectbetweenmediators,forexample,theeffectofM2overlapswithM1,someofeffectswillnotbeestimatedproperly.Forinstance,whenwedecomposethedirecteffectas NDE=TE)]TJ /F1 11.955 Tf 11.96 0 Td[(JNIE, (4) sincethisapproachassumesthatthejointnaturalindirecteffectsofmediators,JNIE,isthesumofallindividualindirecteffects,NIEk's,asin( 4 ),thedirecteffectisre-expressedasNDE=TE)]TJ /F9 7.97 Tf 18.3 14.94 Td[(nXk=1NIEk.However,inthepresenceofoverlappingeffectsbetweenmediators,additivityoftheNIE'sin( 4 )isnotcorrectandonewillobtainabiasedestimateofJNIE;andNDEin( 4 ).Inadditiontoadditivity,causaleffectsonthestandardregressionapproachcanbeidentiedundersequentialignorabilityassumptionwhichstatesfYzm1m2,M1z0,M2z00g?ZYz0m1M2z0?M1jZ=zYz0M1z0m2?M2jZ=z 72

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foranyz,z0,z00,m1,m2.Here,Yz0m1M2z0denotesthepotentialoutcomeundertreatmentstatusz0andthemediatorvaluesm1fortherstmediatorandM2z0,thepotentialsecondmediatorundertreatmentstatusz0.Inotherwords,theassumptionsequentiallyimpliesthattreatmentassignmentZisignorableandthattheobservedmediatorM1isignorablegiventhetreatmentandalsothatobservedmediatorM2isignorableconditionalonthetreatment.ThissequentialignorabilityassumptionisoftenweakenedbyconditioningonbaselinecovariatesX(viaaddingXintoeachlinearmodelin( 4 )).Thismaynotbealwayssuitableforstudyingbehavioralsciencessincemediatorsarepost-randomizationeventsthatgenerallycannotbecontrolled.Inthefollowingsections,weintroduceaBayesianapproach(Kimetal.,2013)whichisbasedonseveralconditionalindependenceassumptionsforcausalitywithinourframework.Thisapproachisdesignedforarandomizedtrial. 4.3InferenceontheCausalEffectsofMultipleMediatorsInthissection,weintroduceassumptions,modelspecicationandposteriorcomputationforaBayesianapproachtoinferringthecausaleffectswithintheframeworkpresentedinSection 4.2 4.3.1AssumptionsforMultipleMediatorsToidentifytheNDE,JNIE,NIE'sandSNIE's,thejointconditionaldistributionofallpotentialoutcomesandthejointdistributionofmediatorsneedtobeidentied.Thefollowingassumptionsaresufcienttoidentifythesedistributions.Again,wedenoteMz=fM1z,,MKzgandYz,Mz=Yz,M1z,M2z,,MKz.Here,weexplainallassumptionsinthepresenceofK=3mediators.Assumption1.Foragivenmediatorsunderinterventionz=1,theconditionaldistributionoftheoutcomeisthesamewhetherthosemediatorvalueswereinducedbyz=1orz=0.Forinstance,thepotentialoutcomesfortheinterventionz=1havethesame 73

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conditionaldistributionwhethercorrespondingmediatorsareinducedbyz=0orz=1asfollowsf1,M10,M20,M30(y1,M0jM0=m0,M1)=f1,M11,M21,M31(y1,M1jM0,M1=m0).Foranytwomediatorsorsinglemediatorwhichareblockedfromtheeffectoftheinterventionz=1,conditionaldistributionsofoutcomesfollowthesamerule.Forexample,thepotentialoutcomeundertheinterventionz=1buttherstandthethirdmediators'pathsfromtheinterventionblockedhasthesameconditionaldistributiontothepotentialoutcomeundertheinterventionz=1andthemediatorsallundertheinterventionz=1;f1,M10,M21,M30(y1,M10,M21,M30jM10=m1,M20,M30=m3,M11,M21=m2,M31)=f1,M11,M21,M31(y1,M1jM0,M11=m1,M21=m2,M31=m3).Similarly,foranysinglemediator'spathfromtheinterventionblocked(e.g.,undernointerventionz=0),itfollowsthesamerelationshipssuchasinthecaseofthethirdmediatorinducedunderz=0,f1,M11,M21,M30(y1,M11,M21,M30jM10,M20,M30=m3,M11=m1,M21=m2,M31)=f1,M11,M21,M31(y1,M1jM0,M11=m1,M21=m2,M31=m3).Assumption2.ThepotentialoutcomeunderZ=zisindependentofmediatorsunderZ=1)]TJ /F4 11.955 Tf 11.96 0 Td[(z,M1)]TJ /F9 7.97 Tf 6.58 0 Td[(z,giventhemediatorsunderZ=z,Mz,forz2f0,1g.Thisimplies fz,M1z,M2z,M3z(yz,Mzjmz,m1)]TJ /F9 7.97 Tf 6.59 0 Td[(z)=fz,M1z,M2z,M3z(yz,Mzjmz),(4)forz=0,1.Theequality( 4 )assumesthatforaxedzmediatorsundertheotherarm(1)]TJ /F4 11.955 Tf 12.09 0 Td[(z)donotcarryanyadditionalinformationforidentifyingtheconditionaldistributionsofthepotentialoutcome(Yz,Mz)givenmediatorsunderz. 74

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Assumption3.Thejointdistributionofpotentialmediatorshasthefollowingform:FM0,M1(m0,m1)=6[)]TJ /F6 7.97 Tf 6.59 0 Td[(11fFM10(m10)g,)]TJ /F6 7.97 Tf 6.58 0 Td[(11fFM20(m20)g,)]TJ /F6 7.97 Tf 6.59 0 Td[(11fFM30(m30)g,)]TJ /F6 7.97 Tf 6.59 0 Td[(11fFM11(m11)g,)]TJ /F6 7.97 Tf 6.59 0 Td[(11fFM21(m21)g,)]TJ /F6 7.97 Tf 6.58 0 Td[(11fFM31(m31)g],where1istheunivariatestandardnormalCDFand6isamultivariatenormalCDFwithmean0andacorrelationmatrixR.ThisspecicationcorrespondstoaGaussiancopulamodel(Nelsen,1999).Throughthismodel,wehaveabenetofexibilityinthemarginaldistributions.Althoughmediatorsunderdifferentz(Mj0,Mk1)cannotbeobservedatthesametime,weareabletoidentifythejointdistributionusingsensitivityparameters.WeconsiderapossiblerestrictionfortheseparametersasCor()]TJ /F6 7.97 Tf 6.59 0 Td[(11fFMk0(Mk0)g,)]TJ /F6 7.97 Tf 6.59 0 Td[(11fFMk1(Mk1)g)=1,Cor()]TJ /F6 7.97 Tf 6.58 0 Td[(11fFMj0(Mj0)g,)]TJ /F6 7.97 Tf 6.58 0 Td[(11fFMk1(Mk1)g)=2forj6=kandk=1,2,3.Thisrestrictionstatesthatpotentialmediatorsshareacommoncorrelation1iftheyarethesamemediator(e.g.,Mk0,Mk1)andtheyshareacommoncorrelation2iftheyaredifferentmediators(e.g.,Mk0,Mj1fork6=j).Assumption4(ConditionalIndependence)AllpotentialoutcomesthatdeneJNIE,NIEk,SNIEjk,etc.(cf.Section 4.2 )areconditionallyindependentgivenallpotentialmediatorsunderz=0,1.ThisassumptionisnotnecessarytoestimateNDE,JNIE.However,itisnecessarytoestimateotherfeaturesoftheposteriordistributionofNDEandJNIE.ItprovidesanupperboundonthevarianceoftheJNIEunderanon-negativecorrelationbetweenpotentialoutcomes.Explicitly,thedifferencebetweenthevarianceofJNIEwithassumption4andwithoutassumption4,A,isboundedbyA2Z(p V)]TJ /F4 11.955 Tf 11.96 0 Td[(W)fM0,M1(m0,m1)dm0dm1, 75

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whereV=E((Y1,M1)2jM1=m0)E((Y1,M1)2jM1=m1)andW=E(Y1,M1jM1=m0)E(Y1,M1jM1=m1).Similarly,thedifferencebetweenthevarianceofNIEkwithassumption4andwithoutassumption4,Ak,isbounded.Forexample,thevarianceofNIE1isboundedbyA12Z(p V1)]TJ /F4 11.955 Tf 11.95 0 Td[(W1)fM0,M1(m0,m1)dm0dm1,whereV1=E((Y1,M1)2jM11=m10,M21=m21,M31=m31)E((Y1,M1)2jM1=m1)andW1=E(Y1,M1jM11=m10,M21=m21,M31=m31)E(Y1,M1jM1=m1).Forderivations,seetheAppendix H 4.3.2IdenticationandModelSpecicationInthefollowing,wewillprovethatAssumption1-4aresufcienttoidentifythejointposteriordistributionofNDE,JNIE,NIE'sandSNIE's.Theorem1.ThejointposteriordistributionofNDE,JNIE,NIEk,SNIEjkandSNIE123isidentiedunderAssumptions1-4.Proof:Fornotationalsimplicity,letYzz0z00z000representYz,M1z0,M2z00,M3z000.Then,thejointdistributionofthetwopotentialoutcomescanbefactoredasfollows:f(0,M10,M20,M30),(1,M11,M21,M31),M0,M1(y0000,y1111,m0,m1)=f(0,M10,M20,M30),(1,M11,M21,M31)(y0000,y1111jm0,m1)fM0,M1(m0,m1).TheRHScanbefurtherfactoredas f(0,M10,M20,M30),(1,M11,M21,M31)(y0000,y1111jm0,m1)fM0,M1(m0,m1)=f(0,M10,M20,M30)(y0000jm0,m1)f(1,M11,M21,M31)(y1111jm0,m1)fM0,M1(m0,m1)(A4)=f(0,M10,M20,M30)(y0000jm0)f(1,M11,M21,M31)(y1111jm1)fM0,M1(m0,m1)(A2)=f(0,M10,M20,M30)f(y0000jm0)f(1,M11,M21,M31)(y1111jm1)fM0,M1(m0,m1), (4) where`A'correspondstoAssumption.Eachcomponentin( 4 )canbeidentiedbyrandomizationand/orAssumption3. 76

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ToobtainthejointposteriordistributionofNDE,JNIE,NIE'sandSNIE's,weneedtoidentify f(y1111,y0000,y1000,y1100,y1010,y1001,y1110,y1101,y1011)= (4) Zf(y1111,y0000,y1000,y1100,y1010,y1001,y1110,y1101,y1011jm0,m1)fM0,M1(m0,m1)dm0dm1.ThesecondtermintheintegrandcanbeidentiedbyAssumption3.Thersttermintheintegrandisfactoredas f(y1111,y0000,y1000,y1100,y1010,y1001,y1110,y1101,y1011jm0,m1)=f1,M11,M21,M31(y1111jm0,m1)f0,M10,M20,M30(y0000jm0,m1)f1,M10,M20,M30(y1000jm0,m1)f1,M11,M20,M30(y1100jm0,m1)f1,M10,M21,M30(y1010jm0,m1)f1,M10,M20,M31(y1001jm0,m1)f1,M11,M21,M30(y1110jm0,m1)f1,M11,M20,M31(y1101jm0,m1)f1,M10,M21,M31(y1011jm0,m1)(A4)=f1,M11,M21,M31(y1111jm1)f0,M10,M20,M30(y0000jm0,)f1,M11,M21,M31(y1111jM1=m0)f1,M11,M21,M31(y1111jM11=m1,M21=m20,M31=m30)f1,,M11,M21,M31(y1111jM11=m10,M21=m2,M31=m30)f1,,M11,M21,M31(y1111jM11=m10,M21=m20,M31=m3)f1,,M11,M21,M31(y1111jM11=m1,M21=m2,M31=m30)f1,,M11,M21,M31(y1111jM11=m1,M21=m20,M31=m3)f1,,M11,M21,M31(y1111jM11=m10,M21=m2,M31=m3), (4) wheretheequalityin( 4 )holdsbyAssumptions1and2.Allcomponentsin( 4 )areidentiedbyrandomizationandobserveddata.ThejointposteriordistributionofNDE,JNIE,NIE'sandSNIE'sisthefunctionofthejointdistributionin( 4 ).Fromthejointdistribution,wecanobtainthemarginaldistributionofeacheffectaswell. 4.3.3SpecicationofModelsTheparametersofthemodelsrequiredforinferencesintheBayesianapproachcanbeestimatedusingtheobserveddata.Here,weassumetheoutcomeisdichotomous 77

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(asinourdataexample).Wetrytominimizeparametricassumptionsinourspecication.Inparticular,forthemarginaldistributionofthepotentialoutcomes,wespecify fz,M1z,M2z,M3z(Y1,Mz=1)Ber(z)z=0,1. (4) withzUnif(0,1)priors.WespecifyDirichletprocesspriorsforthemarginaldistributionsoftheindividualmediators, FMkz(mkz)DP(k0Gk0), (4) whereGk0isabasemeasurewithcorrespondingprecisionparametersk0.WeplaceUnif(0,10)priorsontheprecisionparameters.Thejointdistributionofpotentialmediators,FM0,M1(m0,m1),isspeciedbyAssumption3withthemarginaldistributionsin( 4 ).Additionally,weneedthejointdistributionofallmediatorsandresponseundertheinterventionz=1,F(1,M1),M1(y1,M1,m1),whichisspeciedasF(1,M1),M1(y1,M1,m1)=4[)]TJ /F6 7.97 Tf 6.58 0 Td[(11fF(1,M1)(y1,M1)g,)]TJ /F6 7.97 Tf 6.59 0 Td[(11fFM11(m1)g,)]TJ /F6 7.97 Tf 6.58 0 Td[(11fFM21(m2)g,)]TJ /F6 7.97 Tf 6.59 0 Td[(11fFM31(m3)g],whereweaddtheresponseY1,M1toaCopulamodel. 4.4PosteriorComputationsintheCausalEffectsofMultipleMediatorsTosamplefromtheposteriordistributionofthedirectandindirecteffects,weusethefollowing4steps. 1. Fixthesensitivityparameters,1and2,inAssumption3. 2. Giventhesensitivityparameters,wetakeNsamplesof[FMkz,FMk1jY,z]p(FMkz,FMk1jY,jmobs,yobs)whereMobs=fZiM1+(1)]TJ /F4 11.955 Tf 12.87 0 Td[(Zi)M0,i=1,,ngandYobs=fYzi,Mzi,i=1,,ngusingWinBUGS. 3. ForeachsamplefromStep2,computethedirectandindirecteffectsasfollows a. GenerateQsetsoffM0,M1g(=fM10,M20,M30,M11,M21,M31g)bysamplingfromFM0,M1(m0,m1)whichcanbeobtainedbyAssumption3. 78

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b. Computef(1,M1z1M2z2M3z3)(y)forz1,z2,z32(0,1)viaMonteCarlointegrationusingtheQsetsoffM0,M1gasfollows, f(1,M1z1M2z2M3z3)(y)=1 QQXi=1f(1,M11,M21,M31)(yjm1z1,i,m2z2,i,m3z3,i) (4) Adetailedderivationof( 4 )isinAppendix I .Tocomputef1,M11,M21,M31(yjM11=m1z1,M21=m2z2,M31=m3z3),werstnotethatf(1,M11,M21,M31)(y1,M1jM11=m1,M21=m2,M31=m3)=f(1,M11,M21,M31),M1(y1,M1,m1,m2,m3) fM1(m1,m2,m3),wherefM1(m1,m2,m3)=f(1,M11,M21,M31),M1(y1,M1=1,m1,m2,m3)+f(1,M11,M21,M31),M1(y1,M1=0,m1,m2,m3). c. ComputetheNDE,JNIE,NIE1(otherindirecteffectscanbeestimatedsimilarly),NDE=E(Y1,M0))]TJ /F4 11.955 Tf 11.96 0 Td[(E(Y0,M0)=f(1,M10,M20,M30)(y1,M0=1))]TJ /F4 11.955 Tf 11.96 0 Td[(f(0,M10,M20,M30)(y0,M0=1),JNIE=E(Y1,M1))]TJ /F4 11.955 Tf 11.96 0 Td[(E(Y1,M0)=f(1,M11,M21,M31)(y1,M1=1))]TJ /F4 11.955 Tf 11.96 0 Td[(f(1,M10,M20,M30)(y1,M0=1),NIE1=E(Y1,M1))]TJ /F4 11.955 Tf 11.96 0 Td[(E(Y1,M10,M21,M31)=f(1,M11,M21,M31)(y1,M1=1))]TJ /F4 11.955 Tf 11.95 0 Td[(f(1,M10,M21,M31)(y1,M10,M21,M31=1). 4. Repeatsteps3Ntimestoobtainposteriorsamplesofthedirecteffect(NDE)andindirecteffects(JNIE,NIE,SNIE,etc.).Ifweplacepriorsonthesensitivityparameters,1and2,Step1isreplacedbysamplingfromthepriorandStep4becomesrepeatSteps1-3Ntimes.Inthefollowingsection,weusetheBayesianapproachtoassessmultiplemediatorsintheSTRIDEtrial. 4.5MediationintheSTRIDETrial 4.5.1DescriptionofSTRIDEWeanalyzemediationinuseSTRIDE(Marcusetal.,2007)whichwasdesignedtoevaluatetheefcacyofinterventionstargetingphysicalactivityadoptionandmaintenance.Participantswererandomizedto3treatmentarms:telephone-basedintervention,printbasedinterventionandcontact-controlgroup.Inouranalysis,wecombinetelephone-basedandprint-basedinterventionsintoasingleintervention 79

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group.TheSTRIDEdatahad239subjects:n0=78subjectswereassignedtothecontact-controlgroupandn1=161weresubjectswereassignedtotheintervention.Atbaseline,6,and12months,participants'physicalactivitylogswerereportedbymailalongwithbriefquestionnairesaboutself-efcacy,processesofchangeanddecisionalbalance.Thebinaryprimaryoutcomewasdenedas150minutesperweekormoreofmoderateintensityphysicalactivityofaparticipantsat12months(participantswereassumedactiveiftheyhadactivity150minutesormoreperweekinMarcusetal.2007).Here,potentialmediatorswerebehavioralprocesses,cognitiveprocesses,self-efcacy,anddecisionbalancemeasuredbyquestionnairesatmonths6.Amongthosepsychosocialvariable,weexaminethreepotentialmediators:self-efcacy(M1),behavioralprocesses(M2)andcognitiveprocesses(M3)whichweremeasuredbyasummarystatisticofthequestionnairesona5pointscalerangingfromnever(1)toalways(5).Inthedata,188participantshadcompleteinformation.Forsimplicity,baselinevalueswerecarriedforwardformissingmediatorvaluesatmonth6and6monthvalueswerecarriedforwardformissingoutcomes. 4.5.2PriorsandSensitivityParametersFortheDPPmodelin( 4 ),wespecifyFMkz(mkz)DP(kz,Beta[1,5](kz,kz))forz=0,1;k=1,2,3,wherethebasemeasureisashiftedBetadistributionontheinterval[1,5]sincemediatorshavevaluesrangingfrom1to5.WeplaceUnif(0,10)priorsonkzandkzforz=0,1;k=1,2,3.WeplaceUnif(1,10)priorsontheprecisionparameterskzforz=0,1andk=1,2,3.Sincemediatorsunderz=0andz=1cannotbeobservedatthesametime,weusethesensitivityparameterstoconstructthejointdistributionofmediatorsinAssumption3.Amongmanypossiblespecicationsoftheassociationparameterbetweenmediatorsunderz=0andz=1,weusetherestrictionof1and2suggested 80

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inAssumption3.Tomakethecovariancematrixof(M10,M20,M30,M11,M21,M31)Tpositivedenite,sensitivityparameters1and2areonlyfreetovaryinthecertainranges;seeFigure 4-2 .Here,weconsiderauniformpriorontheregioninFigure 4-2 forsensitivity Figure4-2. Possiblerangesof1and2forthepositivedenitecovariancematrixw/andw/o1>2 parameters.Inadditiontothat,weconsidertwocaseswiththesensitivityparametersxedand1>2:1=0.3and2=0.15;1=0.6and2=0.45. 4.5.3ResultsofMultipleMediatorsForStep2ofthealgorithminSection 4.4 ,weran15000iterationsinWinBUGSanddiscardingtherst5000asburn-in.TheresultsareinTable 4-1 4-2 and 4-3 .Intheanalysis,thetotaleffectis0.181with95%credibleinterval(0.08,0.25)whichimpliesthattheintervention(print-basedandtelephone-based)hasasignicantpositiveeffectonsubject'sphysicalactivity.Wealsoseeadirecteffectrangingfrom-0.004to-0.003with95%C.I.(-0.10,0.07)and(-0.10,0.08),respectively.Thejointindirecteffect 81

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rangesfrom0.184(0.14,0.23)to0.185(0.15,0.22)whichindicatethejointindirecteffecthasastatisticallysignicanteffect.Mediatorspecicindirecteffects,incommonwiththejointindirecteffect,arestatisticallysignicantexcepttheeffectofcognitiveprocesses,textNIE3.Theeffectofself-efcacy,NIE1,variesfrom0.069(0.04,0.09)to0.071(0.05.0.09)andtheeffectofbehavioralprocesses,NIE2,variesfrom0.142(0.11,0.17)to0.143(0.11,0.17)andtheeffectofcognitiveprocesses,NIE3,is0.002(-0.01,0.02),whichindicateself-efcacy,behavioralprocessesmediatetheeffectofinterventiononphysicalactivity.InTable 4-3 ,thesimultaneouseffectbetweenself-efcacyandbehavioralprocesses(SNIE12)isstatisticallysignicantwithestimatesrangingfrom0.022(0.01,0.04)to0.025(0.01,0.04)with95%C.I.sintheparentheses.Theresultswerefairlyrobusttothesensitivityparameter1and2inestimatingTE,JNIEandNDE.However,inNIE'sandSNIE's,whenthedegreeofassociationswasassumedtobe1=0.3and2=0.15,theeffectsdepartedmorefromtheeffectsestimatedunderothercasesfor1and2.Mediator-specicindirecteffectsandsimultaneouseffectsbetweenpairsdecreasedasthecorrelationparameters1and2increased. Table4-1. Bayesianestimatesand95%credibleintervals(inparentheses)ofTE,JNIEandNDEforeachcase:(1)auniformpriorontheregioninFigure 4-2 ;(2)1=0.3and2=0.15;(3)1=0.6and2=0.45. CaseTEJNIENDE (1)0.181(0.08,0.25)0.185(0.15,0.22)-0.004(-0.10,0.07)(2)0.181(0.08,0.25)0.184(0.14,0.23)-0.003(-0.10,0.07)(3)0.181(0.08,0.25)0.184(0.15,0.22)-0.003(-0.10,0.08) Table4-2. Bayesianestimatesand95%credibleintervals(inparentheses)ofNIE'sforeachcase:(1)auniformpriorontheregioninFigure 4-2 ;(2)1=0.3and2=0.15;(3)1=0.6and2=0.45. CaseNIE1NIE2NIE3 (1)0.069(0.04,0.09)0.143(0.11,0.17)0.002(-0.01,0.01)(2)0.071(0.05,0.09)0.142(0.11,0.17)0.002(-0.02,0.02)(3)0.069(0.05,0.09)0.142(0.11,0.17)0.002(-0.01,0.02) 82

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Table4-3. Bayesianestimatesand95%credibleintervals(inparentheses)ofSNIE'sforeachcase:(1)auniformpriorontheregioninFigure 4-2 ;(2)1=0.3and2=0.15;(3)1=0.6and2=0.45. CaseSNIE12SNIE13SNIE23SNIE123 (1)0.022(0.01,0.04)0.009(-0.01,0.03)0.007(-0.01,0.02)0.009(-0.01,0.02)(2)0.025(0.01,0.04)0.012(-0.01,0.03)0.006(-0.02,0.02)0.011(-0.01,0.03)(3)0.022(0.01,0.04)0.011(-0.01,0.02)0.007(-0.01,0.02)0.011(-0.01,0.03) 4.5.4ComparisonwiththeRegressionApproachForcomparison,wealsoestimatedthecausaleffectsinthepresenceofmultiplemediatorsusingtheregressionapproachinSection 4.2.2 undertheassumptionofadditivity.Weusedbootstrappingtoderivethestandarderrorofthecausaleffects.Thenaturaldirecteffectwasestimatedtobe0.084(-0.01,0.18)andthejointindirecteffectswasestimatedtobe0.105(0.06,0.15),verydifferentthantheanalysisinSection 4.5.3 .Allmediator-specicindirecteffectswerenotstatisticallysignicant.However,theassumptionsofadditivityoftheNIE'sisunlikelytobereasonableifthereareoverlappingeffects. Table4-4. Estimatesand95%C.I.'s(inparentheses)ofthetotaleffect(TE),thejointindirecteffect(JNIE)andthedirecteffectfromtheregressionapproach TEJNIENDE 0.188(0.09,0.28)0.105(0.06,0.15)0.084(-0.01,0.18) Table4-5. Estimatesand95%C.I.'s(inparentheses)ofthemediator-specicindirecteffects(NIE)fromtheregressionapproach NIE1NIE2NIE3 0.026(-0.01,0.06)0.102(0.04,0.17)-0.023(-0.06,0.01) 4.6DiscussionontheCausalEffectsofMultipleMediatorsWehaveproposedaBayesianapproachforthecausaleffectsofmultiplemediators.FortheSTRIDEdata,theeffectofthetelephoneandprintbasedinterventionvs.thecontractcontrolwasmarginallysignicant.Basedontheanalysis,threepotentialmediators,self-efcacy,behavioralprocessesandcognitiveprocesses,havea 83

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signicantjointeffectonphysicalactivity.Also,themediatorsexceptcognitiveprocesseshavethemediator-specicindirecteffects.Thereareseveralextensionstothecurrentapproach.First,wemightincorporatesensitivityparametersinAssumption1suchthattwoconditionaldistributionsarenotequalwhentheinterventionhasalargeeffectontheensembleofmediators.Second,wecouldincorporatebaselinecovariatestoweakensomeofourassumptions.Third,wemightconsiderthecaseofcausallyrelatedmultiplemediatorswhichaccommodatesmediatorsmeasuredsequentially.WeareworkingonmakingtheBayesianmethodsavailableasanRpackage. 84

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CHAPTER5CONCLUSIONInthisthesis,wehaveproposedBayesianapproachestoinferringthecausaleffectsofasinglemediator,alongitudinalmediator,andmultiplemediators.Inthesettingofasinglemediator,thecausaleffectsofinterestwereidentiedbyseveralconditionalindependenceassumptionswithsensitivityparameters.AGaussiancopulamodelwasusedtoidentifythejointdistributionoftwomediators,M0andM1,whichallowedforexiblespecicationofthemarginaldistributions.Inthelongitudinalmediationanalysis,weusedaBayesiandynamicmodeltoupdateobserveddatainadditiontousingmodiedassumptionsfromthesinglemediatorcase.Finally,inthecaseofmultiplemediators,anewframeworkwasproposedtodenethejointnaturalindirecteffect,mediator-specicindirecteffectsandsimultaneousindirecteffectsaswellasthetotaleffectandthedirecteffect.WeappliedtheseapproachestotheTOURSweightmanagementtrial,theCTQIIsmokingcessationtrial,andtheSTRIDEphysicalactivitypromotionprojectforasinglemediator,alongitudinalmediatorsandmultiplemediators,respectively.InSection 5.1 ,wesummarizeourresearchcontributions.InSection 5.2 ,futureresearchisdiscussed. 5.1SummaryofContributionsWesummarizethemajorcontributionofthisthesisintothreeparts.First,ourapproachrelaxesparametricassumptionswhichareanessentialcomponentoftheregressionapproach.WeusedBayesiannonparametricstomodelthemarginaldistributionofpotentialmediators.ThejointdistributionofthemediatorswasmodeledwithaGaussiancopulamodelwithsensitivityparameter(s),toallowfortheexibilityinthemarginals.Wealsoprovidestrategiestoelicitthesensitivityparameters.Second,ourapproachdoesnotassumesequentialignorabilitywhichisimplicitlyassumedinmostapproaches.Insteadofassumingsequentialignorability,weassumedtheconditionaldistributionsofunobservablepotentialoutcomesareidentiedby 85

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equatingthemtotheconditionaldistributionoftheobservablepotentialoutcomeuptosensitivityparameters.Inparticular,westratiedthepopulationintothoseforwhomthetreatmenthasalargeeffectandsmalleffectonmediators(asquantiedbyasensitivityparameter).Thismakestheassumptionlessrestrictiveandallowsustoassesssensitivityofinferencestothisassumption.Third,inmostregressionapproachesforlongitudinalmediation(Maxwell,2003;MuchenandCurran,1997;Cheongetal.,2003),theyonlyestimatecross-sectionaldirectandindirecteffects.However,ourapproachestimatesdirectandindirecteffectsateachtimeaswellascross-sectionaleffectsbyusingaBayesiandynamicmodel.Moreover,ourapproachnotonlymeasuresthedirectandindirecteffectsateachtimebutalsoupdatesthedirectandindirecteffectusingthepastinformation. 5.2FutureResearchThereareseveralpossibleextensionstothecurrentapproach.First,wecanincorporatebaselinecovariatestoweakensomeofassumptions.Inparticular,wecansimplyaddthecovariatesintotheconditioningoftheassumptions.Apossiblecomplexationarisesfrommodelingtheconditionaldistributionsofmediatorsgivencovariates.Withinourcurrentnonparametricapproaches,itmaybehardtospecifythedistributionswhencovariatesarecontinuous.WeplantodevelopmethodstoovercomethismatterpossiblyusingideasforMulleretal.(2011).Second,wemightincorporatetime-varyingtreatmentsinthelongitudinalmediationanalysis.ThiswouldrequiremodifyingtheobservationandtheevolutionmodelswithintheBayesiandynamicmodel.Also,currentassumptionswouldneedtobecarefullymodiedwiththetime-varyingtreatments.Third,inthecaseofmultiplemediators,wecanusesensitivityparametersinAssumption1tostratifythepopulationintotwocasesintermsofwhetherthetreatmenthasalargeeffectonthesetofmediators.Forthis,weneedtocarefullyquantifythetreatmenteffectontheensembleofmediatorssincewenowhaveavectorofmediators 86

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undereachtreatmentarm.WewillconsidertheMahalanobisdistanceofmediatorsasonepossibleapproach. 87

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APPENDIXACOMPARISONOFVARIANCES A.1VariancewithoutAssumption5FirstnotethatE(Y1,M1Y1,M0)=Zf(1,M1),(1,M0)(y11=1,y10=1jm0,m1)fM0,M1(m0,m1)dm0dm1,wheref(1,M1),(1,M0)(y11=1,y10=1jm0,m1)=f1,M0(y10=1jy11=1,m0,m1)f1,M1(y11=1jm0,m1).Now,assumef1,M1(y11=1jm0,m1)>0andanon-negativecorrelationbetweenY11andY10givenm0andm1.ThenthecorrelationbetweenY11andY10conditionalon(m1,m0),,is=E(Y10Y11jm0,m1))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y10jm0,m1)E(Y11jm0,m1) s.d.(Y10)s.d.(Y11)=f(1,M0),(1,M1)(y10=1,y11=1jm0,m1))]TJ /F4 11.955 Tf 11.95 0 Td[(f1,M0(y10=1jm0,m1)f1,M1(y11=1jm0,m1) s.d.(Y10)s.d.(Y11)0,wheres.d.(Y10)=p f1,M0(y10=1jm0,m1)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(f1,M0(y10=1jm0,m1))ands.d.(Y11)=p f1,M1(y11=1jm0,m1)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(f1,M1(y11=1jm0,m1)).Notethatf(1,M0),(1,M1)(y10=1,y11=1jm0,m1)=f1,M0(y10=1jm0,m1)f1,M1(y11=1jm0,m1)+s.d.(Y10)s.d.(Y11).Thiscanbere-expressedasf(1,M0),(1,M1)(y10=1,y11=1jm0,m1)=expflogsgn(d)I(jdj)gf1,M1(y11=1jm0)f1,M1(y11=1jm1)+s.d.(Y10)s.d.(Y11),sincef1,M1(y11=1jM0=m0,M1=m1)=f1,M1(y11=1jM1=m1)byAssumption3. 88

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Usingtheseresults,thevarianceoftheNIEwithoutAssumption5is Var(NIE)=E(Y21,M1))]TJ /F5 11.955 Tf 11.96 0 Td[(2E(Y1,M1Y1,M0)+E(Y21,M0))-222(fE(Y1,M1))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y1,M0)g2=E(Y21,M1)+E(Y21,M0))-222(fE(Y1,M1))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y1,M0)g2)]TJ /F4 11.955 Tf 11.96 0 Td[(C1, (A) whereC1=2Rexpfsgn(d)logI(jdj)gf1,M1(y11=1jm0)f1,M1(y11=1jm1)fM0,M1(m0,m1)dm0dm1+2Rs.d.(Y10)s.d.(Y11)fM0,M1(m0,m1)dm0dm1. A.2VariancewithAssumption5NotethatE(Y1,M1Y1,M0)=Zf(1,M1),(1,M0)(y11=1,y10=1jm0,m1)fM0,M1(m0,m1)dm0dm1,wheref(1,M1),(1,M0)(y11=1,y10=1jm0,m1)=expfsgn(d)logI(jdj)gf1,M1(y11=1jm1)f1,M1(y11=1jm0),byAssumption3and5andthefactthat=0.Usingthisresult,thevarianceoftheNIEwithAssumption5,Var(NIE)w,is Var(NIEw)=E(Y21,M1))]TJ /F5 11.955 Tf 11.96 0 Td[(2E(Y1,M1Y1,M0)+E(Y21,M0))-222(fE(Y1,M1))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y1,M0)g2=E(Y21,M1)+E(Y21,M0))-222(fE(Y1,M1))]TJ /F4 11.955 Tf 11.96 0 Td[(E(Y1,M0)g2)]TJ /F4 11.955 Tf 11.95 0 Td[(C2, (A) whereC2=2Rexpflogsgn(d)I(jdj)gf1,M1(y11=1jm0)f1,M1(y11=1jm1)fM0,M1(m0,m1)dm0dm1. A.3ComparisonofVarianceswithandwithoutAssumption5Comparing( A )with( A ),thedifferenceisC1-C2,C1=2Zexpfsgn(d)logI(jdj)gf1,M1(y11=1jm1)f1,M1(y11=1jm0)fM0,M1(m0,m1)dm0dm1+2Zs.d.(Y10)s.d.(Y11)fM0,M1(m0,m1)dm0dm12Zexpfsgn(d)logI(jdj)gf1,M1(y11=1jm0)f1,M1(y11=1jm1)fM0,M1(m0,m1)dm0dm1=C2 89

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whichisnon-negative.Thus,thevarianceoftheNIEwithoutAssumption5hasasmallervariance;Var(NIE)
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APPENDIXBIMPLEMENTATIONOFDIRICHLETPROCESSPRIORSINBUGSWeusethefollowingconstructionoftheDirichletprocessparametersforimplementationinWinBUGS,iBeta(1,Kz),i=ii)]TJ /F6 7.97 Tf 6.59 0 Td[(1Yl=1(1)]TJ /F2 11.955 Tf 11.95 0 Td[(l),iWzBeta[0,350](1z,1z)+(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Wz)Beta[0,350](2z,2z),Gz=MXi=1iiandfMz,y(mzjYz,Mz=y)Gz,wheretheprecisionparameterKzhasauniformprior,DiscUnif[1,20].SincethemediatorinTOURStakesvaluesin[0,350],wespecifyi=Q(i)andGz=MXi=1ii,wherefunctionQ:(0,350)!()]TJ /F5 11.955 Tf 9.3 0 Td[(0.5,350+0.5).WethenspecifyfMz,y(mzjYz,Mz=y)Poisson(S),whereScategorical(1,2,,K). 91

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APPENDIXCDETAILSOFPOSTERIORCOMPUTATIONForeachsample(FM1,1,FM1,0,FM0,1,FM0,0,1,M1,0,M0), 1. GenerateLsets(M0,M1).ThisrequiressamplingfromfMz,Mz0(mz,mz0)=fMz(mzjmz0)fMz0(mz0).NotethatintheTOURStrial,Mzisactuallydiscrete(takingintegervalues0to350),sowecomputefMz(mz)asfollows,fMz(mz)=FMz(mz+0.5))]TJ /F4 11.955 Tf 11.96 0 Td[(FMz(mz)]TJ /F5 11.955 Tf 11.95 0 Td[(0.5);z=0,1whereFMz=FMz,11,M1+FMz,0(1)]TJ /F2 11.955 Tf 11.95 0 Td[(1,M1).WesampleMz0usingFMz0(Mz0)Unif(0,1).Then,givenMz0,weobtainMzusingtheconditionalCDFFMz(mzjmz0)=mzXt=0fMz(tjmz0)=Pmzt=0fMz,Mz0(t,mz0) fMz0(mz0)=FMz,Mz0(mz+0.5,mz0+0.5))]TJ /F4 11.955 Tf 11.95 0 Td[(FMz,Mz0(mz+0.5,mz0)]TJ /F5 11.955 Tf 11.95 0 Td[(0.5) FMz0(mz0+0.5))]TJ /F4 11.955 Tf 11.95 0 Td[(FMz0(mz0)]TJ /F5 11.955 Tf 11.96 0 Td[(0.5)usingthefactFMz(Mzjmz0)Unif(0,1).ThedensityfMz,Mz0(mz,mz0)canbecomputedinthesamemannerwithAssumption4. 2. Computef1,M0(y)viaMonteCarlointegrationusingtheLsets(M0,M1)asfollowsf1,M0(y)=Zf1,M0(yjm0,m1)fM0,M1(m0,m1)dm0dm1=1 CZexpfsgn(d)logyI(jdj)gf1,M1(yjm0,m1)fM0,M1(m0,m1)dm0dm1(A2)=1 CZexpfsgn(d)logyI(jdj)gf1,M1(yjm0)fM0,M1(m0,m1)dm0dm1(A3)1 CLXi=1expfsgn(di)logyI(jdij)gf1,M1(yjm0,i),where'A'correspondsto'Assumption'andthenormalizingconstantCisC=LXi=1expfsgn(di)logI(jdij)gf1,M1(y=1jm0,i)fM0,M1(m0,i,m1,i)+LXi=1f1,M1(y=0jm0,i)fM0,M1(m0,i,m1,i). 92

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Tocomputef1,M1(yjm0),f1,M1(yjm0)=y1,M1(1)]TJ /F2 11.955 Tf 11.95 0 Td[(1,M1)1)]TJ /F9 7.97 Tf 6.58 0 Td[(yfM1,Y1(M1=m0jY1,M1=y) fM1(M1=m0)wherefM1,Y1(M1=m0jY1,M1=y)=FM1,y(m0+0.5))]TJ /F4 11.955 Tf 11.96 0 Td[(FM1,y(m0)]TJ /F5 11.955 Tf 11.95 0 Td[(0.5)andfM1(M1=m0)=fM1,Y1(M1=m0jY1,M1=1)1,M1+fM1,Y1(M1=m0jY1,M1=0)(1)]TJ /F2 11.955 Tf 11.95 0 Td[(1,M1). 3. Computethedirectandindirecteffectsusing1,M0)]TJ /F2 11.955 Tf 12.03 0 Td[(0,M0and1,M1)]TJ /F2 11.955 Tf 12.03 0 Td[(1,M0,where1,M0=f1,M0(y=1). 93

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APPENDIXDORDINALCATEGORICALMEDIATOR D.1Assumption3foranOrdinalMediatorSinceM1tandM0tcannotbeobservedatthesametime,thejointdistributionoftwomediatorsneedsaspecialassumptionwithasensitivityparameterforassociation.IfthereareJlevelsintheordinalmediatorforeachz=0,1,weassumethejointdistributionofthemediatorsateachtime,FM0t,M1t(M0t=i,M1t=jjt+1)canbeobtainedbyaloglinearmodel,logft,ij(t+1)g=0t,i+1t,j+0t,i1t,j.Then,thejointdistributionofmediatorsisfM0t,M1t(M0t=i,M1t=jjt+1)=t,ij(t+1)=exp(0t,i+1t,j+0t,i1t,j) PPexp(0t,+1t,+0t,1t,)fori,j=1,,J.Here,isthesensitivityparameterfortheassociationbetweenthemediatorsfromz=0,1.Notethat=0speciestheindependencemodel.Withoutlossofgenerality,weassumemediatorshave4categories.Sincethereare6identiedmultinomialparameters( 0t,1, 0t,2, 0t,3, 1t,1, 1t,2, 1t,3)fromthemarginaldistributionsofM0tandM1t,wecanestimate0t,iand1t,jbymarginalizingtheabovejointprobabilityandequatingittomarginalprobabilities.Tomakethisidentiable,twoconstraintsareneeded:0t,4=C0(someconstant);1t,4=C1(someconstant),then, 0t,i=Pjexp(0t,i+1t,j+0t,i1t,j) PPexp(0t,+1t,+0t,1t,)fori=1,2,3and 1t,j=Piexp(0t,i+1t,j+0t,i1t,j) PPexp(0t,+1t,+0t,1t,)forj=1,2,3. 94

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ByusingtheNewton-Raphsonmethod,theycouldbecalculatednumerically.Theobservationmodelsfortheresponseandthemediatorcanbespeciedas(Yz,Mzt)]TJ /F13 5.978 Tf 5.76 0 Td[(1t,Yz,Mzt)]TJ /F13 5.978 Tf 5.76 0 Td[(2t)]TJ /F6 7.97 Tf 6.59 0 Td[(1)Mult(z1t,z2t,z3t),logitP(Mzt)]TJ /F6 7.97 Tf 6.59 0 Td[(2jjyzzt,yzzt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)=zjt+z1tyzzt+z2tyzzt)]TJ /F6 7.97 Tf 6.58 0 Td[(1and logitP(Mzt)]TJ /F6 7.97 Tf 6.58 0 Td[(1jjmzt)]TJ /F6 7.97 Tf 6.59 0 Td[(2,yzzt,yzzt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)=zjt+z3tmzt)]TJ /F6 7.97 Tf 6.59 0 Td[(2+z4tyzzt+z5tyzzt)]TJ /F6 7.97 Tf 6.58 0 Td[(1,(D)wherez1tz2tzJtandz1tz2tzJtfort=1,,Tandz=0,1. D.2SpecicationofModelsTheevolutionmodelupdatesparameters,t=(zt,zt,zt,zt),foreachtimet.Wecanspecifythefollowingevolutionmodels,ztjzt)]TJ /F6 7.97 Tf 7.59 0 Td[(1Dir(ztzt)]TJ /F6 7.97 Tf 7.59 0 Td[(1),z1tjz1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1N(z1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,2t),andz1tjz1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1N(z1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1, 2t).Theremainingzjtandzjtforj2canbeupdatedbyzjt=z1t+Pjk=2WktsuchthatWktexponential(Wkt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)andzjt=z1t+Pjk=2VktsuchthatVktexponential(Vkt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)forj=2,3,,J.Inthisway,theordersofparametersarepreserved.Lastly,ztN(zt)]TJ /F6 7.97 Tf 7.59 0 Td[(1,zt).Wespecifyuniformshrinkagepriors(Strawderman,1971;Daniels,1999)forthehyperparametersztwhichcontrolthe`smoothness'oftheevolutionofparameterszt. D.3SensitivityAnalysisToelicitpossiblevaluesof,wenotethattheprobabilityof(M0t=1,M1t=1)fromtheJ2gridsis fM0t,M1t(m0t=1,m1t=1)=t,11=t,11 nt=1 ntexp(0t,1+1t,1+01t,1t,1) (D) 95

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wherentisthenumberofsets(M0t,M1t).Inmodelingofthejointdistributionofordinalcategoricalmediators,itisrequiredtosetrestrictionsonparametersforidentiability(asdescribedearlier).Ifwelet0t,1=c1and1t,1=c2asrestrictionswithsomearbitraryconstantsc1andc2,then( D )canbesimpliedto=logntt,11)]TJ /F4 11.955 Tf 11.96 0 Td[(c1)]TJ /F4 11.955 Tf 11.95 0 Td[(c2 c1c2.Thus,aplausiblerangeofdependsonpreliminaryexpectedvaluesoft,11withpredeterminedrestrictions,c1andc2. 96

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APPENDIXECOMPARISONOFVARIANCESINLONGITUDINALMEDIATION E.1VariancewithoutAssumption4inLongitudinalMediationFirstnotethatE(Y11tY10tjt)=ZE(Y11tY10tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)f(m0t,m1tjt)dm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Now,assumef1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11tjm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)>0forally11tandanon-negativecorrelationbetweenY11tandY10tgivenM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1andt.ThenthecorrelationbetweenY11tandY10tconditionalon(M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t),rt,isrt=E(Y10tY11tjM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t))]TJ /F4 11.955 Tf 11.96 0 Td[(E(Y10tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)E(Y11tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t) p Var(Y10tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)p Var(Y11tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)=E(Y10tY11tjM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t))]TJ /F4 11.955 Tf 11.96 0 Td[(E(Y10tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)E(Y11tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t) s.d.(Y10t)s.d.(Y11t)0,wheres.d.(Y10t)=p Var(Y10tjM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)ands.d.(Y11t)=p Var(Y11tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t).Then,E(Y10tY11tjM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)=E(Y10tjM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)E(Y11tjM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)+rts.d.(Y10t)]TJ /F6 7.97 Tf 6.59 0 Td[(1)s.d.(Y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1).Thiscanbere-expressedasE(Y10tY11tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)=Zy10texpfsgn(dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)logI(jdt)]TJ /F6 7.97 Tf 6.58 0 Td[(1j)y10tgf1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y10tjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)dy10tZy11tf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11tjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)dy11t+rts.d.(Y10t)s.d.(Y11t).Usingtheseresults,thevarianceoftheNIEt(t)withoutAssumption4is Var(NIEtjt)=E(fY11tg2jt))]TJ /F5 11.955 Tf 11.95 0 Td[(2E(Y11tY10tjt)+E(fY10tg2jt))-222(fE(Y11tjt))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y10tjt)g2=E(fY11tg2jt)+E(fY10tg2jt))-222(fE(Y11tjt))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y10tjt)g2)]TJ /F4 11.955 Tf 11.95 0 Td[(C1. (E) 97

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whereC1=2RfRy10texpflogsgn(dt)]TJ /F13 5.978 Tf 5.76 0 Td[(1)I(jdt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j)y10tgf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y10tjM1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)dy10tRy11tf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11tjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)dy11tfM0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)gdm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1+2Rrts.d.(Y10t)s.d.(Y11t)fM0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1jt)dm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1. E.2VariancewithAssumption4inLongitudinalMediationSincert=0,E(Y10tY11tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)=Zy10texpfsgn(dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)logI(jdt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j)y10tgf1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y10tjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)dy10tZy11tf1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y11tjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)dy11t.Usingthisresult,thevarianceoftheNIEwt(t)withAssumption4is Var(NIEwtjt)=E(fY11tg2jt))]TJ /F5 11.955 Tf 11.95 0 Td[(2E(Y11tY10tjt)+E(fY10tg2jt))-222(fE(Y11tjt))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y10tjt)g2=E(fY11tg2jt)+E(fY10tg2jt))-222(fE(Y11tjt))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y10tjt)g2)]TJ /F4 11.955 Tf 11.95 0 Td[(C2, (E) whereC2=2RfRy10texpfsgn(dt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)logI(dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j)y10tgf1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y10tjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)dy10tRy11tf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11tjm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)dy11tfM0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)gdm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1. E.3ComparisonofVarianceswithandwithoutAssumption4inLongitudinalMediationComparing( E )with( E ),thedifferenceisC1-C2,C1=2ZZy10texpfsgn(dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)logI(jdt)]TJ /F6 7.97 Tf 6.58 0 Td[(1j)y10tgf1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(y10tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)dy10tZy11tf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11tjm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)dy11tfM0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)dm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1+2Zrts.d.(Y10t)s.d.(Y11t)fM0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)dm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1ZZy10texpfsgn(dt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)logI(jdt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j)y10tgf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y10tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)dy10tZy11tf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(y11tjm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)dy11tfM0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)dm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=C2 98

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whichisnon-negative.Since0Var(NIEtjt)Var(NIEwtjt)bythefactC1C2,thevarianceofNIEtwithoutAssumption4hasasmallervariance;Var(NIEt)=ZVar(NIEtjt)p(tjVt)dtZVar(NIEwtjt)p(tjVt)dt=Var(NIEwt).Thedifferenceinthevariances,Difft=Var(NIEwt))]TJ /F1 11.955 Tf 11.96 0 Td[(Var(NIEt),isgivenasDifft=2Zrts.d.(Y10t)s.d.(Y11t)fM0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)p(tjVt)dm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dt,wherertisboundedasbelow,rt=E(Y10tY11tjM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t))]TJ /F4 11.955 Tf 11.96 0 Td[(E(Y10tjM0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)E(Y11tjM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t) s.d.(Y10t)s.d.(Y11t)p E(fY10tg2jM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)E(fY11tg2jM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t) s.d.(Y10t)s.d.(Y11t))]TJ /F4 11.955 Tf 10.49 8.44 Td[(E(Y10tjM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)E(Y11tjM0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t) s.d.(Y10t)s.d.(Y11t)bytheCauchy-Schwarzinequality.Thus,thedifferenceinthevariances,DifftisboundedbyDifft2Z(p Q1)]TJ /F4 11.955 Tf 11.96 0 Td[(Q2)fm0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,m1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)p(tjVt)dm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dt,whereQ1=E(fY10tg2jm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)E(fY11tg2jm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)andQ2=E(Y10tjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)E(Y11tjm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t).Especially,forbinaryoutcomes,Q1=Q2. 99

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APPENDIXFMONTECARLOINTEGRATIONINLONGITUDINALMEDIATIONComputef1,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(ytjt)viaMonteCarlointegrationusingtheKsetsofsamples(M0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,V1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1)asfollows,f1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(ytjt)=Zf1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(ytjm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)dm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=1 C(,)Zexpfsgn(dt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)logI(jdt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j)ytgf1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(ytjm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)dm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1(A1)=1 C(,)Zexpfsgn(dt)]TJ /F6 7.97 Tf 6.58 0 Td[(1)logI(jdt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j)ytgf1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(ytjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)dm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1(A2)=1 C(,)ZZexpfsgn(dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1)logI(jdt)]TJ /F6 7.97 Tf 6.58 0 Td[(1j)ytgf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(ytjM1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,v1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)f(v1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1jt)dv1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1dm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1(B)1 C(,)KKXi=1expfsgn(dt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,i)logyI(jdt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,ij)gf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(ytjM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,i,v1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,i,t),where`A'correspondsto`Assumption'and`B'correspondstotheassumptionfromtheBayesiandynamicmodel.ThenormalizingconstantC(,)canbecalculatedbyplugging1and0inytoftheabovesummation(withoutC(,))andaddthemup.ForanordinalmediatorwithJ=f1,,Jg,f1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(yjt)=1 C(,)Xy11t)]TJ /F13 5.978 Tf 5.75 0 Td[(12(0,1)X(i,j,h)2J3expfsgn(dt,(i,h))logI(jdt)]TJ /F6 7.97 Tf 6.59 0 Td[(1,(i,h)j)ytgf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(ytjM1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,i,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(2,j,y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)f(m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(2,j,y11t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jM1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,i,t)f(m0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,i,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,hjt)whereC(,)isthenormalizingconstantwhichcanbecalculatedsimilarly. 100

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APPENDIXGELICITATIONOFTHESENSITIVITYPARAMETERTofacilitateelicitationofinAssumption1,set=0andassumeM1t)]TJ /F4 11.955 Tf 12.02 0 Td[(M0t>withprobability1.Assuchf1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(yt=1jt)=1 C(,0)Zexpflogsgn(dt)]TJ /F13 5.978 Tf 5.76 0 Td[(1)I(jdt)]TJ /F6 7.97 Tf 6.59 0 Td[(1j0)gf1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(yt=1jm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)dm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=1 C(,0)Zf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(yt=1jm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1jt)dm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=f1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(yt=1jt) C(,0)=f1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(yt=1jt) f1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(yt=1jt)+f1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(yt=0jt),wherethesecondequalityisfromthefactP(M1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F4 11.955 Tf 12.14 0 Td[(M0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1>0)=1.ThelastequalityisfromthefactthatC(,0)=Zexpflogsgn(dt)]TJ /F13 5.978 Tf 5.76 0 Td[(1)I(jdt)]TJ /F6 7.97 Tf 6.58 0 Td[(1j0)gf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(yt=1jm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.58 0 Td[(1jt)dm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1+Zf1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(yt=0jm0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,t)fM0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(m0t)]TJ /F6 7.97 Tf 6.59 0 Td[(1,m1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1jt)dm0t)]TJ /F6 7.97 Tf 6.58 0 Td[(1dm1t)]TJ /F6 7.97 Tf 6.59 0 Td[(1=f1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(yt=1jt)+f1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(yt=0jt).Then,wecanexpressasthefollowingoddsratio,=f1,M0t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(yt=1jt)=f1,M0t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(yt=0jt) f1,M1t)]TJ /F13 5.978 Tf 5.76 0 Td[(1(yt=1jt)=f1,M1t)]TJ /F13 5.978 Tf 5.75 0 Td[(1(yt=0jt). 101

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APPENDIXHCOMPARISONOFVARIANCESINMULTIPLEMEDIATORS H.1VariancewithoutAssumption4inMultipleMediatorsFirstnotethatE(Y1111Y1000)=ZE(Y1111,Y1000jm0,m1)fM0,M1(m0,m1)dm0dm1.Now,assumef(1,M11,M21,M31)(y1111jm0,m1)>0andanon-negativecorrelationbetweenY1111andY1000givenM0andM1.Then,thecorrelationbetweenY1111andY1000conditionalon(M1,M0),,is=E(Y1000Y1111jM0,M1))]TJ /F4 11.955 Tf 11.96 0 Td[(E(Y1000jM0,M1)E(Y1111jm0,m1) s.d.(Y1000)s.d.(Y1111)0,wheres.d.(Y1000)=p Var(Y1000jM0,M1)ands.d.(Y1111)=p Var(Y1000jM0,M1)fornotationalsimplicity.Then,E(Y1000Y1111jM0,M1)=E(Y1000jM0,M1)E(Y1111jM0,M1)+s.d.(Y1000)s.d.(Y1111).Thiscanbere-expressedasE(Y1000,Y1111jM0=m0,M1=M1)=Zyf(1,M11,M21,M31)(yjM1=m0)dyZyf(1,M11,M21,M31)(yjM1=m1)dy+s.d.(Y1000)s.d.(Y1111).(A1,2)Usingtheseresults,thevarianceoftheJNIEwithoutassumption4is Var(JNIE)=E(Y21111))]TJ /F5 11.955 Tf 11.96 0 Td[(2E(Y1111Y1000)+E(Y21000))-222(fE(Y1111))]TJ /F4 11.955 Tf 11.96 0 Td[(E(Y1000)g2=E(Y21111)+E(Y21000))-221(fE(Y1111))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y1000)g2)]TJ /F4 11.955 Tf 11.96 0 Td[(C1. (H) whereC1=2RRyf(1,M11,M21,M31)(yjM1=m0)dyRyf(1,M11,M21,M31)(yjM1=m1)dy+s.d.(Y1000)s.d.(Y1111)fM0,M1(m0,m1)dm0dm1. 102

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H.2VariancewithAssumption4inMultipleMediatorsSince=0,E(Y1000,Y1111jM0=m0,M1=M1)=Zyf(1,M11,M21,M31)(yjM1=m0)dyZyf(1,M11,M21,M31)(yjM1=m1)dy.Usingthisresult,thevarianceoftheJNIEwithassumption4,Var(JNIEw),isdenedas Var(JNIEw)=E(Y21111))]TJ /F5 11.955 Tf 11.95 0 Td[(2E(Y1111Y1000)+E(Y21000))-222(fE(Y1111))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y1000)g2=E(Y21111)+E(Y21000))-222(fE(Y1111))]TJ /F4 11.955 Tf 11.96 0 Td[(E(Y1000)g2)]TJ /F4 11.955 Tf 11.95 0 Td[(C2, (H) whereC2=2RRyf(1,M11,M21,M31)(yjM1=m0)dyRyf(1,M11,M21,M31)(yjM1=m1)dyfM0,M1(m0,m1)dm0dm1. H.3ComparisonofVariancesinMultipleMediatorsComparing( H )with( H ),thedifferenceisC1-C2,C1=2ZZyf(1,M11,M21,M31)(yjM1=m0)dyZyf(1,M11,M21,M31)(yjM1=m1)dy+s.d.(Y1000)s.d.(Y1111)fM0,M1(m0,m1)dm0dm12ZZyf(1,M11,M21,M31)(yjM1=m0)dyZyf(1,M11,M21,M31)(yjM1=m1)dyfM0,M1(m0,m1)dm0dm1=C2whichisnon-negative.Thus,thevarianceoftheJNIEwithoutAssumption4hasasmallervariance;Var(JNIE)Var(JNIEw).Thedifferenceinthevariances,A,isgivenasA=2Zs.d.(Y1000)s.d.(Y1111)fM0,M1(m0,m1)dm0dm1, 103

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whereisboundedasbelow,=E(Y1000Y1111jm0,m1))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y1000jm0,m1)E(Y1111jm0,m1) s.d.(Y1000)s.d.(Y1111)p E(Y21000jm0,m1)E(Y21111jm0,m1))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y1000jm0,m1)E(Y1111jm0,m1) s.d.(Y1000)s.d.(Y1111)=p E(Y21111jM1=m0)E(Y21111jM1=m1))]TJ /F4 11.955 Tf 11.95 0 Td[(E(Y1111jM1=m0)E(Y1111jM1=m1) s.d.(Y1000)s.d.(Y1111).wherethesecondinequalityisfromCauchy-SchwarzinequalityandthethirdequalityisfromAssumption1and2.Thus,thedifferenceinthevariances,A,isboundedbyA2Z(p V)]TJ /F4 11.955 Tf 11.96 0 Td[(W)fM0,M1(m0,m1)dm0dm1whereV=E(Y21111jM1=m0)E(Y21111jM1=m1)andW=E(Y1111jM1=m0)E(Y1111jM1=m1). 104

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APPENDIXIMONTECARLOINTEGRATIONINMULTIPLEMEDIATORSForagivenQsetsof(M0,M1),f1,M1z1,M2z2,M3z3(y)=Zf1,M1z1,M2z2,M3z3(yjm0,m1)fM0,M1(m0,m1)dm0dm1=Zf1,M11,M21,M31(yjM11=m1z1,M21=m2z2,M31=m3z3)fM0,M1(m0,m1)dm0dm11 QMXi=1f1,M11,M21,M31(yjM11=m1z1,i,M21=m2z2,i,M31=m3z3,i),wherethesecondequalityisfromAssumption1andAssumption2andthelastapproximationisfromtheMonteCarlointegrationusingQsetsof(M0,M1). 105

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BIOGRAPHICALSKETCH ChanminKimwasbornandspentmostofhislifeinSeoul,Korea.HereceivedbothBachelorofArtsinHistoryandBachelorofBusinessAdministrationinBusinessfromSoganUniversityin2006.AfterworkingshortlyintheTele-communicationdepartmentatSamsungElectronicsfromJan.2006toMay.2006,hejoinedtheDepartmentofStatisticsatColumbiaUniversityasamasterstudentin2006.Afterhismasterdegreebeingawardedin2008,hestartedhisnewlifeatUniversityofFloridaasaPh.D.studentinstatistics.HisdoctoralresearchfocusesondevelopingBayesianmethodologiesformediationanalysis.In2012,hemovedtoAustin,TX.followinghisadvisor,Dr.Daniels.AfterreceivingaDoctorofPhilosophydegreeintheareaofstatisticsfromUFonAugust2013,hejoinedtheUniversityofTexasatAustinasapostdoctoralfellow. 112