SEM - Path Analysis - Heron FINAL

ThiscourseThecourseisfundedbytheESRCRDIandhostedby

ThePsychometricsCentre

TutorsJonHeron,PhD(Bristol)jon.heron@bristol.ac.uk

AnnaBrown,PhD(Cambridge)ab936@medschl.cam.ac.uk

TimCroudace,PhD(Cambridge)tjc39@cam.ac.uk

2Copyright © 2012

3

Day 1 Day 2 Day 39:00- Coffee on arrival

Introductions + Aims of course Lec-6 – Special issues in CFACorrelated errors

Bi-factor modellingMethod factors

Multi-group CFA

Lec-9 – SEMIncorporating latent traits

into path models.

9:00-9:20- 9:20-9:40-

Lec-1Mplus modelling framework

9:40-10:00- 10:00-10:20- 10:20-10:40- 10:40-11:00- Coffee Coffee Coffee 11:00-11:20-

Lec-2 – Regression models Lec-7 – Path models 1The basics / figures /

Identification/ model fit/ equivalent models

Examples 5 – SEMEAS - SEM

11:20-11:40- 11:40-12:00-

Examples 1EAS - regression models

Wrapping up, further reading and questions

12:00-12:20- 12:20-12:40- Examples 3: SZ paper.

Lunch and depart

12:40-13:00-

Lunch Lunch13:00-

13:20- 13:20-13:40- 13:40-14:00-

Lec-3 - CFA with continuous variables Lec-8 – Path models 2

Model refinementDirect and indirect effects

Binary mediators - logit/probit

14:00-14:20- 14:20-14:40- 14:40-15:00-

Lec-4 – EFA with continuous variables

15:00-15:20- 15:20-15:40- 15:40-16:00- Coffee Coffee 16:00-16:20- Lec-5 - CFA and EFA with

categorical variablesExamples 4

Path model using EAS

16:20-16:40- 16:40-17:00-

Examples 2EAS – CFA/EFA

17:00-17:20- 17:20-17:40- 17:40-

CFA+PathAnalysis=SEMSonowit’stimeforPathAnalysis

4

Beforelunch

0PathAnalysisModels[1]

0 Modelspecificationandidentification0 Modelestimation0 Modelfit0 Equivalentmodels

0Examples3– Schizophreniamodel

5

Afterlunch

0PathAnalysisModels[2]

0 Modelrefinement(pathtesting)0 DirectandIndirecteffects(mediation)0 Mediationwithbinarymeasures0 Skeweddataandbootstrapping

0Examples4– PathAnalysis~EAStemperament

6

PathAnalysis1TheBasics

7

StepsofSEM(fromKline)

1. Specifymodel2. Modelidentified?(ifno,goto1)3. Collectdata4. Assessmodelfit5. Ifmodelfitpoorthenre‐specify6. Ifmodelfitgood1. Interpretestimates2. Considernearequivalentmodels3. Reportresults

Modelspecification

0 HowdoesTHEORY sayourconceptsshouldrelatetoeachother??

0 DothisBEFORE lookingatthedata0 Orevenbetter,beforeCOLLECTING thedata

0 Knowingwhatdatayouhavecaninfluenceyourmodel– “ooh,howcanIusemytenmeasuresofemotionalsymptoms….?”

Relatingstufftootherstuff

0 Single/Multiplecauses0 Direct/Indirecteffects0 Uni‐ /Bi‐directionaleffects0 Independent/correlatederrorsorresiduals

Singlecause

Beer Happiness

e1

Multiplecorrelatedcauses

Beer Happiness

e1

Peanuts

Multipleoutcomes

Beer Happiness

e11

Windy‐pops

e21

Multipleoutcomes/correlatederrors

Beer Happiness

e11

Windy‐pops

e21

UnmeasuredexposuresexplainPartoftheresidualassociationbetweenhappinessandwindy‐pops

IndirectEffects

Beer Peanuts

e11

Happiness

e21

Bi‐directionaleffects

Beer

Peanuts

Bi‐directionaleffects– thereality?

Beer

e41

Peanuts

Beer

Peanuts

Beer

e21

e31

e11

Bi‐directionaleffects– thereality?

Beer

e41

Peanuts

Beer

Peanuts

Beer

e21

e31

e11

Hunger HungerThirst Thirst

Recursivemodels

Beer Happiness

e11

Windy‐pops

e21

Alsoconsideredrecursive

Beer Happiness

e11

Windy‐pops

e21

Considerednon‐recursive

Beer Happiness

e11

Windy‐pops

e21

Pathwayfromhappinesstowindy‐pops andbackagaintohappiness

Identification

0 Theaimofamodelistosimplify thedata0 TheinformationweputIN shouldideallybemorethantheparameterswegetOUT

0 Otherwisewe’vejustre‐packagedwhatwestartedwith

0 Atbestwehaveamodelthatteachesuslittle0 Atworstwedon’tevengetthat

Asimpleexample

0 Theequation

X1 +X2 =5

hasmoreunknowns(X1,X2)thaninformation(5)

0 Thereareaninfinitenumberofsolutions(valuesofX1,X2)thatwouldsatisfythis

Asimpleexample

0 Whatifweaddanotherequation?

X1 +X2 =52X1 +2X2 =10

0 Thereisstillnouniquesolutionasequationsarelinearlydependent

Asimpleexample

105

2211

2

1

XX

A*X=B

(A)‐1A*X=(A)‐1B

X=(A)‐1B CannotsolveforXAisnon‐invertible ornon‐positivedefinite

Asimpleexample

0 Whatiftheyweren’tlinearlydependent??

X1 +X2 =52X1 +X2 =8

0 Thereisnowauniquesolution:X1 =3,X2 =2

0 Thismodelis just‐identified0 Informationinandparametersoutisbalanced0 Giventheequations&theXi the5,8arereproducible

Asimpleexample

0 Threeequations:‐X1 +X2 =52X1 +X2 =83X1 +X2 =12

0 Thereisnowmoreinformationthanunknownparameters

0 Thismodelisover‐identified

Asimpleexample

Observed X1=2,X2=3 X1=3,X2=3 X1=2.5,X2=3 X1=2.75,X2=3

X1 +X2 5 5 6 5.5 5.75

2X1 +X2 8 7 9 8 8.5

3X1 +X2 12 9 12 10.5 11.25

Sumofsquared

differences‐ 0+1+9=10 1+1+0=2 2.5 1.375

Asimpleexample

0 Iteratetowardsasolutionthatminimises chosenstatistic– thesumsofsquareddifferencesbetweenobservedandpredictedvalues

0 Over‐identified=>onedegreeoffreedom totestadequateofsimplifiedmodel(assumingdistributionofsumofsquaresisknown)

Whataboutinpathanalysis/SEM?

0 Thedataisthecovariancematrix0 Andsometimesthemeansaswell

0 Covariancematrixfor5variablescontains(5*6)/2=15elements

0 Samplesizedoesnotaffectthisnumber!

IdentificationinSEM

0 Ifeverymodelparametercanbeexpressedasauniquefunctionofthetermsofthepopulationcovariancematrix suchthatthestatisticalcriteriontobeminimised(e.g.thesumofsquareddifferences)isalsosatisfied.

0 Recursivemodels– alwaysidentified0 Non‐recursivemodels– morecomplicated

EmpiricalIdentification

0 Modelidentificationcanbeassessedpriortodatacollection

0 Thedatacanbringanastysurprise!

0 Twomeasuresstronglycollinear0 Dataveryweaklycorrelated(~zerocellsincovmatrix)0 Outofboundselements(pairwisedeletion)0 Empiricallyunder‐identified

Timeforanexample

Population

0 Atotalof102persons(87menand15women)haddiagnosesofschizophreniaspectrumdisorders(68withschizophreniaand34withschizoaffectivedisorder),confirmedwiththeStructuredClinicalInterviewforDSM‐IV.

0 TheywererecruitedfromacomprehensivedayhospitalataVeteransAffairsmedicalcenter(N=70)andlocalcommunitymentalhealthcenter(N=32)forastudyoftheeffectsofcognitive‐behavioraltherapyonvocationalrehabilitation.

Measures0 SUMDawareness

0 ScaleforAssessingUnawarenessofMentalDisorder0 Internalstigma

0 InternalizedStigmaofMentalIllnessScale0 Hopeandself‐esteem

0 BeckHopelessnessScale/RosenbergSelf‐EsteemScale0 Avoidantcoping

0 WaysofCopingQuestionnaire0 PANNSsocialavoidance (singleitem)0 PANNSdepression (singleitem)0 PANNSpositivesymptoms

0 afactor‐analyticallyderivedcomponent(positivesymptoms,suchashallucinationsanddelusions)

Thedata

Thedata– warning!!

Aproposedmodel(model2inpaper)

Atweakedmodeldiagram

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

No effects flowing upstreamResiduals included for dependent variables

FOURestimatedresidualvariances

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

2

1 4

3

THIRTEENestimatedassociations

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

1

2

3

4

5

6

7

89

1011

12

13

SIXestimatedexogenous(co)variances

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

1

2

3

4

5

6

Identified?

Covariancematrixhas7+6+5+4+3+2+1=(7*8)/2=28 uniqueitems

Proposedmodelhas13+4+6=23 parameters

Modelisover‐identified (provideditisrecursive)

5 degreesoffreedomleftovertotestmodel

Justtoaddalittleconfusion….

0 WhenfittingthismodelinMplus,only17 parameterswouldbepresentedandnot23

0 Exogenouscovariancematrixnotpartofdefault output

0 Thesameoccurswhenfittingaregressionmodel– wearenotusuallyinterestedintheassociationswithinourcovariates

0 Thisdoesn’tmeantheyareconstrainedtobezero

0 Thesevaluescan berequested+themodelwillnotbeaffected,neitherwillthed.f.formodeltesting(inthiscase5)

socavoid

aware

stigma

positive

hope

avoidcop

depress

socavoid onavoidcop;socavoid onhope;socavoid onpositive;

avoidcop onaware;avoidcop onpositive;avoidcop onhope;

hope onaware;hope onstigma;hope onpositive;

depress onsocavoid;depress onhope;depress onaware;depress onpositive;

socavoid

aware

stigma

positive

hope

avoidcop

depress

socavoid onavoidcophopepositive;avoidcop onawarepositivehope;hope onawarestigmapositive;

depress onsocavoidhopeawarepositive;

Fullsyntax

DATA:FILE="szinputmatrix2.txt";TYPE=STDCORRELATION;NGROUPS=1;NOBSERVATIONS=102;

VARIABLE:NAMES=awarestigmahopeavoidcopsocavoiddepresspositive;USEVARIABLES=awarestigmahopeavoidcopsocavoiddepresspositive;

MODEL:socavoidonavoidcophopepositive;avoidcoponawarepositivehope;hopeonawarestigmapositive;depressonsocavoidhopeawarepositive;

!residualvariancesforendogenousvariables‐ unnecessaryhopeavoidcopsocavoiddepress;

!exogenouscovariancematrix‐ unnecessaryawarestigmapositive;awarewithstigmapositive;stigmawithpositive;

OUTPUT:standardizedresidualmodindices(3.8);;

Modelfit

TESTS OF MODEL FIT

Chi-Square Test of Model FitValue 3.475Degrees of Freedom 5P-Value 0.6271

Chi-Square Test of Model Fit for the Baseline ModelValue 156.188Degrees of Freedom 18P-Value 0.0000

CFI/TLI

CFI 1.000TLI 1.040

Modelfit

LoglikelihoodH0 Value -1251.477H1 Value -1249.739

Information CriteriaNumber of Free Parameters 23Akaike (AIC) 2548.954Bayesian (BIC) 2609.329Sample-Size Adjusted BIC 2536.680

RMSEA (Root Mean Square Error Of Approximation)Estimate 0.00090 Percent C.I. 0.000 0.114Probability RMSEA <= .05 0.742

SRMR (Standardized Root Mean Square Residual)Value 0.027

Covariances/Correlations/ResidualCorrelations

HOPE AVOIDCOP SOCAVOID DEPRESS AWARE STIGMA POSITIVEHOPE 3.229AVOIDCOP -0.440 0.240SOCAVOID -1.107 0.142 1.580DEPRESS -1.219 0.178 0.832 2.739AWARE 0.778 -0.040 -0.544 -1.120 7.322STIGMA -1.147 0.127 0.381 0.304 -0.527 1.171POSITIVE -2.544 -0.195 1.946 1.391 -0.120 1.149 19.57

ModelEstimatedCovariances/Correlations/ResidualCorrelations

HOPE AVOIDCOP SOCAVOID DEPRESS AWARE STIGMA POSITIVEHOPE 3.198AVOIDCOP -0.436 0.238SOCAVOID -1.096 0.140 1.565DEPRESS -1.207 0.150 0.791 2.691AWARE 0.770 -0.039 -0.222 -1.009 7.250STIGMA -1.136 0.144 0.404 0.461 -0.522 1.159POSITIVE -2.519 -0.193 1.927 1.378 -0.119 1.138 19.38

Standardizedmeanresidual

52From Mplus tech appendix

Standardizedcovarianceresidual

53From Mplus tech appendix

Problemwithstandardizedresiduals

54From Mplus tech appendix

StandardizedresidualsStandardized Residuals (z-scores) for Covariances/Correlations/Residual Corr

HOPE AVOIDCOP SOCAVOID DEPRESS AWARE STIGMA POSITIVEHOPE 999.000AVOIDCOP -0.019 0.019SOCAVOID -0.012 0.009 0.012DEPRESS 0.000 0.453 0.806 0.412WARE 0.010 -0.002 -1.123 -0.942 0.027STIGMA -0.057 -0.551 -0.297 -1.318 -0.004 0.000POSITIVE 999.000 999.000 999.000 -0.001 0.011 999.000 999.000

Normalizedresiduals

Normalized Residuals for Covariances/Correlations/Residual Correlations

HOPE AVOIDCOP SOCAVOID DEPRESS AWARE STIGMA POSITIVEHOPE 0.000AVOIDCOP 0.000 0.000SOCAVOID 0.000 0.000 0.000DEPRESS 0.000 0.322 0.150 0.054AWARE 0.000 0.000 -0.937 -0.219 0.000STIGMA 0.000 -0.345 -0.193 -0.898 0.000 0.000POSITIVE 0.000 0.000 0.000 0.000 0.000 0.000 0.000

56

ModificationIndices

Minimum M.I. value for printing the modification index 1.000

M.I. E.P.C. Std E.P.C. StdYX E.P.C.

ON Statements

HOPE ON DEPRESS 1.534 -0.215 -0.215 -0.197AVOIDCOP ON SOCAVOID 1.302 0.567 0.567 1.453SOCAVOID ON AWARE 1.301 -0.045 -0.045 -0.097DEPRESS ON STIGMA 1.545 -0.204 -0.204 -0.134

WITH Statements

SOCAVOID WITH AVOIDCOP 1.301 0.634 0.634 1.495DEPRESS WITH HOPE 1.545 -0.453 -0.453 -0.226AWARE WITH SOCAVOID 1.410 -0.331 -0.331 -0.116AWARE WITH DEPRESS 1.547 -3.033 -3.033 -0.789STIGMA WITH DEPRESS 1.545 -0.215 -0.215 -0.140POSITIVE WITH DEPRESS 1.544 3.694 3.694 0.588

InterpretEstimatesSTDYX Standardization

Two-TailedEstimate S.E. Est./S.E. P-Value

SOCAVOID ON AVOIDCOP 0.057 0.102 0.562 0.574HOPE -0.388 0.102 -3.798 0.000POSITIVE 0.231 0.091 2.532 0.011

AVOIDCOP ON AWARE 0.063 0.082 0.766 0.444POSITIVE -0.281 0.084 -3.334 0.001HOPE -0.600 0.074 -8.116 0.000

HOPE ON AWARE 0.062 0.079 0.788 0.431STIGMA -0.533 0.071 -7.530 0.000POSITIVE -0.191 0.079 -2.416 0.016

DEPRESS ON SOCAVOID 0.239 0.100 2.394 0.017HOPE -0.260 0.099 -2.614 0.009AWARE -0.171 0.086 -1.976 0.048POSITIVE 0.022 0.094 0.237 0.813

AWARE WITH STIGMA -0.180 0.096 -1.879 0.060POSITIVE -0.010 0.099 -0.101 0.920

STIGMA WITH POSITIVE 0.240 0.093 2.572 0.010

InterpretEstimatesSTDYX Standardization

Two-TailedEstimate S.E. Est./S.E. P-Value

VariancesAWARE 1.000 0.000 999.000 999.000STIGMA 1.000 0.000 999.000 999.000POSITIVE 1.000 0.000 999.000 999.000

Residual VariancesHOPE 0.614 0.076 8.132 0.000AVOIDCOP 0.676 0.076 8.878 0.000SOCAVOID 0.716 0.076 9.477 0.000DEPRESS 0.758 0.074 10.281 0.000

Estimatedassociations

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

0.062

-0.533

-0.191

-0.281

0.063

0.231

-0.388

-0.600

0.057

-0.171

-0.260

0.022

0.239

-0.180

-0.010

0.240

Considernearequivalentmodels

Beer Happiness

e1

PeanutsBeer Happiness

Peanuts

e1

e1

“Themaindifferencebetweenthetwomodelsisthatthefirstmodeltreatspositivesymptomsasanoutcomewhereasthesecondtreatsitasaninput,orpredictorofoutcome.”

“Modelfitindicessuggestthatthealternativemodelalsofitthedatawell.”

Lee‐Hershbergerreplacingrule1

Withinablockofvariablesatthebeginningofamodelthatisjust‐identifiedandwithunidirectionalrelationstosubsequentvariables,directeffects,correlateddisturbances,andequality‐constrainedreciprocaleffectsareinterchangeable

Lee‐Hershbergerreplacingrule1

socavoid

aware

stigma

positive

hope

avoidcop

depress

Minortweaks?

socavoid

aware

stigma

positive

hope

avoidcop

depress

Ormajorrevisionscontrarytotheory?

socavoid

aware

stigma

positive

hope

avoidcop

depress

Lee‐Hershbergerreplacingrule2

Atsubsequentplacesinthemodelwheretwoendogenousvariableshavethesamecausesandtheirrelationsareunidirectional,allofthefollowingmaybesubstitutedforoneanother:Y1→Y2,Y2→Y1,D1D2,andtheequality‐constrainedreciprocaleffectY1Y2

EquivalentModels

0 Modelswithanentirelydifferentinterpretationmayfitthedataequallywell.

0 Agoodmodelfitdoesnotgiveyouevidencethatyourswasthemodelthatgeneratedthedata

0 Shouldalwaysconsideralternativemodels0 Theremaybemanyequivalentmodels,particularlyifyourmodeliscomplex

0 Theremaybemanymanymore near‐equivalentmodels

Practicaltime

0 Convertmodel1fromtheschizophreniapaperintoMplusmodelsyntax

0 Howmanyparametersdoyouexpectandofwhattype?

0 Interprettheoutput(thatwe’reproviding)

PathAnalysis2

70

ThisSession

0PathAnalysisModels[2]

0 Modelrefinement(pathtesting)0 DirectandIndirecteffects(mediation)0 Mediationwithbinarymeasures

0Examples4– PathAnalysis~EAStemperament

71

Hegivethandhetakethaway

0Removingpaths0 Wald/LRtests0 Couldbekeypartofhypothesis

0 DoesXaffectY?0 IsthereadirecteffectofXonYwhenaccountingforZ?

0Addingpaths0 Modificationindices0 Canbeabused⇾improvemodelfit

72

Removingpaths

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

0.062

-0.533

-0.191

-0.281

0.063

0.231

-0.388

-0.600

0.057

-0.171

-0.260

0.022

0.239

-0.180

-0.010

0.240

?

Indirectroute1

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

0.062

-0.533

-0.191

-0.281

0.063

0.231

-0.388

-0.600

0.057

-0.171

-0.260

0.022

0.239

-0.180

-0.010

0.240

Indirectroute2

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

0.062

-0.533

-0.191

-0.281

0.063

0.231

-0.388

-0.600

0.057

-0.171

-0.260

0.022

0.239

-0.180

-0.010

0.240

Indirectroute3

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

0.062

-0.533

-0.191

-0.281

0.063

0.231

-0.388

-0.600

0.057

-0.171

-0.260

0.022

0.239

-0.180

-0.010

0.240

Indirectroute4

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

0.062

-0.533

-0.191

-0.281

0.063

0.231

-0.388

-0.600

0.057

-0.171

-0.260

0.022

0.239

-0.180

-0.010

0.240

Aswellasbyassociation

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

0.062

-0.533

-0.191

-0.281

0.063

0.231

-0.388

-0.600

0.057

-0.171

-0.260

0.022

0.239

-0.180

-0.010

0.240

LRTest

MODEL:socavoid on avoidcop hope positive;avoidcop on aware positive hope;hope on aware stigma positive;depress on socavoid hope aware positive@0;

hope avoidcop socavoid depress;

aware stigma positive;aware with stigma positive;stigma with positive;

79

80

MODEL FIT INFORMATION

Number of Free Parameters 23

LoglikelihoodH0 Value -1251.477H1 Value -1249.739

Chi-Square Test of Model FitValue 3.475Degrees of Freedom 5P-Value 0.6271

MODEL FIT INFORMATION

Number of Free Parameters 22

LoglikelihoodH0 Value -1251.505H1 Value -1249.739

Chi-Square Test of Model FitValue 3.531Degrees of Freedom 6P-Value 0.7398

Unconstrained

Constrained

WaldTest

MODEL:socavoid on avoidcop hope positive;avoidcop on aware positive hope;hope on aware stigma positive;depress on socavoid hope aware;depress on positive (to_test);

hope avoidcop socavoid depress;

Model test:to_test = 0;

81

WaldTest‐ resultsNumber of Free Parameters 23

LoglikelihoodH0 Value -1251.477H1 Value -1249.739

Chi-Square Test of Model FitValue 3.475Degrees of Freedom 5P-Value 0.6271

Wald Test of Parameter ConstraintsValue 0.056Degrees of Freedom 1P-Value 0.8129

82

Removingpaths‐ Summary

0 Testing>1parameteratonce0 Testingequalitytoothernon‐zerovalues0 Testingequalityoftwoparameters(e.g.acrossgroups)

0 Don’tgomad!0 Stepwise/p‐valueapproachtomodelrefinementneveragoodidea

83

Addingpaths

84

socavoid

aware

stigma

positive

hope

avoidcop

depress

Startwithareducedmodel(otherwisenopoint!):‐

Syntaxforreducedmodel

MODEL:socavoid on avoidcop hope positive;avoidcop on aware;hope on stigma;depress on socavoid aware;

OUTPUT:modindices(3.8);

85

Fitispoor

Number of Free Parameters 11

Chi-Square Test of Model FitValue 56.204Degrees of Freedom 11P-Value 0.0000

RMSEA (Root Mean Square Error Of Approximation)Estimate 0.20190 Percent C.I. 0.151 0.254Probability RMSEA <= .05 0.000

CFI/TLICFI 0.673TLI 0.465

86

Modindices outputMinimum M.I. value for printing the modification index 3.800

M.I. E.P.C. Std E.P.C. StdYX E.P.C.ON StatementsHOPE ON AVOIDCOP 20.099 -1.314 -1.314 -0.358HOPE ON DEPRESS 9.205 -0.295 -0.295 -0.269HOPE ON POSITIVE 5.284 -0.077 -0.077 -0.189AVOIDCOP ON HOPE 25.321 -0.137 -0.137 -0.501AVOIDCOP ON SOCAVOID 12.686 0.288 0.288 0.719AVOIDCOP ON DEPRESS 4.493 0.068 0.068 0.227AVOIDCOP ON STIGMA 5.807 0.110 0.110 0.242SOCAVOID ON DEPRESS 3.925 -0.262 -0.262 -0.350DEPRESS ON HOPE 6.192 -0.227 -0.227 -0.249

WITH StatementsAVOIDCOP WITH HOPE 19.935 -0.311 -0.311 -0.442DEPRESS WITH HOPE 6.984 -0.589 -0.589 -0.276Etc.

87

Syntaxforreducedmodel2

MODEL:socavoid on avoidcop hope positive;avoidcop on aware hope;hope on stigma;depress on socavoid aware;

OUTPUT:modindices(3.8);

88

Model“improvement”

M.I. E.P.C. Std E.P.C. StdYX E.P.C.ON StatementsAVOIDCOP ON HOPE 25.321 -0.137 -0.137 -0.501

First modelNumber of Free Parameters 11Loglikelihood

H0 Value -588.665

Revised modelNumber of Free Parameters 12Loglikelihood

H0 Value -573.865

89

Theothermodindices havechanged!

90

M.I. E.P.C. M.I. E.P.C.ON StatementsHOPE ON AVOIDCOP 20.099 -1.314HOPE ON DEPRESS 9.205 -0.295 8.747 -0.290 HOPE ON POSITIVE 5.284 -0.077 5.284 -0.077 AVOIDCOP ON HOPE 25.321 -0.137AVOIDCOP ON SOCAVOID 12.686 0.288 7.114 -0.380 AVOIDCOP ON DEPRESS 4.493 0.068AVOIDCOP ON STIGMA 5.807 0.110SOCAVOID ON DEPRESS 3.925 -0.262 DEPRESS ON HOPE 6.192 -0.227 6.356 -0.233 AVOIDCOP ON POSITIVE 8.852 -0.028

Addingpaths‐ Summary

0Modindices canbeusedtoindicateplaceswheremodelfitcanbeimproved

0Usewithcaution0Alwaysbeledbytheory

91

MediationDirectandIndirectpaths

92

Whatdowemeanbymediation?

0 Mediationinobservationalstudies0 Mediatorassumedtobepartofcausalsequence0 Improvesourunderstanding

0 AntenataldepressionassociatedwithchildIQ0 Whymightthatbe?

0 Parenting0 Postnataldepression

93

Direct andIndirectpaths

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

0.062

-0.533

-0.191

-0.281

0.063

0.231

-0.388

-0.600

0.057

-0.171

-0.260

0.022

0.239

-0.180

-0.010

0.240

Simplerexample

95

Positive symptoms

Social avoidance

Depressive symptoms

BaronandKenny– causalsteps

96

Positive symptoms

Social avoidance

Depressive symptomsp < 0.05

Positive symptoms

Social avoidance

Depressive symptoms

p < 0.05

Positive symptoms

Social avoidance

Depressive symptoms

p < 0.05

p > 0.05

(i) (ii)

(iii) (iv)

Positive symptoms

Social avoidance

Depressive symptoms

BaronandKenny– causalsteps

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Positive symptoms

Social avoidance

Depressive symptoms

Positive symptoms

Social avoidance

Depressive symptoms

Positive symptoms

Social avoidance

Depressive symptoms

Positive symptoms

Social avoidance

Depressive symptoms

BaronandKenny– causalsteps

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Positive symptoms

Social avoidance

Depressive symptoms

(i)

(iv)

Positive symptoms

Social avoidance

Depressive symptoms

Totaleffect

Directeffect

BaronandKenny– causalsteps

0 Verywidelyused0 Simpletodo(e.g.InSPSS)

0 Lowpowertodetect0 Reliesonp‐values(frommultiplestests)0 Canhavemediationwithouta andb bothbeingstrong

0 Non‐significantdirect‐effecteasierwithsmallsample0 Shouldwereallyberewardingsmallsamples?

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Alternative

0 Directlyquantifyindirecteffecta*b

0 Sobel test:a*b/(SE(a*b)0 OKinlargesamples0 Assumessamplingdistributionisnormal0 BootstrappingfavouredtoderiveSE’s

0 Evidenceofnon‐zeroindirecteffect⇾mediation

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Ratioofindirecttototaleffect(ab/c)

0Proportionofthetotaleffectthatismediated0DavidMackinnon

0Canbegreaterthanone0Canbenegative0Getsabitfunnyroundc=00Ratioofindirecttodirect– stillnotaproportion

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InMplus

VARIABLE:NAMES = aware stigma hope avoidcop socavoiddepress positive;USEVARIABLES = socavoid depress positive;

MODEL:socavoid on positive;depress on socavoid positive;

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Mplusresults

Two-TailedEstimate S.E. Est./S.E. P-Value

SOCAVOID ONPOSITIVE 0.099 0.026 3.773 0.000

DEPRESS ONSOCAVOID 0.500 0.127 3.930 0.000POSITIVE 0.021 0.036 0.589 0.556

Residual VariancesSOCAVOID 1.373 0.192 7.141 0.000DEPRESS 2.270 0.318 7.141 0.000

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InMplus– Modelindirect

VARIABLE:NAMES = aware stigma hope avoidcop socavoiddepress positive;USEVARIABLES = socavoid depress positive;

MODEL:socavoid on positive;depress on socavoid positive;

Model indirect:depress IND positive;

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Extraoutputobtained:‐TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS

Two-TailedEstimate S.E. Est./S.E. P-Value

Effects from POSITIVE to DEPRESSTotal 0.071 0.036 1.955 0.051Total indirect 0.050 0.018 2.722 0.006

Specific indirectDEPRESSSOCAVOIDPOSITIVE 0.050 0.018 2.722 0.006

DirectDEPRESSPOSITIVE 0.021 0.036 0.589 0.556

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Extraoutput:‐TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS

Two-TailedEstimate S.E. Est./S.E. P-Value

Effects from POSITIVE to DEPRESSTotal 0.071 0.036 1.955 0.051Total indirect 0.050 0.018 2.722 0.006

Specific indirectDEPRESSSOCAVOIDPOSITIVE 0.050 0.018 2.722 0.006

DirectDEPRESSPOSITIVE 0.021 0.036 0.589 0.556

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Routetaken

107

SOCAVOID ONPOSITIVE 0.099 (0.026)

DEPRESS ONSOCAVOID 0.500 (0.127)POSITIVE 0.021 (0.036)

Residual VariancesSOCAVOID 1.373 (0.192)DEPRESS 2.270 (0.318)

Positive symptoms

Social avoidance

Depressive symptoms

0.099

0.021

0.500

Effects from POSITIVE to DEPRESSTotal 0.071 (0.036)Total indirect 0.050 (0.018)Direct 0.021 (0.036)

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SOCAVOID ONPOSITIVE 0.099 (0.026)

DEPRESS ONSOCAVOID 0.500 (0.127)POSITIVE 0.021 (0.036)

Residual VariancesSOCAVOID 1.373 (0.192)DEPRESS 2.270 (0.318)

Positive symptoms

Social avoidance

Depressive symptoms

0.099

0.021

0.500

IndirectEffect=productofpaths=0.099*0.500

Sohowdoweinterpretthisthen?

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Effects from POSITIVE to DEPRESSTotal 0.071 (0.036)Total indirect 0.050 (0.018)Direct 0.021 (0.036)

Positive symptoms

Social avoidance

Depressive symptoms

0.099

0.021

0.500

Strongevidenceofanon‐zeroindirecteffect

Substantialpartoftotaleffectofpositivesymptomsondepressionismediatedthroughsocialavoidance(giventhecurrentmodel)

Takeadeepbreath!

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Nowforamorecomplexexample

Awareness

Internalized stigma

Positive symptoms

Hope and self‐esteem

Social avoidance

Avoidant coping

Depressive symptoms

0.062

-0.533

-0.191

-0.281

0.063

0.231

-0.388

-0.600

0.057

-0.171

-0.260

0.022

0.239

-0.180

-0.010

0.240

depressINDpositive;

Effects from POSITIVE to DEPRESSTotal 0.053 (0.035)Total indirect 0.045 (0.017)

Specific indirect

POSITIVE ⇾HOPE ⇾DEPRESS 0.019 (0.011)POSITIVE ⇾SOCAVOID ⇾DEPRESS 0.021 (0.012)POSITIVE ⇾HOPE ⇾SOCAVOID ⇾DEPRESS 0.007 (0.004)POSITIVE ⇾AVOIDCOP ⇾SOCAVOID ⇾DEPRESS -0.001 (0.003)POSITIVE ⇾HOPE ⇾AVOIDCOP ⇾SOCAVOID ⇾DEPRESS 0.531 (0.595)

Direct

POSITIVE ⇾DEPRESS 0.008 (0.035)

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PositivetoDepressVIA Hope

0Modelindirect:0 depressVIAhopepositive;

Effects from POSITIVE to DEPRESS via HOPE

Sum of indirect 0.026 (0.013)

Specific indirect

POSITIVE ⇾HOPE ⇾DEPRESS 0.019 (0.011)POSITIVE ⇾HOPE ⇾SOCAVOID ⇾DEPRESS 0.007 (0.004)POSITIVE ⇾HOPE ⇾AVOIDCOP ⇾SOCAVOID ⇾DEPRESS 0.531 (0.595)

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Summary– direct/indirecteffects

0 INDandVIA0 providesinformationondirect/indirectpathways0 Ideallyshouldbeusedwithbootstrapping

0Modeldependent0 Directeffectwilldiminishwithmodelcomplexity

0Mediation0 Extenttowhichatotaleffectispartitionedintoindirectanddirectcomponents

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Mediationmodels2Includingbinarymeasures

115

Binarydatainmediationmodels

0Asamediator/intermediatevariable

0Asanoutcome

0Asanexogenousvariable0Makesnodifference0 Categoricaltreatedascontinuous(dummies)

116

Withcontinuousdata

117

SOCAVOID ONPOSITIVE 0.099 (0.026)

DEPRESS ONSOCAVOID 0.500 (0.127)POSITIVE 0.021 (0.036) Positive 

symptoms

Social avoidance

Depressive symptoms

0.099

0.021

0.500

Effects from POSITIVE to DEPRESSTotal 0.071 (0.036)Total indirect 0.050 (0.018)Direct 0.021 (0.036)

WithacontinuousoutcomeY

0 VarianceofoutcomeYisknown0 Fixedacrossmodelswithdifferentcovariates0 Ordinaryregressionmodelshaveafixedscale

0 Canfitanumberofregressionmodels0 Indirect/mediatedeffect=totaleffect– directeffect=c‐c’

0 OrcanfitasingleSEMmodel0 Indirect/mediatedeffect=productofpaths=a*b

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WithabinaryoutcomeY

0 UnobservedcontinuousvariableY*underliesbinaryY0 VarianceofY*isunknown0 Residualvarianceforlogit/probit modelsfixed (1,π2/3)0 Scaledependsonvariablesinthemodel

0 Regressionapproach(c‐c’)0 Misleadingresults0 Rescalingispossible

0 SEMapproachwithcategoricaloptionstillvalid

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Parameterrescaling– quickcomment

0 Parametersfromseparateregressionnotcomparable

0 MultiplyeachcoefficientbytheSDofthepredictorvariableintheequationandthendividingbytheSDoftheoutcomevariable.

0 Excelspreadsheet0 http://nrherr.bol.ucla.edu/Mediation/logmed.html

0 Stata function“binary_mediation”doesthesamething0 Andallowsbootstrappingtobeincorporated

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Mplus– probit&logitwithabinaryY

0ML(logit/probit)

0 YismodelledasY* whenYisthedependentvariable

0 YismodelledasY whenYistheindependentvariable

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0WLSMV(probit)

0 YismodelledasY* whenYisthedependentvariable

0 YismodelledasY* whenYistheindependentvariable

Sowhatdoesthatmean?

0 Instandardbinaryoutcomeregression,logitandprobitmodelsareroughlyequivalent

0 InSEMmediationmodelsconclusionsmaydifferdependingonmethodandestimatorused

0 EffectofbinaryMonoutcomeYwillnotbecomparableacrossmodellingapproaches

0 IrrespectiveofwhetherYiscontinuousorbinary

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Alogit/probitexample

123

Postnataldepression

Emotionality

Adolescentdepression

Postnataldepression(mdep_pn)isbinary,treatedascontinuousEmotionality(emo_bin)isbinaryandtreatedassuchAdolescentdepression(mfqsum18)iscontinuous

Probitmodel‐ WLSMVDefine:

emo_bin = (emotott3 >10);mfqsum18 = mfq18_01 + mfq18_02 + mfq18_03 + ...+ mfq18_13;

Variable:Usevariables = mdep_pn emo_bin mfqsum18;Categorical = emo_bin;

Analysis:estimator = WLSMV;

Model:mfqsum18 on mdep_pn emo_bin;emo_bin on mdep_pn;

Model indirect:mfqsum18 IND mdep_pn;

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Probitmodel‐ WLSMV

Two-TailedEstimate S.E. Est./S.E. P-Value

MFQSUM18 ONMDEP_PN 0.988 0.339 2.911 0.004EMO_BIN 0.551 0.186 2.959 0.003

EMO_BIN ONMDEP_PN 0.666 0.090 7.386 0.000

TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS

Effects from MDEP_PN to MFQSUM18Total 1.355 0.318 4.255 0.000Specific indirect 0.367 0.133 2.757 0.006Direct 0.988 0.339 2.911 0.004

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Probit model‐ ML

Two-TailedEstimate S.E. Est./S.E. P-Value

MFQSUM18 ONMDEP_PN 1.145 0.341 3.358 0.001EMO_BIN 1.100 0.361 3.048 0.002

EMO_BIN ONMDEP_PN 0.666 0.090 7.386 0.000

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Logit model‐ ML

Two-TailedEstimate S.E. Est./S.E. P-Value

MFQSUM18 ONMDEP_PN 1.145 0.341 3.358 0.001EMO_BIN 1.100 0.361 3.048 0.002

EMO_BIN ONMDEP_PN 1.162 0.154 7.548 0.000

LOGISTIC REGRESSION ODDS RATIO RESULTSEMO_BIN ON

MDEP_PN 3.195

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Scaledparametersapproach(e.g.Stata)Logit: emo_bin on iv (a1 path)------------------------------------------------------------------------------

emo_bin | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------

mdep_pn | 1.161604 .1539016 7.55 0.000 .859962 1.463245

_cons | -1.924484 .0859839 -22.38 0.000 -2.093009 -1.755959------------------------------------------------------------------------------OLS regression: dv on iv (c path)------------------------------------------------------------------------------

mfqtot18 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

mdep_pn | 1.355169 .335292 4.04 0.000 .697477 2.01286

_cons | 6.047658 .1456367 41.53 0.000 5.761985 6.333332------------------------------------------------------------------------------OLS regression: dv on mv & iv (b & c' paths)------------------------------------------------------------------------------

mfqtot18 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

emo_bin | 1.100295 .3613163 3.05 0.002 .3915547 1.809035

mdep_pn | 1.145388 .3413925 3.36 0.001 .4757294 1.815046_cons | 5.907522 .1523523 38.78 0.000 5.608675 6.206369

------------------------------------------------------------------------------ 128

BinaryMediationsummary

0 Withprobit/WLSMVtheindirecteffectcanbedirectlyoutputtedusing“modelindirect”

0 Howeverthisyieldsmaineffectsthataremoredifficulttointerpret(notlikeoddsratios)

0 OutputusingMLisnotscaledsopathestimatescannotsimplybemultipliedtoprovideestimateofindirecteffect

0 Re‐scalingshouldbepossibletogetbestofbothworldsandyieldresultsthatagreewithStata – watchthisspace...

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Furthermediationreading

0 AndrewF.Hayes(2009):BeyondBaronandKenny:StatisticalMediationAnalysisintheNewMillennium,CommunicationMonographs,76:4,408‐420.

0 Mackinnon,DavidPeter.Introductiontostatisticalmediationanalysis.LawrenceErlbaumandAssociates(2008).

0 DavidP.Mackinnon,Lockwood,C.M.,Brown,C.H.,Wang,W.&Hoffman,J.M..Theintermediateendpointeffectinlogisticandprobitregression.ClinicalTrials(2007).

0 http://nrherr.bol.ucla.edu/Mediation/logmed.html0 Alsosee“binary_mediation”Stata command

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