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ThiscourseThecourseisfundedbytheESRCRDIandhostedby
ThePsychometricsCentre
TutorsJonHeron,PhD(Bristol)[email protected]
AnnaBrown,PhD(Cambridge)[email protected]
TimCroudace,PhD(Cambridge)[email protected]
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
97
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
98
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
100
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
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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
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
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Binarydatainmediationmodels
0Asamediator/intermediatevariable
0Asanoutcome
0Asanexogenousvariable0Makesnodifference0 Categoricaltreatedascontinuous(dummies)
116
Withcontinuousdata
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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
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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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