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psychometrika
/s11336-021-09767-0
TIMETOINTERVENE:ACONTINUOUS-TIMEAPPROACHTONETWORKANALYSIS
ANDCENTRALITY
Oisí
UTRECHTUNIVERSITY
,discrete-
time(DT)vectorauto-regressive(VAR)modelsdefinethenetworkstructurewithcentralitymeasuresused
,VARmodelssufferfromtime--
time(CT)modelshavebeensuggestedasanalternativebutrequireaconceptualshift,implyingthat
DT-,weproposeandillustrateaCT
networkapproachusingCT-
.
Keywords:dynamicalnetworkanalysis,continuous-timemodeling,centrality,intensivelongitudinaldata,
experiencesamplingmethodology.
Dynamicalnetworksareapopularapproachfortheanalysisofexperiencesamplingdatain
psychology(Bringmannetal.,2013;Borsboom&Cramer,2013).Inthisapproach,researchers
typicallymakeuseofthediscrete-time(DT)first-ordervectorauto-regressive(VAR)model,with
theestimatedlaggedparametersofthismodeltreatedasedgesdirectlyconnectingnodesinthe
,dynamicalnetworkanalyseshavebeenpromoted
,central-
itymeasuresbasedonparameterestimatesareusedtoidentifywhichvariableinthenetwork
representsthemostpromisingtargetforfutureinterventions(Bringmannetal.,2013;Fisher&
Boswell,2016;Kroezeetal.,2017;Epskampetal.,2018;Rubeletal.,2018;Baketal.,2016;
Bringmannetal.,2015;Bastiaansenetal.,2019;Fisheretal.,2017;Christianetal.,2019).
However,itiswellknownthattheDT-VARmodelsuffersfromtheproblemoftime-interval
dependency(Gollob&Reichardt,1987),whichentailsthattheestimatedlaggedparametersare
canberesolvedbymodelingpsychologicalprocessesasunfoldingcontinuouslyovertimeusing
continuous-time(CT)modelsthatexplicitlyaccountforthetime-intervaldependencyoflagged
parameters(.,Boker,2002;Oud&Delsing,2010;vanMontfortetal.,2018;Ryanetal.,2018;
Voelkleetal.,2012).Suchmodelscaneasilydealwithunequalintervals,andcanbeusedtoderive
howlaggedparametersareexpectedtoevolveoverawholerangeoftime-,
takingaCTperspectivealsoentailsaconceptualshift,inthatlaggedregressionparametersatany
intervalshouldbeinterpretedastotalratherthandirectrelationships(Aalenetal.,2012;Aalenet
al.,2016;Deboeck&Preacher,2016).Whilethegeneralconsequencesofthishavebeendiscussed
elsewhere,
numberofopenquestions,mostnotably:whataretheimplicationsoftheCTperspectivefor
currentcentralitymeasures?HowcanCTmodelsbeusedtoyieldnovelinsightsintoadynamical
network?AndhowcanweuseCTmodelstochooseinterventiontargets?
SupplementaryInformationTheonlineversioncontainssupplementarymaterialavailableat.
1007/s11336-021-09767-0.
CorrespondenceshouldbemadetoOisínRyan,UtrechtUniversity,Padualaan14,3584CH,Utrecht,The
:o.******@
©2021TheAuthor(s):.
PSYCHOMETRIKA
.
Inthefirstsection,weprovideanoverviewoftheDT-VARmodelandhowpath-specificeffects
,wediscuss
thetime-,wepresent
theCT-VARmodelasanalternativeapproachtodynamicalnetworkanalysis,andexplorethe
,weintroducenewfit-for-purposecentralitymeasures
thatbothreflecttheCTnatureoftheunderlyingprocessandhaveaclearanddirectconceptual
,wedemonstratethe
,thedevelopmentsin
thispaperfocusonsingle-subjectmodels,thoughthecritiquesandmeasuresdevelopedhere
generalizeinastraightforwardwaytowithin-subjectsparametersofmultilevelmodels.
:DT-VARNetworks
Researcherswhoadoptanetworkperspectiveonpsychologicalphenomenaoftenusethe
parametersof(single-subjectormultilevel)first-orderdiscrete-timevectorauto-regressive(DT-
VAR)modelstosuggestinterventiontargets(Bringmannetal.,2013;Peetal.,2015;Fisher&
Boswell,2016;Kroezeetal.,2017;Rubeletal.,2018;Baketal.,2016).Inthissection,we
haveusedthismodeltofindinterventiontargets,thatis,throughconsideringpath-specificeffects
,
asthisinsightwillproveusefullaterwhenconsideringhowCTmodelscouldbeusedinan
,weelaborateonthetime-intervalproblem,anddiscusshowthis
issuecastsdoubtontheappropriatenessofcurrentpractice,whichmotivatesthedevelopments
presentedintheremainderofthispaper.
-VARModel
TheDT-VARmodelisasingle-subjecttime-seriesmodelthatdescribesdynamicrelationships
ofavariableonitself(anauto-regressiveeffect)oranothervariable(across-laggedeffect)atthe
nextmeasurementoccasion(.,atalagofone).Thismodelcanbewrittenas
Yτ=c+Yτ−1+τ(1)
wheregivenpvariables,thep×1vectorofrandomvariablesYatoccasionτisregressedonthe
p×1vectorofthosesamevariablesatthepreviousoccasion,Yτ−×pmatrixoflagged
regressionparametersisdenoted,whilethep×1vectorscandτdenotetheinterceptsand
randomshocks,respectively,thelatterbeingnormallydistributedwithmeanzeroandvariance-
covariancematrix(Hamilton,1994).ThemultivariatemeanoftheDT-VARmodelμcanbe
expressedasafunctionoftheinterceptsandlaggedparameters(μ=(I−)−1c)andcanbe
centered,theintercepttermcanbeomitted(c=0),aconventionwewilladoptthroughoutthe
remainderofthepaper.
Inqualitativeterms,themodeldescribesasystemwheretherandomshocksτpushthesystem
awayfromitsequilibrium,andthelaggedparametersdeterminehowthevariablesreacttothese
shocks,eventuallyreturningthemtoequilibriumovertime(formoredetails,see,amongothers,
Ryanetal.,2018;Strogatz,2015;Haslbecketal.,inpress).Adistinctioncanbemadebetween
DT-VARmodelswhichexhibit“positive”and“negative”auto-regression:intheformercase,the:.
OISÍ
(a)
(b)
Figure1.
Pathmodel(left-handside)andnetwork(right-handside)representationsoftwofour-variableDT-
pathmodels,thepresenceofanarrowlinkingtwovariablesdenotessomenonzerodependencybetweenthem,conditional
,thearrowsrepresentauto-regressiveandcross-laggedregression
parametersinafirst-orderDT-
parameters.
systemreturnstoequilibriuminanexponentialfashionovertime;Inthelatter,variablesswitch
theirsign(frompositivetonegativeandviceversa)
case,thisisdeterminedbythesignoftheauto-regressiveparameterφ,butinthemultivariate
casebythesignoftheeigenvaluesof.Positiveauto-regressionsystemsarefoundinmany
psychologicaltime-seriesapplications(.,Bringmannetal.,2013;Koval&Kuppens,2012;
Oravecz&Tuerlinckx,2011).Inthispaper,wewillassumethatoursystemofinterestexhibits
positiveauto--VARmodelingeneralarethatthe
sameamountoftime(denotedt)elapsesbetweentwosubsequentmeasurementoccasions,and
thattheunderlyingprocessisstationary,whichentailsthatthemeans,varianceandcovariances,
TheDT-VARmodelcanberepresentedaseitherapathmodel,asshownintheleft-hand
,orasadynamicalnetworkstructure,asshownintheright-handpanel,wherethe
nodesrepresenttherandomvariables,andtheedgesrepresentthevaluesofthelaggedparameters
(Bringmannetal.,2013;Epskampetal.,2018).Thelaggedparametersinaretypically
,takeit
(repeatedmeasurementsof)Stress(Y1),Anxiety(Y2),
Self-Consciousness(Y3)andfeelingsofPhysicalDiscomfort(Y4).Wewillreferthroughoutto
thedynamicalsystemcomposedofthesefourtime-varyingprocessesastheStress-Discomfort
1Inorderforaprocesstobestationary,thematrixeigenvaluesmustliewithintheunitcircle(Hamilton,1994).:.
PSYCHOMETRIKA
reciprocalcross-laggedrelationshipswithallothervariables,resultinginacompletelyconnected
,across-laggedparametersuchasφ41=
effectofcurrentStress(Y1,τ)onPhysicalDiscomfortatthenextmeasurementoccasion(Y4,τ+1),
conditionalon(.,controllingfor)currentfeelingsofAnxiety,Self-ConsciousnessandPhysical
Discomfort(Y2,τ,Y3,τ,Y4,τ).Thisparameterisweaklypositive,leadingtotheinterpretationthat
ahighlevelofcurrentStresshasasmallpositivedirecteffectonfeelingsofPhysicalDiscomfort
atthenextoccasion.
-VARModels
ToidentifywhichvariablesshouldbeconsideredtargetsforaninterventionbasedonaDT-
VARmodel,psychologyresearchershavemainlyusedtwoapproaches:(a)path-specificeffects,
whichareinspiredbytheSEMliterature(Bollen,1987);and(b)centralitymeasures,whichcome
fromthenetworkanalysisliterature(Freeman,1978;Opsahletal.,2010).
Path-specificeffectshavebeenusedtodescribethetotal,directandindirecteffectsofone
variableonanother,andcanbecalculatedusingthewell-knownpath-tracingrulesfromtheSEM
literature(Bollen,1987).Forinstance,thetotaleffectofStresslevelsnow(Y1,τ)onPhysical
Discomforttwomeasurementoccasionslater(Y4,τ+2)isthesumofthedirecteffectpathways
(.,Y1,τ→Y4,τ+1→Y4,τ+2,andY1,τ→Y1,τ+1→Y4,τ+2),andtheindirecteffectpathways
throughthemediatingvariablesAnxietyandSelf-Consciousness(.,Y1,τ→Y2,τ+1→Y4,τ+2,
andY1,τ→Y3,τ+1→Y4,τ+2,respectively;Cole&Maxwell,2003).Ifweinterpretparameters
asdirectcausaleffects,wemaysuggestthatinterventionsshouldtargetvariablesthathavestrong
,wecouldsearchforthosemediatorsthroughwhich
thestrongestindirecteffectspass(Groenetal.,2020;Bernatetal.,2007;Bramsenetal.,2013).
Forinstance,,wemightsuggestAnxietyasanintervention
targetduetotherelativelystronglag-onedirecteffectonPhysicalDiscomfort(φ42=.08),or
becauseitisamediatorofthelargestlag-twoindirecteffect,fromStresstoPhysicalDiscomfort
(Y1,τ→Y2,τ+1→Y4,τ+2=−.005).
Analternativeapproachtofindingatargetforinterventioncomesfromthenetworkapproach
andisbasedoncentralitymeasures(.,Bringmannetal.,2013;Fisher&Boswell,2016;Kroeze
etal.,2017;Epskampetal.,2018;Rubeletal.,2018;Baketal.,2016;Bringmannetal.,2015;
Bastiaansenetal.,2019).Centralitymeasuresareusedtosummarizetherelationsaparticular
variablehaswiththenetworkasawhole,typicallysummingovertheindividualrelationsthat
-
specificeffectsandcentralitymeasuresforDT-VARmodelshasnotyetbeendescribedinthe
literatureacloseinspectionofthecomputationandinterpretationofmanypopularcentrality
measuresrevealsthattheyareverysimilartopath-tracingeffects:specifically,manycentrality
measuresareinterpretedascapturingeithertotal,directorindirecteffects,andinturn,these
measuresareoftencloselyrelatedtosummariesofthecorrespondingpath-,
wewillmentionthreesuchmeasures;fortheexactconnectionbetweenthesemeasuresand
path-tracingeffects,thereaderisreferredtoAppendixA.
First,theTwo-StepExpectedInfluencemeasure(EI(2);Robinaughetal.,2016;Kaiser&
i
Laireiter,20