Ho June 11 2003 Abstract This article discusses the use of symmetric ultiplicativ in teraction e57355ect to capture cer tain yp es of thirdorder dep endence patterns often presen in so cial net orks and other dy adic datasets Suc an e57355ect along ID: 80878
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BilinearMixedE ectsModelsforDyadicDataPeterD.Ho June11,2003AbstractThisarticlediscussestheuseofasymmetricmultiplicativeinteractione ecttocapturecer-taintypesofthird-orderdependencepatternsoftenpresentinsocialnetworksandotherdyadicdatasets.Suchane ect,alongwithstandardlinear xedandrandome ects,isincorporatedintoageneralizedlinearmodel,andaMarkovchainMonteCarloalgorithmisprovidedforBayesianestimationandinference.Inanexampleanalysisofinternationalrelationsdata,accountingforsuchpatternsimprovesmodel tandpredictiveperformance.KEYWORDS:socialnetwork,balance,innerproductscaling,generalizedlinearmodel.1IntroductionDyadicdataconsistofmeasurementsthataremadeonpairsofobjectsorunderapairofconditions,sothatyi;jdenotesthevalueofthe(possiblydirected)measurementfromitoj.Examplesincludesocialnetworkanalysis,\roundrobin"experimentsinpsychology,andcomparativedatainwhichyi;jmightbeameasureofsimilaritybetweenunitsiandj.Inthesocialnetworksliterature,modelinghasfocusedonthebinarycasewhereyi;jiseitherzeroorone,indicatingthepresenceorabsenceofa\link"fromitoj.Thishasledtothedevelopmentofdataanalysistoolsbasedondirectedgraphsandthestudyofexponentiallyparameterizedrandomgraphmodels(WassermanandPattison1996).Forvalued(non-binary)dyadicdatasets,aperceivedlackofstatisticaltoolshassometimesledtoad-hocreductionsofvaluedresponsestobinarydata.However,ANOVAmethodsareavailableforvalueddyadicdata:theso-calledsocialrelationsmodel(Warner,Kenny,andStoto1979;Wong1982)allowsforthedecompositionofthevarianceintosenderandreceiverspeci ce ects,aswellasallowsforcorrelationbetweenresponseswithinadyad.SuchamodelhasbeenstudiedinthecontextofalineargroupsymmetrymodelbyLi(2002),andadvancesinvariancecomponentanalysishavebeenmadebyandGillandSwartz(2001)andLiandLokenPeterD.Ho isAssistantProfessorofStatistics,Box354322,UniversityofWashington,SeattleWA98195-4322,Email:ho @stat.washington.edu,Web:www.stat.washington.edu/ho .ThisresearchwassupportedbyOceofNavalResearchgrantN00014-02-1-1011.TheauthorthanksMarkHandcockandMichaelWardforhelpfuldiscussions.1 (2002).Thesemodelsgenerallypresumenormallydistributeddataandadditivee ects,andthusthelackofanysortofdependencebeyondthosespeci edbysecond-ordermoments.Incontrast,manyobserveddyadicdatasetsexhibitcertainformsofthird-orderdependence,andoftenitisofscienti cinteresttoquantifythesehigherorderpatterns.Inthisarticleweproposeaclassofgeneralizedadditivemodelsbasedonthesocialrelationsmodel,butincorporatethirdorderdependenceviaabilineare ect.Thebilineare ectforapair(i;j)issimplytheinnerproductofunobservedcharacteristicvectorsziandzj,speci ctounitsiandjrespectively.ThisapproachissimilarinspirittothelatentvariablemethodsproposedbyHo ,Raftery,andHandcock(2002)tocapturetransitivityinasocialnetworkdataset,buthassomecomputationalandconceptualadvantages.Thebilineare ectisalsoatypeofmultiplicativeinteraction(Gabriel1978;MarasingheandJohnson1982;Oman1991).ThemodelspresentedinthisarticlearesimilartothegeneralizedbilinearregressionmodelsstudiedbyGabriel(1998),whoconsideredapproximatemaximumlikelihoodestimationinthecontextoffactorialdesigns.Inthisarticle,weshowhowabilineare ectcanbeusedtorepresentcertainformsofdependenceoftenseenindyadicdata,anddevelopaMarkovchainMonteCarloalgorithmbasedonGibbssampling,providingarbitrarilyexactBayesianinference.Withsomemodi cations,thealgorithmcanbeusedasameansofmakingBayesianinferenceforabroadclassofgeneralizedbilinearregressionmodelswithmixede ects.Inthenextsection,wediscussthebasiclinearmixede ectsmodelfordyadicdataandtheresultingdependencestructure.InSection3,wediscusstypesofthird-orderdependenceoftenseeninnetworkdatasetsandtheuseofabilineare ecttocapturesuchdependence.Section4givesaMarkovchainMonteCarlo(MCMC)algorithmwhichcanbeusedtoobtainsamplesfromtheposteriordistributionoftheparameters.Issuessuchasmodel t,modelselectionandinterpretationarediscussedinthecontextofadataanalysisoninternationalrelationsinSection5.AdiscussionfollowsinSection6.2LinearMixedE ectsModelsforExchangeableDyadicDataSupposeweareonlyinterestedinestimatingthelinearrelationshipsbetweenresponsesyi;jandapossiblyvectorvaluedsetofvariablesxi;j,whichcouldincludecharacteristicsofuniti,character-isticsofunitj,orcharacteristicsspeci ctothepair.Inthiscasewemightconsidertheregressionmodelyi;j= 0xi;j+i;j;(1)whereyi;iistypicallynotde ned.Thegeneralizedleastsquaresestimate^ anditscovariancematrixdependonthejointdistributionofthei;j'sonlythroughtheircovariance.Itisoftenassumedinregressionproblemsthattheregressorsxi;jcontainenoughinformationsothatthedistributionoftheerrorsisinvariantunderpermutationsoftheunitlabels.Thisassumptionisequivalenttothennmatrixoferrors(withanunde neddiagonal)havingadistributionthatis2 invariantunderidenticalrowandcolumnpermutations,sothatfi;j:i6=jgisequalindistributiontof(i);(j):i6=jgforanypermutationoff1;:::;ng.Thisconditioniscalledweakrow-and-columnexchangeabilityofanarray.Forundirecteddata,suchexchangeabilityimpliesa\randome ects"representationoftheerrors,inthati;jisequalindistributiontof(;ai;aj;\ri;j)where;ai;aj;\ri;jareindependentrandomvariablesandfisafunctiontobespeci ed(Aldous1985,Theorem14.11).IfinadditiontotheaboveinvarianceassumptionwealsomodeltheerrorsasGaussian,thenthejointdistributioncanberepresentedintermsofalinearrandome ectsmodel.Inthemoregeneralcaseofdirectedobservations,wecanrepresentthejointdistributionofthei;j'sasfollows:i;j=ai+bj+\ri;j(2)(ai;bi)0multivariatenormal(0;a;b);ab= 2aabab2b!(\ri;j;\rj;i)0multivariatenormal(0;\r);\r= 2\r2\r2\r2\r!;withe ectsotherwisebeingindependent.Thecovariancestructureoftheerrors(andthustheobservations)isasfollows:E(2i;j)=2a+2ab+2b+2\rE(i;ji;k)=2aE(i;jj;i)=2\r+2abE(i;jk;j)=2bE(i;jk;l)=0E(i;jk;i)=abandso2arepresentsthedependenceofobservationshavingacommonsender,2bthatofobser-vationshavingacommonreceiver,andrepresentsthecorrelationofobservationswithinadyad(ofteninterpretedas\mutuality"or\reciprocity").Thishasbeencalledthe\socialrelations"or\roundrobin"model(Warneretal.1979;Wong1982),andisrelatedtoamodelfordiallelcrossdatausedbyCockerhamandWeir(1977).Themodelisaspecialcaseofalineargroupsymmetrymodel(AnderssonandMadsen,1998),andhasbeenstudiedinthiscontextbyLi(2002).RecentadvancesinvariancecomponentestimationhavebeenmadebyGillandSwartz(2001)andLiandLoken(2002).Toanalyzeresponsesinparticularsamplespaces,theerrorstructuredescribedabovecanbeaddedtoalinearpredictorinageneralizedlinearmodel:i;j= 0xi;j+ai+bj+\ri;j(3)E(yi;jji;j)=g(i;j)p(y1;2:::;yn;n 1j1;2:::;n;n 1)=Yi6=jp(yi;jji;j):Thisisageneralizedlinearmixed-e ectsmodelwithinverse-linkfunctiong(),inwhichtheobser-vationsaremodeledasconditionallyindependentgiventherandome ects,butareunconditionally3 dependent.ThecovariancepatternfortheobservationsisgivenapproximatelyasCov(yi1;j1;yi2;j2)=E[Cov(yi1;j1;yi2;j2ji1;j1;i2;j2)]+Cov[E(yi1;j1ji1;j1);E(yi2;j2ji2;j2)]=E[0]+Cov[g(i1;j1);g(i2;j2)]Cov(i1;j1;i2;j2)g0( 0xi1;j1)g0( 0xi2;j2);wherethepatternforCov(i1;j1;i2;j2)isthesameasthatforthei;j'sgivenabove.However,unlikethelinearregressioncase,^ isnotgivenbylinearcombinationsoftheobservations,andE(^ )andCov(^ )arenotfunctionsofonlythe rstandsecondordermomentsofthedata.Modellackof t,orthirdandhigherorderdependence,willa ectourinferenceon .Manydyadicdatasetsexhibitcertainformsofthirdorderdependence.Indeed,itisthesehigherorderpatternsofdependencethatareoftenofinterest,andmayalsoprovideinformationusefulforpredictiveinference.3ModelingThirdOrderDependencePatternsSomedependencepatternscommonlyseenindydaicdatasetshavebeengiventhedescriptivetitlesoftransitivity,balance,andclusterability.Inthecontextofbinarydata,graphtheoreticde nitionsoftheseconceptsappearinWassermanandFaust(1994,chapter6)andareasfollows:Transitivity:Fordirectedbinarydata,anorderedtriadi;j;kistransitiveifwheneveryi;j=1andyj;k=1,wehaveyi;k=1,i.e.\afriendofafriendisafriend."Balance:Forsignedunorderedrelations,atriadi;j;kissaidtobebalancedifyi;jyj;kyk;i�0.Theideaisthatiftherelationshipbetweeniandjis\positive"thentheywillrelatetoanotherunitkinanidenticalfashion,sothatifyi;j�0thenyj;kandyk;iareeitherbothpositiveorbothnegative.Clusterability:Thisisarelaxationoftheconceptofbalance.Atriadisclusterableifitisbalancedortherelationsareallnegative.Theideaisthataclusterabletriadcanbedividedintogroupswherethemeasurementsarepositivewithingroupsandnegativebetweengroups.Inastatisticalsense,adatasetwilldisplayvaryingdegreesoftransitivity,balance,orclusterability.Oftenitisfoundthattherearemoretransitive,balanced,orclusterabletriadsthanwouldbeexpectedundermodels(2)or(3).Anotherindicationofthirdorderdependencewouldbeifafter ttingaregressionmodelandobtainingtheresiduals^i;j,theaveragevalueof^i;j^j;k^k;iissubstantiallylargerthanzero,theexpectedvaluepresumedbymodel(2).Ho etal.(2002)usedsimplefunctionsoflatentcharacteristicvectorsina xede ectssettingtocapturesomeformsoftransitivity,balance,andclusterability.Forexample,theyconsideredmodelsinwhichi;j= 0xi;j+f(zi;zj)wheref(zi;zj)= jzi zjj(\thedistancemodel")orf(zi;zj)=z0izj=jzjj(\theprojectionmodel").Inwhatfollows,weconsiderasimilarapproachusingtheinnerproductkernelf(zi;zj)=z0izj,andgiverandomand xede ectsinterpretations.4 Addingthebilineare ectz0izjtothelinearrandome ectsinmodels(2)and(3)givesi;j=ai+bj+\ri;j+i;j(4)i;j=z0izjwheretherandome ectsai;bjand\ri;jaremodeledwiththemultivariatenormaldistributionsdescribedabove.Wehavewritteni;j=z0izjtosuggesttheinterpretationofz0izjasamean-zerorandome ect:Ifthez'saremodeledasindependentk-dimensionalmultivariatenormalrandomvectorswithmeanzeroandcovariancematrixz,thentheresultingdistributionforthe'shasthefollowingmomentproperties:E(i;j)=0;E(2i;j)=trace2z;E(i;jj;kk;i)=trace3z;withallothersecondandthirdordermomentsequaltozero.Notethatanorthogonaltransfor-mationofthez'sleavesz0iziinvariant,sowecanassumezisadiagonalmatrix(otherwise,theo -diagonaltermsarenon-identi able).Forsimplicitywefocusonthecasez=2zIkk,forwhichtheabovemomentsare0;k4z,andk6zrespectively.Withi;jaddedtotheerrorterm,thenonzerosecondandthirdordermomentsareE(2i;j)=2a+2ab+2b+2\r+k4zE(i;ji;k)=2aE(i;jj;i)=2\r+2ab+k4zE(i;jk;j)=2bE(i;jj;kk;i)=k6zE(i;jk;i)=ab:Thusthee ecti;j=z0izjcanbeinterpretedasamean-zerorandome ectabletoinduceaparticularformofthird-orderdependenceoftenfoundindyadicdatasets.Marginally,askincreasesthedistributionofi;jwillconvergetoanormaldistribution,duetothecentrallimittheorem.Jointly,theMarkovdependencegraphforthe'shastwodyadsasneighborsiftheyhaveatleastoneunitincommon.Consideredas xede ects,the'scanbeviewedasinteractiontermsthatarehighlyconstrainedduetothefunctionaldependenceonthez's.Theconstraintiseasytovisualizeintermsofthez's:Ifziandzjarevectorsofsimilardirectionandmagnitude,thenz0izkandz0jzkwillnotbetoodi erent.Thisfeaturecanberelatedtotransitivity,whichisconceptuallyameasureofhowi;kisafunctionofi;jandj;k.Consideringforthemomentz'sscaledtohaveunitlengthsothatjzi zjj=p2(1 z0izj),bythetriangleinequalitywehave1 z0izk1 z0izj+1 z0jzk+2q(1 z0izj)(1 z0jzk);ori;ki;j+j;k 1+2q(1 i;j)(1 j;k);5 whichgivesalowerboundfori;kintermsofi;jandj;k.Balanceandclusterabilitydescribehowsimilari;kandj;kareasafunctionofi;j.Forscaledz's,wehaveji;k j;kj=jz0k(zi zj)jjzkjjzi zjj=jzi zjj:Notingthatz0izj=cos( i j),where iistheangleofzifroma xedaxis,wehavejzi zjj=2sin[( i j)=2]=2sin12cos 1(z0izj)=q2(1 i;j);andsoji;k j;kjp2(1 i;j).Ifi;jislarge,thedi erencebetweeni;kandj;kmustbesmall.Ifi;jisnegativeone,thedi erenceisunconstrainedandcouldrangefromzerotoamaximumoftwo(inthisscaledcase).4ParameterEstimationInthefrequentistsetting,approximateestimationforgeneralizedlinearmixede ectsmodelsoftenproceedsviaTaylorexpansionsanditerativelyreweightedleastsquaresforthe xede ects,alongwithapproximaterestrictedmaximumlikelihoodestimationforthevariancecomponents(Schall1991;BreslowandClayton1993;Wol ngerandO'Connell1993;McGilchrist1994).Theaccuracyoftheseapproximatemethodsisgenerallydependentonthesamplesize,seeBoothandHobert(1998)foradiscussion.Gabriel(1998)suggestsanalgorithmalongtheselinesforthegeneralizedbilinearmixede ectsmodel.Alternatively,ZegerandKarim(1991),Gelfand,SahuandCarlin(1996),andNatarajanandKass(2000)haveproposedGibbssamplingapproachestoparameterestimationforgeneralizedlinearmixede ectsmodels.However,estimationismoredicultforthecomplicateddependencestructureoftherandome ectsintheinvariantnormalmodel(2).GillandSwartz(2001)haveproposedaGibbssamplingschemeforestimationofrandome ectsinthelinearcasewiththeidentitylink,althoughwehavefoundthattheiralgorithmdoesnotmixwellwhencovariatesareincluded,duetoaweakidenti abilityoftheunitlevelrandome ectsandcertainregressioncoecients:AsdiscussedinGelfand,Sahu,andCarlin(1995)therandome ectsaandbwillbeconfoundedtoadegreewitheachotherandtoregressionparametersassociatedwithpredictorsthatdonotvaryacrossreceivers(i.e.sender-speci ce ects)oracrosssenders(receiver-speci ce ects).Forexample,apopulation-levelinterceptisonesuchparameter.Toobtaina\cleaner"partitionofthevarianceandamoreecientMCMCsamplingscheme,wedecomposexi;jintoxi;j=(xd;i;j;xs;i;xr;j),i.e.intodyadspeci cregressorsxd;i;j,senderspeci cregressorsxs;iandreceiverspeci cregressorsxr;j.Thegeneralizedbilinearmodelisthenrewrittenasi;j= 0dxd;i;j+( 0sxs;i+ai)+( 0rxr;j+bj)+\ri;j+z0izj6 orequivalentlyi;j= 0dxd;i;j+si+rj+\ri;j+z0izjsi= 0sxs;i+airi= 0rxr;i+bi:Thisparameterizationforthelinearunit-levele ectsissimilartothe\centered"parameterizationssuggestedbyGelfandetal.(1995,1996).Notethataninterceptcanbethoughtofasbothasenderorreceiverspeci ce ect.Forsymmetry,weincludetheconstant1/2atthebeginningofeachxs;iandxr;jvector,andestimatethe rstcomponentsof sand rasbeingequal.Usingtheabovereparameterizationfori;j,weestimatetheparametersforthegeneralizedbilinearregressionmodelbyconstructingaMarkovchaininf d; s; r;ab;Z;2z;\rg(whereZdenotestheknmatrixoflatentvectors),havingp( d; s; r;ab;Z;2z;\rjY)astheinvariantdistribution.ThisisobtainedviaanalgorithmbasedonGibbssampling,whichalsosampless;randthe's.Thebasicalgorithmistoiteratethefollowingsteps:1.Samplelineare ects:(a)Sample d;s;rj s; r;ab;\r;;Z(linearregression);(b)Sample s; rjs;r;ab(linearregression);(c)Sampleaband\rfromtheirfullconditionals.2.Samplebilineare ects:(a)Fori=1;:::;n:samplezijfzj;j6=ig;; ;s;r;z;\r(alinearregression);(b)Samplezfromitsfullconditional.3.Sampledyadspeci cparameters:Updatefi;j;j;igusingaMetropolis-Hastingsstep:(a)Propose(i;jj;i)MVN(( 0xi;j+ai+bj+z0izj 0xj;i+aj+bi+z0jzi);\r);(b)Accept(i;jj;i)withprobabilityp(yi;jji;j)p(yj;ijj;i)p(yi;jji;j)p(yj;ijj;i)^1:Variouscombinationsoftheabovestepscanbeusedtoestimatedi erentmodels.Thestepsin1aloneprovideaBayesianestimationprocedureforthelinearregressionproblemhavinganerrorcovarianceasin(2).Bayesianestimationofthenormalbilinearmodelwiththeidentitylinkcouldproceedbyreplacingeachi;jwithyi;jandonlyiteratingsteps1and2.Estimationofageneralizedlinearmixede ectsmodelwithrandome ectsstructuregivenby(2)couldproceedbyiteratingsteps1and3.Thefullconditionaldistributionsrequiredtoperformsteps1and2aregivenbelow.Notethatthe'sareessentiallyunrestrictedintheabovesamplingscheme.Atthislevelthe tissaturatedanddoesnotdependontheregressors,atleasttothedegreethatthepriorfor7 \risdi use.WhattheMCMCalgorithmaboveprovidesisessentiallyasaturated tforthe's(althoughsomewhatsmoothedbythecommonvariance)andanANOVA-likedecompositionofthe'sintoregressor,sender,receiverandinner-producte ects.4.1ConditionalDistributionsfortheLinearE ectsComponents:Notingthati;j z0izj= 0dxi;j+si+rj+\ri;j,weseethatconditionalonthe'sandz's,theotherparameterscanbesampledusingastandardBayesiannormal-theoryregressionapproach,althoughwithacomplicatedcovariancestructure.Fullconditionalof( d;s;r):SimilartoWong's(1982)approachtotheinvariantnormalmodel,weletui;j=i;j+j;i 2z0izjandvi;j=i;j j;iforij.Wethenhave uv!= XuXv!0BB@ dsr1CCA+ uv!;(5)whereXuandXvaretheappropriatedesignmatricesanduandvarevectorsofindependenterrortermswithvariances2u=22\r(1+)and2v=22\r(1 )respectively.Thefullconditionaldistributionof( d;s;r)isthenproportionaltop(u;vj d;s;r;\r)p(s;rj s; r;ab)p( d).Foramultivariatenormal( d; d)priordistributionon d,thetermintheexponentofthefullconditionalis 0" 1 d d 1srXsr sr!+X0uu=2u+X0vv=2v# 12 0" 1 d00 1sr!+X0uXu=2u+X0vXv=2v# where 0=( 0ds0r0),Xsrand srarethecombineddesignmatrixandregressionparametersforsandr,andsristhecovariancematrixof(s0r0)0,whichiseasilyderivedfromab.Theconditionaldistributionisthusmultivariatenormal(;)where=" 1 d d0 1srXsr sr!+X0uu=2u+X0vv=2v#=" 1 d00 1sr!+X0uXu=2u+X0vXv=2v# 1:Notethattheinverseofsrisgivenby 1sr= (2b=)Inn (ab=)Inn (ab=)Inn(2a=)Inn!;=2a2b 2ab:Fullconditionalof( s; r):Thefullconditionalof( s; r)isproportionaltop(s;rj s; r;ab)p( s; r).Assumingamultivariatenormal( sr; sr)priordistributionforthecombinedregression8 parameters,thefullconditionalisamultivariatenormaldistributionwithmeanandvariance(;)givenby=" 1 sr sr+Xsr 1sr sr!#=( 1 s;r+X0sr 1srXsr) 1Fullconditionalofab:Thefullconditionalofabisproportionaltop(s;rj s; r;ab)p(ab).UsingapriordistributionofabinverseWishart(ab0;)(parameterizedsothatE(ab)=ab0=( 3)),thefullconditionalofabisabja;binverseWishart(ab0+(ab)0(ab);+n),wherea=(s Xs s)andb=(r Xr r).Fullconditionalof\r:Usingpriordistributionsof2uinversegamma( u1; u2)and2vinversegamma( v1; v2),thefullconditionalsaregivenby2ujuinversegamma( u1+12 n2; u2+12P[ui ^ui;j]2)and2vjvinversegamma( v1+12 n2; v2+12P[vi ^vi;j]2),where^ui;j=E[ui;jj d;xi;j;si;rj]= 0d(xi;j+xj;i)+si+sj+ri+rj,and^vi;jisgivensimilarly.Thecovariancematrix\rcanbere-constructedfrom2uand2vvia2\r=(2u+2v)=4and=(2u 2v)=(2u+2v).4.2ConditionaldistributionsfortheBilinearE ectsComponent:Letei;j=(i;j+j;i ^ui;j)=2,theresidualofthesymmetricpartofthematrixof'safter ttingthelineare ects,andletu;i;j=\ri;j+\rj;i.Consideringthefullconditionalofzi,wehaveei;1=z0iz1+u;i;1=2ei;2=z0iz2+u;i;2=2...ei;n=z0izn+u;i;n=2;andweseethatsamplingzifromitsfullconditionalisequivalenttoa(Bayesian)linearregressionproblem.Modelingthez'sasaprioriindependentmultivariatenormal(0;z)variables,thefullconditionalofziismultivariatenormal(;)with=4Z iei; i=2u=( 1z+4Z iZ0 i=2u) 1whereZ idenotesthek(n 1)matrixobtainedbyremovingtheithcolumnofZ,andei; idenotesthevectorofresidualsfei;j:j6=ig.Usinganinverse-Wishart(z0;)prior,thefullconditionalofzisinverse-Wishart(z0+ZZ0;+n).Alternatively,ifwerestrictztobe2zIkkanduseaninversegamma( 0; 1)prior,thenthefullconditionalisgivenby2zjZinversegamma( 0+(nk)=2; 1+trace(Z0Z)=2).9 5DataAnalysis:InternationalRelationsinCentralAsiaWeanalyzedataoninternationalrelationsincentralAsiaasrecordedbytheKansasEventDataProject(http://www.ku.edu/keds/project.html)anddescribedbySchrodt,Simpson,andGerner(2001).NewsstoriesaredownloadedfromtheReutersBusinessBrie ngServiceonAfghanistan,Armenia,Azerbaijan,andtheformerSovietRepublicsofCentralAsia,andpoliticalinteractionsbetweencountriesarerecordedandcategorized.Wetakeourresponseyi;jtobethetotalnumberof\positive"actionsreportedlyinitiatedbycountryiwithtargetjfrom1992to1999(i.e.afterthebreakupoftheSovietUnion),asrecordedbytheKEDSproject.Positiveactionshereincludesucheventsasapproval,endorsementorpraiseofonegovernmentbyanother,militaryassistance,formationofalliances,promisesof nancialorpolicysupportandothers(essentiallyalleventshavingGoldsteinscalevaluesgreaterthan2.5,exceptcease- reorcedingofpower.SeetheKEDSprojectwebpageformoredetails).Weincludeinourpopulationthe99countriesclosestingeographicdistancetoAfghanistan,plustheUnitedStates,givingatotalofn=100countriesforanalysis.Wenotethatseventeenoftheone-hundredcountrieshadzeroactionsaseitherinitiatorsortargetsofactionsoverthesevenyearperiod.5.1DataDescriptionSomedescriptiveplotsoftherawdataaregiveninFigure1.Panel(a)plotslog(1+Pj:j6=iyi;j)versuslog(1+Pj:j6=iyj;i)foreachcountryi.ThequantitiesPj:j6=iyi;jandPj:j6=iyj;iaretypicallycalledtheoutdegreeandindegreeofuniti,respectively.Notethestrongcorrelation,whichsuggestsalargevalueofab=(ab)intherandome ectsmodelbeingconsidered.Inpanel(b)weplotthelogofeachcountry'soutdegreeplusone,log(1+Pj:j6=iyi;j),versuslogpopulation,whichsuggestsapositiverelationshipbetweenresponseandpopulation(aplotoflog-indegreeversuspopulationissimilar).Inpanel(c)weplottheresponseonalogscaleversusthegeographicdistanceinthousandsofmilesbetweencountriesiandj.Moreprecisely,thisdistanceisthe\minimumdistance"betweentwocountries,andiszeroifiandjshareaborder.Onaverage,thenumberofeventsbetweentwocountriesdecreasesasgeographicdistanceincreases.ThispatternismademoreclearbyseparatingoutthemeasurementsinvolvingtheUnitedStates(whicharecircled).5.2ModelandPriorsNotethatthedataarefromanobservationalstudy,andthatthedataarenotrandomlysampled.Rather,wehavede nedapopulationofunitsbasedongeographicdistanceandhavemeasurementsonallpairs.Forthisanalysis,weprimarilyinterpretaprobabilitymodelasatoolfordescribingthevarianceinthedataset,andtheregressioncoecientsasmeasuresofthemultiplicative,orlog-linear,componentsoftherelationshipbetweenresponseandregressors.We ttherandome ectsmodel(4)tothedatausingaPoissondistributionandthelog-link,10 012345670123456log(indegree+1)log(outdegree+1)AfghanistanAlbaniaAlgeriaArmeniaAustriaAzerbaijanBahrainBangladeshBelarusBelgiumBhutanBosnia and HerzegovinaBrunei DarussalamBulgariaBurundiCambodiaComorosCroatiaCyprusCzech RepublicDenmarkDjiboutiEgyptEstoniaEthiopiaFinlandFranceGeorgiaGermanyGreeceHong KongHungaryIndiaIndonesiaIranIraqIsraelItalyJordanKazakstanKenyaKoreaKuwaitKyrgyzstanLaosLatviaLebanonLibyaLithuaniaMacauMacedoniaMalaysiaMaldivesMaltaMauritaniaMoldovaMongoliaMyanmarNepalNetherlandsNorth KoreaNorwayOmanPakistanPalestinePhilippinesPolandPR ChinaQatarRomaniaRussiaRwandaSaudi ArabiaSenegalSerbiaSeychellesSingaporeSlovakiaSloveniaSomaliaSri LankaSudanSwedenSwitzerlandSyrian Arab RepublicTaiwanTajikistanTanzaniaThailandTunisiaTurkeyTurkmenistanUgandaUkraineUnited Arab EmiratesUnited KingdomUnited StatesUzbekistanViet NamYemen(a)1012141618200123456log(population)log(outdegree+1)AfghanistanAlbaniaAlgeriaArmeniaAustriaAzerbaijanBahrainBangladeshBelarusBelgiumBhutanBosnia and HerzegovinaBrunei DarussalamBulgariaBurundiCambodiaComorosCroatiaCyprusCzech RepublicDenmarkDjiboutiEgyptEstoniaEthiopiaFinlandFranceGeorgiaGermanyGreeceHong KongHungaryIndiaIndonesiaIranIraqIsraelItalyJordanKazakstanKenyaKoreaKuwaitKyrgyzstanLaosLatviaLebanonLibyaLithuaniaMacauMacedoniaMalaysiaMaldivesMaltaMauritaniaMoldovaMongoliaMyanmarNepalNetherlandsNorth KoreaNorwayOmanPakistanPalestinePhilippinesPolandPR ChinaQatarRomaniaRussiaRwandaSaudi ArabiaSenegalSerbiaSeychellesSingaporeSlovakiaSloveniaSomaliaSri LankaSudanSwedenSwitzerlandSyrian Arab RepublicTaiwanTajikistanTanzaniaThailandTunisiaTurkeyTurkmenistanUgandaUkraineUnited Arab EmiratesUnited KingdomUnited StatesUzbekistanViet NamYemen(b)0510150123456geographic distancelog(yij+1)(c)Figure1:Relationshipsbetween(a)Outdegreeandindegree;(b)Outdegreeandpopulation;(c)Responseandgeographicdistance.ResponsesinvolvingtheUnitedStatesarecircled.sothateachresponseyi;jisassumedtohavecomefromaPoissondistributionwithmeanei;j,andthatthey'sareconditionallyindependentgiventhe's.Wedecomposethevarianceinthe'sasfollows:i;j= 0+ dxi;j+ sxi+ rxj+i;ji;j=ai+bj+\ri;j+z0izj;wherexi;jisthegeographicdistancebetweeniandjandxiisthelogpopulationofi.Forestimationofvariancecomponents,wemodeltherandome ectsashavingthefollowingmultivariatenormaldistributions:(ai;bi)0MVN(0;ab),(\ri;j;\rj;i)0MVN(0;\r),ziMVN(0;2zIkk).Priordistributionsoftheparametersaretakentobe multivariatenormal(0;100I44);abinverseWishart(I22;4);2u;2vi.i.d.inversegamma(1;1),2\r=(2u+2v)=4,=(2u 2v)=(2u+2v).PosteriorcalculationsproceedasdescribedinSection4.5.3SelectingtheLatentDimension:Oneissueinmodel ttingistheselectionofthedimensionkofthelatentvariablesz.Selectionofkcoulddependonthegoaloftheanalysis.Forexample,ifthegoalisdescriptive,i.e.thedesiredendresultisadecompositionofthevarianceintointerpretablecomponents,thenachoiceofk=1;2or3wouldallowforasimplegraphicalpresentationofamultiplicativecomponentofthevariance.11 kLLP(k)logp(yj^ ;^a;^b;^Z;^)AIC^20-3558.78-2432.67-2638.672.381-3351.76-2317.47-2623.471.662-3078.79-2214.68-2620.681.233-3076.73-2127.26-2633.260.874-3077.30-2038.95-2644.950.54Table1:SelectionofkAlternatively,onecouldexaminemodel tasafunctionofkbasedonthelog-likelihood,oruseacross-validationcriterionifoneisprimarilyconcernedwithpredictiveperformance.Consideringlikelihood-basedmeasuresof t,thelog-probabilityofthedatagiventhevaluesoftheparametersgetsevaluatedforeachupdateofthe's,andsologp(Yj)=Pi6=jlogp(yi;jji;j)canbecalculatedwithnoextrae ort.However,suchaquantityisnotappropriateforselectingbetweenmodels.AsdescribedinSection4,themodelisessentiallyunrestrictedinthe's,givinganearlysaturated twhichdoesnotdependmuchonthechoiceofkortheregressors(providedthepriorfor\rissucientlydi use).Alikelihoodthatismoreappropriateisthemarginalprobabilityofdatawithinapair,logp(Yj ;a;b;Z;\r)=P(i;j)logp(yi;j;yj;ij ;ai;bj;aj;bi;zi;zj;\r),wherethesumisoverunorderedpairs.Thisisessentiallythelog-likelihoodtreatingthea;b,andz'sas xede ects.Notethatingenerallogp(yi;j;yj;ij ;ai;bj;aj;bi;zi;zj;\r)isanintegralover\ri;jand\rj;ithatneedstobeapproximated,exceptinthecaseofthenormalmodelwiththeidentitylink.Insomesituationsthepurposeofthemodelistomakepredictionsofunobserveddata.Forexample,supposeonlyasubsetofthen(n 1)responseswererandomlychosentobemeasured.Aslongaswehavesomemeasurementsforeachunit,wecanestimatethee ectsa;bandzforeachunitandmakepredictionsformissingresponsesbasedontheseestimates.Althoughpredictionisnotthegoalforthesedata,forillustrativepurposeswecomparethemarginalprobabilitycriteriondiscussedabovetothefollowingfour-foldcrossvalidationprocedure:1.Randomlysplitthesetoforderedpairsfi;j:i6=jgintofourtestsetsA1;A2;A3;A4.2.Fork=0;1;2;3;4:(a)Forl=1;2;3;4:i.performtheMCMCalgorithmusingonlyfyi;j:fi;jg62Alg,butsamplevaluesofi;jforallorderedpairs.ii.Basedonthesampledvaluesofi;jcomputetheposteriormean^i;jforfi;jg2Alandthelogpredictiveprobabilitylpp(Al)=Pfi;jg2Allogp(yi;jj^i;j).(b)MeasurethepredictiveperformanceforkasLPP(k)=P4l=1lpp(Al).12 3.SelectkbasedonLPP(k).Forthesedata,themarginallikelihoodandcross-validationcriteriaforselectingkaregiveninTable2.Thecrossvalidationproceduresuggeststhatmodelshavingadimensionofk=2;3or4haveroughlythesamepredictiveperformance.Intermsofthemarginallikelihoodcriterion,thebiggestimprovementsin tareingoingfromk=0tok=1andfromk=1tok=2.Theimprovementsin tingoingfrom2to3andfrom3to4dimensionsaresmaller.UsinganAIC-likecriterionandpenalizingtheimprovementinlikelihoodbythenumberofadditionalparameters(100peradditionaldimension),wewouldchoosek=2.Basedontheseresults(andourabilitytoplotresultsintwo-dimensions)wechoosetopresenttheresultsforthek=2modelinmoredetail.050000100000150000200000-11-9-7-5iterationb0050000100000150000200000-0.30-0.20-0.10iterationbd0500001000001500002000000.40.81.21.6iterationbs0500001000001500002000000.40.81.21.6iterationbrFigure2:MarginalMCMCoutputforregressioncoecients.SolidlinesarefromtheMarkovchainwithdata-informedstartingvalues,dashedlinesfromthechainwithuninformedstartingvalues.5.4Resultsfork=2TwoMarkovchainsoflength200,000eachwereconstructedusingthealgorithmdescribedabove.The rstchainusedstartingvaluesofzeroforallregressioncoecientsandcountry-speci cin-tercepts,theidentitymatrixforaband\r,avalueof0:1for2z,andcomponentsofZsampledindependentlyfromanormal(0;2z)distribution.Thesecondchainusedstartingvaluesobtainedfromthefollowingprocedure:Maximumlikelihoodestimatesof d,sandrwereobtainedby t-tinganordinarygeneralizedlinearmodelusinggeographicdistanceasaregressorandsenderandreceiverlabelsasfactorvariables.Estimatesof 0; s; r,andabwereobtainedfromtheesti-matesofsandr.Theiterativelyreweightedleast-squares ttingprocedureproducesamatrixRofworkingresiduals,withtheo diagonalelementsunde ned.Anestimate^ZofZwasthenobtained13 0500001000001500002000000246810iterationsa20500001000001500002000000246810iterationsb20500001000001500002000001.01.21.41.6iterationse20500001000001500002000001.01.52.02.53.0iterationsz2Figure3:MarginalMCMCoutputforvariancecomponentparameters. d s r2a2bab2\r2zmean-0.181.000.946.466.376.41.230.951.99sd0.040.170.171.231.21.210.140.010.27Table2:Posteriormeansandstandarddeviationsfork=2byapproximatingRwithamatrixproductoftheformZ0Z.Thiscanbedonewithaniterativeleast-squaresprocedure,similartotheGibbssamplingprocedureoutlinedinSection4.2:seetenBergeandKiers(1989)formoredetailsonthisproblem.Anestimateof\risthenobtainedfromR ^Z0^Z.SamplesofparametervaluesweresavedfromtheMarkovchainsevery100iterations,andareplottedinFigures2and3.Bothchainsappeartohaveachievedstationarityafterabout50,000iterations,andsowebaseourinferenceonthesavedsamplesafterthispoint.Posteriormeansandstandarddeviationsofthemodelparameters,basedonthe3000savedMCMCsamples(1500fromeachchain),aregiveninTable2.Asintherawdata,weseeanegativerelationbetweenresponseandgeographicdistance(E[ djy]= 0:18),andapositiverelationbetweenresponseandcountrypopulations(E[ sjy]=1:00;E[ rjy]=0:94).Wealsoestimateastrongpositivecorrelationofwithin-dyadresponsesaswellasthewithin-countryrandome ectsaandb.Next,weanalyzetheposteriordistributionofthetheknmatrixoflatentvectorsZ.NotethattheprobabilitymodeldependsonZonlythroughthematrixofinnerproductsZ0Z,whichisinvariantunderrotationsandre\rectionsofZ.Therefore,logPr(YjZ; ;X)=logPr(YjZ; ;X)foranyZwhichisequivalenttoZundertheoperationsofrotationorre\rection.ValuesofZ14 -4-202-3-2-1012AfghanistanAlbaniaAlgeriaArmeniaAustriaAzerbaijanBahrainBangladeshBelarusBelgiumBhutanBosnia and HerzegovinaBrunei DarussalamBulgariaBurundiCambodiaComorosCroatiaCyprusCzech RepublicDenmarkDjiboutiEgyptEstoniaEthiopiaFinlandFranceGeorgiaGermanyGreeceHong KongHungaryIndiaIndonesiaIranIraqIsraelItalyJordanKazakstanKenyaKoreaKuwaitKyrgyzstanLaosLatviaLebanonLibyaLithuaniaMacauMacedoniaMalaysiaMaldivesMaltaMauritaniaMoldovaMongoliaMyanmarNepalNetherlandsNorth KoreaNorwayOmanPakistanPalestinePhilippinesPolandPR ChinaQatarRomaniaRussiaRwandaSaudi ArabiaSenegalSerbiaSeychellesSingaporeSlovakiaSloveniaSomaliaSri LankaSudanSwedenSwitzerlandSyrian Arab RepublicTaiwanTajikistanTanzaniaThailandTunisiaTurkeyTurkmenistanUgandaUkraineUnited Arab EmiratesUnited KingdomUnited StatesUzbekistanViet NamYemenFigure4:PosteriormeanofZsampledfromtheposteriordistributionmayseemat rsttobehighlyvariable,butperhapsarenearlyrotationsofeachotherandarethusnothighlyvariableintermsoftheresultinginnerproductmatrices.ToappropriatelycomparesamplevaluesofZ,wemust rstrotatethemtoacommonorientation.Forthesedatathisisdoneusinga\Procrustean"transformation(Sibson1978),inwhichforeachsampleZwe ndtherotationZofZthathasthesmallestsumofsquareddeviationsfromanarbitrary xedreferencematrixZ0.TherotatedmatrixZwhichminimizesthesumofsquaresisgivenbyZ=Z0Z0(ZZ00Z0Z0) 1=2Z.SeeHo etal.(2002)forfurtherdiscussion.TheresultingmeanofZisgiveninFigure4.Marginaluncertaintyinthez'scouldbedisplayedbyplottingsamplez'sovertheplotofthemeans,usingcolorstodistinguishbetweencountries.Generally,twocountrieswillbemodeledashavingz'sinthesamedirectioniftheyhavelargeresponsestooneanotherrelativetotheirtotalnumberofactionsandcovariatevalues,and/oriftheirresponsesinvolvingothercountriesaresimilar(amodelwhichcandistinguishbetweenthesetwophenomenaisproposedinthediscussion).Forexample,CroatiaandSloveniaareeachrecordedastheinitiatorofanactionwiththeotherasatarget,andeachinitiatesanactionwithSerbiaaswell.WiththeexceptionofoneactionfromSloveniatoItaly,thesearetheonlyeventsrecordedforCroatiaandSlovenia,andsothesecountriesare\similar"inthattheyhaveactionsinvolvingeachotherandtoSerbia,andonlyoneotheractioninvolvinganothercountry.Bosnia-HerzegovinaandDenmarkhavenoactionswithCroatiaorSlovenia,butlikeCroatiaandSloveniatheyeach15 haveoneactionwithSerbiaandveryfewactionsotherwise(eachhasoneactionwithAzerbaijan,andnootheractions),andarethuslocatedinasimilardirection.Serbia,althoughactivewiththisgroupofcountries(onthescaleoftheirresponserates),hasactionswith10othercountries,andisthereforeplacedmoretowardsthecenter.Ofcourse,theposteriorvariancesofthez'sforCroatia,Slovenia,Bosnia-Herzegovina,andDenmarkarequitehigh,asourinformationaboutthemiscomingprimarilyfromthefewnonzeroresponsesamongthem.var[log(yij+1)]Density0.180.200.220.240.260.2805101520(a)0.00.51.01.50.00.51.01.52.0sender specific variancepredicted sender specific variance(b)Figure5:Goodnessof ttests:(a)Posteriorpredictivedistributionofpopulationvariance.(b)Posteriorpredictivecon denceregionsforcountry-speci cvarianceinactioninitiation.Finally,weevaluatesomeaspectsofmodeladequacyviagoodnessof tstatistics.ThisisdonebycomparingtheobservedvalueofastatisticofinterestT(Y)toitsposteriorpredictivedistributionp(T(Ypred)jY).SamplesfromtheposteriorpredictivedistributionareobtainedbysimulatingdatasetsusingtheparameterssampledbytheMarkovchain(see,forexampleGelman,Carlin,SternandRubin1995chapter6).InthepresentcasewemightbeinterestedinanyoverorunderdispersionofthedatarelativetothePoissonmodel.Weevaluateanysuchlackof tbyconsideringasteststatisticstheoverallsamplevarianceoflog(yi;j+1),aswellasthesamplevarianceofflog(yi;j+1):j6=igforeachi,thatis,thevarianceofresponsesfromeachsender,onalogscale.Theposteriorpredictivedistributionsofthesequantitieswereestimatedbysub-sampling1000valuesof( d;s;r;Z;\r)fromthetwoMarkovchains,generatingadatasetfromeachsub-sampledsetofparametervalues,andthencomputingthestatisticsfromeachgenerateddataset.TheresultsareplottedinFigure5.The rstpanelgivesahistogramof1000samplesfromtheposteriorpredictivedistributionoftheoverallvariance.Theposteriorpredictivedistributioniscenteredaroundtheobservedoverallvariance,givenbytheverticalline,andnolackof tisindicatedbythisstatistic.ThesecondpanelofFigure5plotstheobservedsender-speci cvariancesforeachcountryversusa95%posteriorpredictive16 intervalforthatquantity.Thecon denceintervalscontaintheobservedvaluesfor97ofthe100countries,andthusdonotindicatemuchlack-of- t.ThePoissonmodelseemsto tthevarianceinresponsereasonablywell,atleastintermsofthesestatistics.6DiscussionThisarticlehaspresentedanapproachtomodelingthirdorderdependencepatternsoftenseenindyadicdatasets,suchassocialnetworks.Themodelsarebasedongeneralizedlinearmixede ectsmodelswiththeadditionofareduced-rankinteractiontermcomposedofinnerproductsoflatentcharacteristicvectors.Suchanapproachallowsfortheanalysisofdyadicdatausingfamiliarregressiontools,butalsoallowsonetocapturepatternssuchastransitivity,balance,andclusterabilitywhichareoftenofinteresttosocialscienceresearchers.Otherapproachestocapturingsuchdependencepatternshaveusedmetricdistances(Ho etal.2002)andultrametricdistances(SchweinbergerandSnijders,2003),althoughnotinthepresenceofthecovariancestructure(2).Whilesuchlatentdistancemodelsmaybeeasytounderstand,theinner-productapproachhassomeconceptualappeal,asthetermz0izjcanbeviewedasamean-zerorandome ect.Anotherdependencepatternoftenofinteresttoresearchersisthatofstochasticequivalence,inwhichtwounitsiandjaresaidtobestochasticallyequivalentiftheirresponseshavethesameprobabilitydistribution,i.e.p(yi;1;:::;yi;n)=p(yj;1;:::;yj;n).Themodelconsideredinthispaper,aswellasthelatentdistanceapproachesmentionedabove,potentiallyconfoundstochasticequivalencepatternswiththoseofclusterabilityandbalance:twounitswillgenerallybeestimatedtohavesimilarlatentcharacteristicvectorsiftheyhavestrongrelationstoeachother,orhavesimilarrelationstoothersunitunits.However,insomedatasetstheremaybeclustersofunitsthatrelatesimilarlytoothers,butnotstronglytoeachother.NowickiandSnijders(2001)consideredalatentclassmodelwhichidenti edclustersofsuchstochasticallyequivalentunits,butdidnotseparatelyconsiderclusteringbasedonstrengthofrelations.Apossibleapproachtomodelingbothtypesofpatternsistoextendthebilineare ectdiscussedinthispapertoamoregeneralasymmetricbilineare ectsuchasz0iRzj,whereRisakkmatrix.Estimationofsimilartypesofe ectshasbeenconsideredbybyGabriel(1998),andleastsquaresrepresentationsofanasymmetricmatrixYbyZ0RZhasbeenconsideredbytenBergeandKiers(1989),Kiers(1989)andTrenda lov(2002),amongothers.Inthepresentapplication,thevectorzicouldbeinterpretedasgivinggradesofmembershipforunititoeachofkclasses,andRlmastheresponseratefromclassltom.Interestingly,therestrictionofeachzitobeunityatonecomponentandzeroattheothersgivesarepresentationofthelatentclassmodelofNowickiandSnijders(2000).Unrestrictedestimationofz0iRzj,inthepresenceoftheerrorstructure(2),isatopicofcurrentresearchbytheauthor.17 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