Regression models with correlated binary response

RegressionModelswithCorrelatedBinaryResponseVariables:AComparisonofDierentMethodsinFiniteSamplesMartinSpiessandAlfredHamerleAbstractThepresentpaperdealswiththecomparisonoftheperformanceofdierentestimationmethodsforregressionmodelswithcorrelatedbinaryresponses.Throughout,weconsiderprobitmodelswhereanunderlyinglatentcontinousrandomvariablecrossesathreshold.Theerrorvariablesintheunobservablelatentmodelareassumedtobenormallydistributed.Theestimationproceduresconsideredare(1)marginalmaximumlikeli-hoodestimationusingGauss-Hermitequadrature,(2)generalizedestima-tionequations(GEE)techniqueswithanextensiontoestimatetetrachoriccorrelationsinasecondstep,and,(3)theMECOSAapproachproposedbySchepers,ArmingerandKusters(1991)usinghierarchicalmeanandcovariancestructuremodels.Wepresenttheresultsofasimulationstudydesignedtoevaluatethesmallsamplepropertiesofthedierentestima-torsandtomakesomecomparisonswithrespecttotechnicalaspectsoftheestimationproceduresandtobiasandmeansquarederroroftheestima-tors.TheresultsshowthatthecalculationoftheMLestimatorrequiresthemostcomputingtime,followedbytheMECOSAestimator.ForsmallandmoderatesamplesizesthecalculationoftheMECOSAestimatorisproblematicbecauseofproblemsofconvergenceaswellasatendencyofunderestimatingthevariances.Inlargesampleswithmoderateorhighcorrelationsoftheerrorsinthelatentmodel,theMECOSAestimatorsarenotasecientasMLorGEEestimators.Thehigherthe‘true’valueofanequicorrelationstructureinthelatentmodelandthelargerthesamplesizesare,themoreistheeciencygainoftheMLestimatorcomparedtotheGEEandMECOSAestimators.UsingtheGEEapproach,theMLestimatesoftetrachoriccorrelationscalculatedinasecondsteparebiasedtoasmallerextentthanusingtheMECOSAapproach.Keywords:maximumlikelihood;Gauss-Hermitequadrature,general-izedestimationequations;meanandcovariancestructureanalysis;tetra-choriccorrelations;simulationstudyLehrstuhlfurStatistik,WirtschaftswissenschaftlicheFakultat,UniversitatRegensburg121.INTRODUCTION1IntroductionThesubjectofthepresentpaperistodiscussandcomparethreedierentmeth-odsfortheestimationofregressionmodelsappliedtodatasetswithcorrelatedbinaryresponsevariables.Thiskindofdatasetsoftenariseinappliedsciences,forexampleinstudieswith(t=1;:::;T)repeatedmeasureson(n=1;:::;N)individualsorwithTnmeasuresondierentindividualswithinthesamefamiliesorblocks.Theselectionofastatisticalmodelaswellasanestimationmethodthenhingese.g.uponthenumberofobservationsperrealizationofthevectorofcovariatesorwhetherthestructureofassociationbetweentheresponsevariablesisofscienticinterestornot.Ingeneral,seriouscomputationaldicultiesariseintheapplicationofthemethodofmaximumlikelihood(ML)tothesemodelsbecauseofthelackofarichclassofdistributionssuchasthemultivariateGaus-sianinthecaseofcontinuousresponsevariables.Hencelikelihoodmethodsareonlyavailableinafewcases.Anexampleistherandomeectsprobitmodel(e.g.BockandLieberman,1970;Heckman,1981).Startingwithalatentlinearregressionmodel,withtheobservableresponsevariabletakingonthevalue1if(andonlyif)thelatent,notobservableresponsevariablecrossesathreshold,and0otherwise(Pearson,1900),itisoftenassumedthattheerrortermofthelatentlinearmodelhasacomponentsofvariancestructurewhichinturnimpliesanequicorrelationstructureinthecorrelationmatrixofthelatenterrors.Assumingthisassociationstructurethecomputationofthelog-likelihoodfunctionandtheirderivativesbecomesfeasiblebecauseonlyone-foldintegralshavetobeevaluated.ThiscanapproximatelybedoneusingGauss-Hermitequadrature(AndersonandAitkin,1985;BockandLieberman,1970;ButlerandMot,1982).ProvidedthatenoughevaluationpointsareusedtheMLestimatorsandtheirestimatedvariancesareunbiased(Butler,1985).Toestimateabroaderclassofmodelsalternativeapproacheshavebeensug-gested.Oneapproachisthe‘generalizedestimationequations’approach(LiangandZeger,1986;ZegerandLiang,1986)whichleadstoconsistentandasymp-toticallynormallydistributedestimatesforthevectorofregressionparametersgivenonlythecorrectspecicationoftherstmomentsoftheresponsevariables(GEE1approach).Inaddition,consistentestimatesforthevariancesofthere-gressionparameterestimatorsareavailable.Althoughthedependenciesbetweentheobservableormanifestresponsevariablesaretakenintoaccounttoincreaseeciency,theassociationistreatedasanuisance.Incontrast,ZhaoandPren-tice(1990;seealsoPrentice(1988)andLiang,ZegerandQaqish(1992))denea‘generalizedestimationequations’approachforsimultaneousinferenceonre-gressionandassociationparameters(GEE2approach).Theconsistencyoftheseparametersdependsuponthecorrectspecicationoftherstmomentsoftheresponsevariablesandthecorrectmodellingoftheassociationstructure.NotethattheregressionparameterestimatesusingGEE1areconsistentwhetheror3nottheassociationstructureiscorrectlyspecied,whilethisnotnecessarilyholdsfortheGEE2approach.Liang,ZegerandQaqish(1992)foundtheregressionparameterestimatesusingbothapproachestobesimilarecientiftheyareesti-matedundercorrectspecications.Ontheotherhand,theGEE1approachmayleadtoinecientestimationoftheassociationparameters.TheresultsofsimulationstudiesusingtheGEE1appr