多水平模型英文原著chap9.docx

多水平模型英文原著chap9.docx

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多水平模型英文原著chap9

Chapter9

Multileveleventhistorymodels

9.1Eventhistorymodels

Thisclassofmodels,alsoknownassurvivaltimemodelsoreventdurationmodels,haveastheresponsevariablethelengthoftimebetween'events'.Sucheventsmaybe,forexample,birthanddeath,orthebeginningandendofaperiodofemploymentwithcorrespondingtimesbeinglengthoflifeordurationofemployment.Thereisaconsiderabletheoreticalandappliedliterature,especiallyinthefieldofbiostatisticsandausefulsummaryisgivenbyClayton（1988）.Weconsidertwobasicapproachestothemodellingofdurationdata.Thefirstisbasedupon'proportionalhazard'models.Thesecondisbasedupondirectmodellingofthelogduration,oftenknownas'acceleratedlifemodels'.Inbothcaseswemaywishtoincludeexplanatoryvariables.

Themultilevelstructureofsuchmodelsarisesintwogeneralways.Thefirstiswherewehaverepeateddurationswithinindividuals,analogoustoourrepeatedmeasuresmodelsofchapter5.Thus,individualsmayhaverepeatedspellsofvariouskindsofemploymentofwhichunemploymentisone.Inthiscasewehavea2-levelmodelwithindividualsatlevel2,oftenreferredtoasarenewalprocess.Wecanincludeexplanatorydummyvariablestodistinguishthesedifferentkindsofemploymentorstates.Thesecondkindofmodeliswherewehaveasingledurationforeachindividual,buttheindividualsaregroupedintolevel2units.Inthecaseofemploymentdurationthelevel2unitswouldbefirmsoremployers.Ifwehadrepeatedmeasuresonindividualswithinfirmsthenthiswouldgiverisetoa3-levelstructure.

9.2Censoring

Acharacteristicofdurationdataisthatforsomeobservationswemaynotknowtheexactdurationbutonlythatitoccurredwithinacertaininterval,knownasintervalcensoreddata,waslessthanaknownvalue,leftcensoreddata,orgreaterthanaknownvalue,rightcensoreddata.Forexample,ifweknowatthetimeofastudy,thatsomeoneenteredherpresentemploymentbeforeacertaindatethentheinformationavailableisonlythatthedurationislongerthanaknownvalue.Suchdataareknownasrightcensored.Inanothercasewemayknowthatsomeoneenteredandthenleftemploymentbetweentwomeasurementoccasions,inwhichcaseweknowonlythatthedurationliesinaknowninterval.ThemodelsdescribedinthischapterhaveproceduresfordealingwithcensoringInthecaseoftheparametricmodels,wheretherearerelativelylargeproportionsofcensoreddatatheassumedformofthedistributionofdurationlengthsisimportant,whereasinthepartiallyparametricmodelsthedistributionalformisignored.Itisassumedthatthecensoringmechanismisnoninformative,thatisindependentofthedurationlengths.

Insomecases,wemayhavedatawhicharecensoredbutwherewehavenodurationinformationatall.Forexample,ifwearestudyingthedurationoffirstmarriageandweendthestudywhenindividualsreachtheageof30,allthosemarryingforthefirsttimeafterthisagewillbeexcluded.Toavoidbiaswemustthereforeensurethatageofmarriageisanexplanatoryvariableinthemodelandreportresultsconditionalonageofmarriage.

Thereisavarietyofmodelsfordurationtimes.Inthischapterweshowhowsomeofthemorefrequentlyusedmodelscanbeextendedtohandlemultileveldatastructures.Weconsiderfirsthazardbasedmodels.

9.3Hazardbasedmodelsincontinuoustime

Theunderlyingnotionsarethoseofsurvivorandhazardfunctions.Considerthe（singlelevel）casewherewehavemeasuresoflengthofemploymentonworkersinafirm.Wedefinetheproportionoftheworkforceemployedforperiodsgreaterthantasthesurvivorfunctionanddenoteitby

where

isthedensityfunctionoflengthofemployment.Thehazardfunctionisdefinedas

andrepresentstheinstantaneousrisk,ineffectthe（conditional）probabilityofsomeonewhoisemployedattimet,endingemploymentinthenext（small）unitintervaloftime.

Thesimplestmodelisonewhichspecifiesanexponentialdistributionforthedurationtime,

whichgives

sothatthehazardrateisconstantand

.Ingeneral,however,thehazardratewillchangeovertimeandanumberofalternativeformshavebeenstudied（seeforexample,CoxandOakes,1984）.AcommononeisbasedontheassumptionofaWeibulldistribution,namely

ortheassociatedextremevaluedistributionformedbyreplacingby

.Anotherapproachtoincorporatingtime-varyinghazardsistodividethetimescaleintoanumberofdiscreteintervalswithinwhichthehazardrateisassumedconstant,thatisweassumeapiecewiseexponentialdistribution.Thismaybeusefulwherethereare'natural'unitsoftime,forexamplebasedonmenstrualcyclesintheanalysisoffertility,andthiscanbeextendedbyclassifyingunitsbyotherfactorswheretimevariesovercategories.Wediscusssuchdiscretetimemodelsinalatersection

Themostwidelyusedmodels,towhichweshalldevoteourdiscussion,arethoseknownasproportionalhazardsmodels,andthemostcommondefinitionis

.Thetermdenotesalinearfunctionofexplanatoryvariableswhichweshallmodelexplicitlyinsection9.5.Itisassumedthat

thebaselinehazardfunction,dependsonlyontimeandthatallothervariationbetweenunitsisincorporatedintothelinearpredictor.Thecomponentsofmayalsodependupontime,andinthemultilevelcasesomeofthecoefficientswillalsoberandomvariables.

9.4Parametricproportionalhazardmodels

Forthecasewherewehaveknowndurationtimesandrightcensoreddata,definethecumulativebaselinehazardfunction

andavariablewithmean

takingthevalueoneforuncensoredandzeroforcensoreddata.Itcanbeshown（McCullaghandNelder,1987）thatthemaximumlikelihoodestimatesrequiredarethoseobtainedfromamaximumlikelihoodanalysisforthismodelwherewistreatedasaPoissonvariable.ThiscomputationaldeviceleadstotheloglinearPoissonmodelforthei-thobservation

（9.1）

wheretheterm

istreatedasanoffset,thatis,aknownfunctionofthelinearpredictor..

Thesimplestcaseistheexponentialdistribution,forwhichwehave

.Equation（9.1）thereforehasanoffset

andtheterm

isincorporatedinto.WecanmodeltheresponsePoissoncountusingtheproceduresofchapter6,withcoefficientsinthelinearpredictorchosentoberandomatlevels2orabove.Thisapproachcanbeusedwithotherdistributions.FortheWeibulldistribution,ofwhichtheexponentialisaspecialcase,theproportionalhazardsmodelisequivalenttothelogdurationmodelwithanextremevaluedistributionandweshalldiscussitsestimationinalatersection.

9.5ThesemiparametricCoxmodel

Themostcommonlyusedproportionalhazardmodelsareknownassemiparametricproportionalhazardmodelsandwenowlookatthemultilevelversionofthemostcommonoftheseinmoredetail.

Considerthe2-levelproportionalhazardmodelforthejk-thlevel1unit

（9.2）

whereistherowvectorofexplanatoryvariablesforthelevel1unitandsomeorallofthearerandomatlevel2.Weadoptthesubscriptsj,kforlevelsoneandtwoforreasonswhichwillbeapparentbelow.

Wesupposethatthetimesatwhichalevel1unitcomestotheendofitsdurationperiodor'fails'areorderedandateachoftheseweconsiderthetotal'riskset'.Atfailuretimetherisksetconsistsofallthelevel1unitswhichhavebeencensoredorforwhichafailurehasnotoccurredimmediatelypreceedingtime.Thentheratioofthehazardfortheunitwhichexperiencesafailureandthesumofthehazardsoftheremainingrisksetunitsis

whichissimplytheprobabilitythatthefailedunitistheonedenotedby

（Cox,1972）.Itisassumedthat,conditionalonthe,theseprobabilitiesareindependent.

Severalproceduresareavailableforestimatingtheparametersofthismodel（seeforexampleClayton,1991,1992）.Forourpurposesitisconvenienttoadoptthefollowing,whichinvolvesfittingaPoissonorequivalentmultinomialmodelofthekinddiscussedinchapter7.

Ateachfailuretimewedefinearesponsevariateforeachmemberoftheriskset

whereiindexesthemembersoftheriskset,andj,klevel1andlevel2units.Ifwethinkofthebasic2-levelmodelasoneofemployeeswithinfirmsthenwenowhavea3-levelmodelwhereeachlevel2unitisaparticularemployeeandcontaininglevel1unitswhereisthenumberofrisksetstowhichtheemployeebelongs.Level3isthefirm.Theexplanatoryvariablescanbedefinedatanylevel.Inparticulartheycanvaryacrossfailuretimes,allowingsocalledtime-varyingcovariates.Overallproportionality,conditionalontherandomeffects,canbeobtainedbyorderingthefailuretimesacrossthewholesample.Inthiscasethemarginalrelationshipbetweenthehazardandthecovariatesgenerallyisnotproportional.Alternatively,wecanconsiderthefailuretimesorderedonlywithinfirms,sothatthemodelyieldsproportionalhazardswithinfirms.Inthiscasewecanstructurethedataasconsistingoffirmsatlevel3,failuretimesatlevel2andemployeeswithinrisksetsatlevel1.Inbothcases,becausewemaketheassumptionofindependenceacrossfailuretimeswithinfirms,thePoissonvariationisatlevel1andthereisnovariationatlevel2.Inotherwordswecancollapsethemodeltotwolevels,withinfirmsandbetweenfirms.

AsimplevariancecomponentsmodelfortheexpectedPoissoncountiswrittenas

（9.3）

wherethereisa'blockingfactor'foreachfailuretime.Infactwedonotneedgenerallytofitallthesenuisanceparameters:

insteadwecanobtainefficientestimatesofthemodelparametersbymodellingasasmoothfunction