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.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
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