多水平模型(英文原著)chap

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.

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9.3Hazardbasedmodelsincontinuoustime

Theunderlyingnotionsarethoseofsurvivorandhazardfunctions.Considerthe(singlelevel)casewherewehavemeasuresoflengthofemploymentonworkersinafirm.Wedefinetheproportionoftheworkforceemployedforperiodsgreaterthantasthesurvivorfunctionanddenoteitby

whereisthedensityfunctionoflengthofemployment.Thehazardfunctionisdefinedas

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

Thesimplestmodelisonewhichspecifiesanexponentialdistributionforthedurationtime,

whichgives,sothatthehazardrateisconstantand=.In

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

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

Themostwidelyusedmodels,towhichweshalldevoteourdiscussion,arethoseknownas

proportionalhazardsmodels,andthemostcommondefinitionisThetermn

denotesalinearfunctionofexplanatoryvariableswhichweshallmodelexplicitlyinsection9.5.It

isassumedthatthebaselinehazardfunction,dependsonlyontimeandthatallothervariation

betweenunitsisincorporatedintothelinearpredictorn.Thecomponentsofnmayalsodepend

upontime,andinthemultilevelcasesomeofthecoefficientswillalsoberandomvariables.

9.4Parametricproportionalhazardmodels

Forthecasewherewehaveknowndurationtimesandrightcensoreddata,definethecumulative

baselinehazardfunctiontandavariablevwithmean,takingthevalue

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

(9.1)

wherethetermgistreatedasanoffset,thatis,aknownfunctionofthelinearpredictor.

Thesimplestcaseistheexponentialdistribution,forwhichwehave.Equation(9.1)

thereforehasanoffsetⅡandthetermIQisincorporatedinton.WecanmodeltheresponsePoissoncountusingtheproceduresofchapter6,withcoefficientsinthelinearpredictorchosentoberandomatlevels2orabove.Thisapproachcanbeusedwithotherdistributions.FortheWeibulldistribution,ofwhichtheexponentialisaspecialcase,theproportionalhazardsmodelisequivalent

tothelogdurationmodelwithanextremevaluedistributionandweshalldiscussitsestimationina

latersection.

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9.5ThesemiparametricCoxmodel

Themostcommonlyusedproportionalhazardmodelsareknownassemiparametricproportionalhazardmodelsandwenowlookatthemultilevelversionofthemostcommonoftheseinmore

detail.

Considerthe2-levelproportionalhazardmodelforthejk-thlevellunit

(9.2)

whereistherowvectorofexplanatoryvariablesforthelevel1unitandsomeorallofthe厭

arerandomatlevel2.Weadoptthesubscriptsj,kforlevelsoneandtwoforreasonswhichwillbeapparentbelow.

Wesupposethatthetimesatwhichalevellunitcomestotheendofitsdurationperiodor'fails'areorderedandateachoftheseweconsiderthetotal'riskset'.Atfailuretimet.therisksetconsistsofallthelevellunitswhichhavebeencensoredorforwhichafailurehasnotoccurredimmediatelypreceedingtimeti.Thentheratioofthehazardfortheunitwhichexperiencesafailureandthesumofthehazardsoftheremainingrisksetunitsis

whichissimplytheprobabilitythatthefailedunitistheonedenotedby(Cox,1972).Itis

assumedthat,conditionalonthe,theseprobabilitiesareindependent.

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

Ateachfailuretimelwedefinearesponsevariateforeachmemberoftheriskset

whereiindexesthemembersoftheriskset,andj,klevel1andlevel2units.Ifwethinkofthebasic2-levelmodelasoneofemployeeswithinfirmsthenwenowhavea3-levelmodelwhereeachlevel2unitisaparticularemployeeandcontainingnalevellunitswherenzisthenumberofrisksetstowhichtheemployeebelongs.Level3isthefirm.Theexplanatoryvariablescanbedefinedatanylevel.Inparticulartheycanvaryacrossfailuretimes,allowingsocalledtime-varyingcovariates.Overallproportionality,conditionalontherandomeffects,canbeobtainedbyorderingthefailuretimesacrossthewholesample.Inthiscasethemarginalrelationshipbetweenthehazardandthecovariatesgenerallyisnotproportional.Alternatively,wecanconsiderthefailuretimesorderedonlywithinfirms,sothatthemodelyieldsproportionalhazardswithinfirms.Inthiscasewecanstructurethedataasconsistingoffirmsatlevel3,failuretimesatlevel2andemployeeswithinrisksetsatlevel1.Inbothcases,becausewemaketheassumptionofindependenceacrossfailuretimeswithinfirms,thePoissonvariationisatlevel1andthereisnovariationatlevel2.Inotherwordswecancollapsethemodeltotwolevels,withinfirmsandbetweenfirms.

AsimplevariancecomponentsmodelfortheexpectedPoissoncountiswrittenas

(9.3)

wherethereisa'blockingfactor'cforeachfailuretime.Infactwedonotneedgenerallytofitallthesenuisanceparameters:insteadwecanobtainefficientestimatesofthemodelparametersbymodellingarasasmoothfunctionofthetimepoints,using,say,aloworderpolynomialorasplinefunction(Efron,1988).

Forthemodelwhichassumesoverallproportionalityanestimatorofthebaselinesurvivingfractionforanindividualinthek-thfirmattimeh,where,is

andtheestimateforanindividualwithspecificcovariatevaluesX;xis

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(9.4)

Forthemodelwhichassumesproportionalitywithinfirmsthesetwoexpressionsbecomerespectively

Wherewefitpolynomialstotheblockingfactors,thea,areestimatedfromthepolynomial

coefficients,andthesurvivingfractioncanbeplottedagainstthetimeassociatedwitheachinterval.

9.6Tiedobservations

Wehaveassumedsofarthateachfailuretimeisassociatedwithasinglefailure.Inpracticemanyfailureswilloftenoccuratthesametime,withintheaccuracyofmeasurement.Sometimes,datamayalsobedeliberatelygroupedintime.InthiscaseallthefailuresattimesIhavearesponseequalto1.ThisprocedureforhandlingtiesisequivalenttothatdescribedbyPeto(1972)(seealsoMcCullaghandNelder,1989).

9.7Repeatedmeasuresproportionalhazardmodels

Asinthecaseofordinaryrepeatedmeasuresmodelsdescribedinchapter6wecanconsiderthecaseofmultipleepisodesordurationswithinindividualswithbetweenandwithinindividualvariationandpossiblyfurtherlevelswhereindividualsmaybenestedwithinfirms,etc.Themodelsofprevioussectionscanbeappliedtosuchdata,buttherearefurtherconsiderationswhicharise.Whereeachindividualhasthesamefixednumbernofepisodes.Wecantreatthese,asinchapter5,asconstitutingnvariatessothatwehaveann-variatemodelwithan(nxn)covariancematrixbetweenindividuals.Thevariatesmaybeeitherreallydistinctmeasurementsorsimplythedifferentepisodesinafixedordering.ThisisthemodelconsideredbyWeietal(1989)whodefineproportionalityaswithinindividuals.Wecanalsomodelamultivariatestructurewhere,withinindividuals,therearerepeatedepisodesforanumberofdifferenttypesofinterval.Foreachtypeofintervalwemayhavecoefficientsrandomattheindividuallevelandthesecoefficientswillgenerallyalsocovaryatthatlevel.

Oftenwithrepeatedmeasuresmodelsthefirstepisodeisdifferentinnaturefromsubsequentones.Anexamplemightbethefirstepisodeofadiseasewhichmaytendtobelongerorshorterthansubsequentepisodes.Ifthefirstepisodeistreatedasifitwereaseparatevariatethenthesubsequentepisodescanberegardedashavingthesamedistribution,asintheprevioussection.

Anotherpossiblecomplicationinrepeatedmeasuresdata,asinchapter5isthatwemaynotbeabletoassumeindependencebetweendurationswithinindividuals.Thiswillthenleadtoserialcorrelationmodelswhichcanbeestimatedusingtheproceduresdiscussedinchapter6fortheparametriclogdurationmodelsdiscussedbelow.

9.8Exampleusingbirthintervaldata

Thedataareaseriesofrepeatedbirthintervalsfor379HutteritewomenlivinginNorthAmerica(LarsenandVaupel,1993;Egger,1992).Theresponseisthelengthoftimeinmonthsfrombirthtoconception,rangingfromlto160,withthefirstbirthintervalignoredandnocensoredinformation.Thisgives2235birthsinall.

Thereisinformationavailableonthemother'sbirthyear,herageinyearsatthestartofthebirthinterval,whetherthepreviouschildwasaliveordead,andthedurationofmarriageatthestartofthebirthinterval.Sincewehavealargenumberofwomeneachwitharelativelysmallnumberofintervalswehaveassumedoverallproportionality,withfailuretimesorderedacrossthewholesample.Table1givestheresultsforavariancecomponentsanalysisandonewhereseveralrandomcoefficientsareestimated.Afourthorderpolynomialwasadequatetosmooththeblockingfactors.

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Table9.1ProportionalhazardsmodelforHutteritebirthintervals.Inthe

randompartsubscript0referstointercept,1topreviousdeath.

Parameter

Fixed

Intercept

Mother'sbirthyear-1900

Mother'sage(year-20)

Previousdeath

Marriageduration(Months)

Estimate(s.e.)

A

-3.65

0.026(0.003)

-0.008(0.014)

0.520(0.118)

-0.003(0.001)

Estimate(s.e.)

B

-3.64

0.026(0.003)

-0.004(0.014)

0.645(0.144)

-0.004(0.001)

Random

c

。

〇aol

0.188(0.028)

0.188(0.028)

0.005(0.088)

0.381(0.236)

Theonlycoefficientestimatedwithanon-zerovarianceatlevel2waswhetherornotthepreviousbirthdied,butalargesamplechisquaredtestforthetworandomparametersforthiscoefficientgivesaP-valueof0.01on2degreesoffreedom.Anincreaseonthelinearscaleisassociatedwithashorterinterval.Thusthebirthintervaldecreasesforthelaterbormothersandalsoifthepreviousbirthisadeath.Theintervalissomewhatshorterthelongerthemarriagedurationwithlittleadditionaleffectofmaternalage.Thisapparentlackofasubstantialageeffectseemstobeaconsequenceofthehighcorrelation(0.93)betweendurationofmarriageandage.Higherordertermsfordurationandagewerefittedbuttheestimatedcoefficientsweresmallandnotsignificantatthe10%level.Thebetween-individualstandarddeviationisabout0.4whichiscomparableinsizetotheeffectofapreviousdeath.Thebetween-individualstandarddeviationforamodelwhichfitsnocovariatesis0.45sothatthecovariatesexplainonlyasmallproportionofthebetween-individualvariation.Figure9.1showstwoaverageestimatedsurvivingfractioncurvesforawomanaged20,bornin1900withmarriageduration12months.Thehigheroneisforthosewheretherewasapreviouslivebirthandthelowerwheretherewasapreviousdeath.

Figure9.1Probabilityofexceedingeachbirthintervallength;livebirthupper,previousdeathlower.

9.9Thediscretetime(piecewise)proportionalhazardsmodel

Wheretimeisgroupedintopreassignedcategorieswewritethesurvivorfunctionattimeintervall,theprobabilitythatfailureoccursafterthisinterval,ass.Thisgives

Thisgives

whichcanbeusedtoestimatethesurvivorfunctionfromasetofestimatedhazards.

Fortheproportionalhazardsmodel(9.2)anda2-levelmodeltheexpectedhazardisgiven(Aitkinet

al,1989)by

where,asbefore,theaareconstantstobeestimated,oneforeachtimeinterval.Thisleadstoamodelwheretheresponseisabinomialvariate,beingthenumberofdeathsdividedbythenumberintherisksetatthestartoftheinterval(seealsoEgger,1992).Anycensoredobservationsinanintervalareexcludedfromtheriskset.Theestimationfollowsthatforthelogitbinomialmodel

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describedinchapter7,exceptthatwenowrequirethefirstandseconddifferentialsoftheloglogfunction,namely

AsintheCoxmodel,wecanfitapolynomialfunctiontothesuccessivetimeintervals,ratherthanthefullsetofblockingfactors.Thedatawillbeorderedwithinlevel2unitssothatarisksetingeneralwillextendoverseveralsuchunits.Ageneralprocedureistospecifytheresponseforeachlevel1unitasbinary,thatiszeroiftheunitsurvivestheintervalandoneifnot,withtheappropriatecinthefixedpart.Thusa2-levelmodelwillbecomespecifiedasa3-levelmodelwiththebinomialvariationatlevel1andtheactuallevellunitsatlevel2.Themodelcanbefurtherextendedtopolytomousoutcomes,or'competingrisks',whereseveraldifferentkindsoffailurecanoccur.Theanalysisfollowsthesamepattern,butwiththeresponsebeingamultinomial

variateandthecorrespondingmodelsofChapter7canbeappliedwithadifferentlinearpredictorforeachoutcomecategory.

9.10Logdurationmodels

Fortheacceleratedlifemodelthedistributionfunctionfordurationiscommonlyassumedtobeof

theform

where天isabaselinefunction(CoxandOakes,1984).Fora2-levelmodelthiscanbewrittenas

(9.5)

whichisinthestandardformfora2-levelmodel.WeshallassumeNormalityfortherandomcoefficientsatlevel2(andhigherlevels)butatlevel1weshallstudyotherdistributionalformsforthee.Thelevel1distributionalformisimportantwheretherearecensoredobservations.WefirstconsiderthecommonchoiceofanextremevaluedistributionforthelogdurationL,conditionalon ,whichaswenotedabove,impliesanequivalencewiththeproportionalhazardsmodel.Omittinglevelsubscriptswewrite

For(9.5)thisgives

(9.6)

(9.7)

Wherethedifferentialisforuseintheestimationofcensoreddataandiswithrespecttoβinthe

expressionbelow.

ThemeanofLisincorporatedintothefixedpredictor.Ifwehavenocensoreddataweestimatetheparametersforthemodelgivenby(9.5)bytreatingitasastandardmultilevelmodel.Wenotethattheestimationisstrictlyquasilikelihoodsinceweareusingonlythemeanandvariancepropertiesofthelevel1distribution.Ifweassumeasimplelevel1variancethenwecaniterativelyestimatecfromtheaboverelationshipandwealsoobtainforthe2-levelmodel(9.5)

Wherethereiscomplexvariationatlevel1thencwillvarywiththelevellunits.Toestimatethesurvivalfunctionforagivenlevel2unitwefirstconditiononthecovariatesandrandomcoefficients,thatisX,B,andthenuse(9.7).

Wecanchooseotherdistributionalformsforthelogdurationdistribution.Theseincludetheloggammadistribution,theNormalandthelogistic.Thus,forexample,fortheNormaldistributionwehave

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whereφarethecumulativeanddensityfunctionsofthestandardNormaldistribution.

Quasilikelihoodestimatescanbeobtainedforanysuitabledistributionwithtwoparameters.Thepossibilityoffittingcomplexvariationatlevel1canbeexpectedtoprovidesufficientflexibilityusingthesedistributionsformostpurposes.ingthesedistributionsformostpurposes.

Table9.2.LogdurationofbirthintervalforHutteritewomen.Subscript1referstobirthyear,2

toageand3topreviousdeath.

Parameter

Fixed

Intercept

Mother's

Mother'sPrevious

Marriage

Random

Level2

birthyear-1900

age-20

death

duration(Months)

Cmo

o

〇m?

Level1

。

3

-2loglikelihood

Estimate(s.e.)Estimate(s.e.)Estimate(s.e.)

ABC

1.971.961.97

-0.021(0.002)-0.005(0.010)-0.435(0.079)

0.003(0.001)

-0.021(0.002)-0.005(0.010)

0.436(0.079)

0.003(0.001)

-0.021(0.002)-0.005(0.010)-0.438(0.089)

0.003(0.001)

0.127(0.017)

0.121(0.054)

-0.001(0.002)0.0001(0.0001)

-0.005(0.003)

0.0001(0.0001)

0.0006(0.0003)

0.114(0.052)

-0.001(0.002)

0.0001(0.0001)

-0.004(0.003)

0.0001(0.0001)

0.0005(0.0003)

0.549(0.018)0.533(0.018)0.522(0.018)

0.200(0.108)

5305.95295.55290.8

9.11Censoreddata

Wheredataarecensoredinlogdurationmodelswerequirethecorrespondingprobabilities.Thus,forrightcensoreddatawewoulduse(9.7)withcorrespondingformulaeforintervalorleftcensoreddata.Foreachcensoredobservationwethereforehaveanassociatedprobability,sayg,withtheresponsevariablevalueofone.

Thisleadstoabivariatemodel,inwhichforeachlevellunittheresponseiseitherthecontinuouslogdurationtimeortakesthevalueoneifcensoredwithcorrespondingexplanatoryvariablesineachcase.Therearebasicallytwoexplanatoryvariablesforthelevel1variation,oneforthecontinuouslogdurationresponseandoneforthebinomialresponse.Intheformercasewecanextendthisforcomplexlevel1variation,asintheexampleanalysisbelow.Forthelatterweusethestandardlogitmodelasdescribedinchapter7,possiblyallowingforextra-binomialvariation.Therandomparametersatlevel1forthetwocomponentsareuncorrelated.Whencarryingoutthe

computations,wemayobtainstartingvaluesfortheparametersusingjusttheuncensored

observations.

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Sincethesamelinearfunctionoftheexplanatoryvariablesentersintoboththelinearandnonlinearpartsofthismodel,werequireonlyasinglesetoffixedpartexplanatoryvariables,althoughthesewillrequiretheappropriatetransformationforthelogitresponseasdescribedinchapter7.Wealsonotethatanykindsofcensoreddatacanbemodelled,solongasthecorrespondingprobabilitiesarecorrectlyspecified.

Figure9.2.Level1residualsbyNormalscoresforAnalysisBinTable9.2

Wecanreadilyextendthismodeltothemultivariatecasewhereseveralkindsofdurationsaremeasured.Thiswillrequireoneextralowestleveltobeinsertedtodescribethemultivariatestructure,withlevel2becomingthebetween-observationlevelandlevel3theoriginallevel2.Forthelogitpartofthemodelwewillallowcorrelationsatlevel2wherethesecanbeinterpretedaspoint-biserialcorrelations.

Forrepeatedmeasuresmodelswheretherearedifferenttypesofdurationwecanchoosetofitamultivariatemodel.Alternatively,asdiscussedinchapter4,wemaybeabletospecifyasimplermodelwherethetypesdifferonlyintermsofafixedpartcontribution,orperhapswheretherearedifferentvariancesforeachtypewithacommoncovariance.Aspointedoutearlier,wemay

sometimeswishtotreatthefirstdurationlengthseparatelyandthisisreadilydonebyspecifyingit

asaseparateresponse.

9.12Infinitedurations

Itissometimesfoundthatforaproportionofindividuals,theirdurationlengthsareextremelylong.Thus,someemployeesremaininthesamejobforlifeandsomepatientsmayacquireadiseaseandretainitfortherestoftheirlives.Instudiesofsocialmobility,someindividualswillremaininaparticularsocialgroupforafinitelengthoftimewhileothersmayneverleaveit:suchmodelsaresometimesreferredtoasmover-stayermodels.Wecantreatsuchdurationsasiftheywereinfinite.Sinceanygivenstudywilllastonlyforafinitetime,itisimpossibletodistinguishinfinitetimesfromthosewhicharerightcensored.Nevertheless,ifwemakesuitabledistributionalassumptionswecanobtainanestimateoftheproportionofinfinitesurvivaltimes.

Foraconstantθ,givenanunobserveddurationtime,theobservationiseitherrightcensoredwithfinitedurationorhasinfinitedurationsothatwereplacetheprobability,by

Ingeneralθwilldependonexplanatoryvariablesandanobviouschoiceforamodelis

(9.8)

Thecoefficientsin(9.8)mayalsovaryacrosslevel2units.

Wheretheobservationisnotcensoredweknowthatithasafinitedurationsothatfortheinfinite

durationparameterswehavearesponsevariabletakingthevaluezerowithpredictorgivenby 三.Thefullmodelcanthereforebespecifiedasabivariatemodelwhereforobserveddurationswehavetworesponses,onefortheuncensoredcomponentl,andtheonefortheparameters9.Forthecensoredobservationsthereisasingleresponsewhichtakesthevalueone

withpredictorfunction

Wecanextendtheproceduresofchapter7tothejointestimationofB9,notingthatforthe

censoredobservationswhenestimatingβ,wehave

andforestimatingwehave

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9.13Exampleswithbirthintervaldataandchildren'splayepisodes

WefirstlookagainattheHutteritebirthintervaldata.Sinceallthedurationsareuncensoredweapplyastandardmodeltothelog(birthinterval)values.ResultsaregiveninTable9.2.

Weseethatwecannowfittheyearofbirthandageasrandomcoefficientsatlevel2.Ajointtestgivesachi-squaredvalueof10.4with5d.f.P=0.065,andtheyareeachseparatelysignificantwithasignificancelevelof6%.Wehavesignificantheterogeneityatlevel1wherethevariancewithinwomenisgreaterwheretherehasbeen