多水平模型英文原著chap7.docx
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多水平模型英文原著chap7.docx
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多水平模型英文原著chap7
Chapter7
Discreteresponsedata
7.1Modelsfordiscreteresponsedata
Allthemodelsofpreviouschaptershaveassumedthattheresponsevariableiscontinuouslydistributed.Wenowlookatdatawheretheresponseisessentiallyacountofevents.Thiscountmaybethenumberoftimesaneventoccursoutofafixednumberof‘trials’inwhichcaseweusuallydealwiththeresultingproportionasresponse:
anexampleistheproportionofdeathsinapopulation,classifiedbyage.Wemayhaveavectorofcountsrepresentingthenumbersofeventsofdifferentkindswhichoccuroutofatotalnumberofevents:
anexampleisgiveninchapter3wherewestudiedthenumberofresponsestoeach,ordered,categoryofaquestiononabortionattitudes.
Statisticalmodelsforsuchdataarereferredtoas‘generalisedlinearmodels’(McCullaghandNelder,1989).A2-levelmodelcanbewritteninthegeneralform
(7.1)
whereistheexpectedvalueoftheresponsefortheij-thlevel1unitandfisanonlinearfunctionofthe‘linearpredictor’
.Notethatweallowrandomcoefficientsatlevel2.Themodeliscompletedbyspecifyingadistributionfortheobservedresponse
.WheretheresponseisaproportionthisistypicallytakentobebinomialandwheretheresponseisacounttakentobePoisson.Equation(7.1)isaspecialcaseofthenonlinearmodelstudiedinchapter5andweshallbeusingtheresultsgiventhere.Itremainsforustospecifythenonlinear‘link’functionf.Table7.1listssomeofthestandardchoices,withlogarithmschosentobasee.
Inadditiontothesewecanalsohavethe‘identity’function
butthiscancreatedifficultiessinceitallows,inprinciple,predictedcountsorproportionswhicharerespectivelylessthanzerooroutsidetherange(0,1).Nevertheless,inmanycases,usingtheidentityfunctionproducesacceptableresultswhichmaydifferlittlefromthoseobtainedwiththenonlinearfunctions.Inthefollowingsectionsweconsidereachcommontypeofmodelinturnwithexamples.
Table7.1Somenonlinearlinkfunctions.
Response
Name
Proportion
logit
Proportion
complementaryloglog
Vectorofproportions
multivariatelogit
Count
log
7.2Proportionsasresponses
Considerthe2-levelvariancecomponentsmodelwithasingleexplanatoryvariablewheretheexpectedproportionismodelledusingalogitlinkfunction
(7.2)
Theobservedresponsesareproportionswiththestandardassumptionthattheyarebinomiallydistributed
(7.3)
whereisthedenominatorfortheproportion.Wealsohave
(7.4)
Wenowwritethemodelinthestandardwayincludingthelevel1variationas
(7.5)
Usingthisexplanatoryvariableandconstrainingthelevel1varianceassociatedwiththistobeoneweobtaintherequiredbinomialvarianceinequation(7.4).Whenfittingamodelwecanalsoallowthelevel1variancetobeestimatedandbycomparingtheestimatedvariancewiththevalue1.0obtainatestfor‘extrabinomial’variation.Suchvariationmayariseinanumberofways.
Ifwehaveomittedalevelinthemodel,forexampleignoredhouseholdclusteringinasurveywithoneormoreindividualssampledfromahousehold,wewouldexpectagreaterthanbinomialvariationattheindividuallevel.Likewise,supposetheindividualsandhouseholdswerenestedwithinareasandwechosetoclassifyindividuals,saybygenderand3socialclassgroupsgiving6cellsineacharea.Ifwetreattheseasthelevel1unitssothattheresponseisaproportion,thenwenolongerhaveabinomialvariancesincetheseproportionsarebaseduponthesumofseparatebinomialvariableswithdifferingprobabilities.Herethevarianceforcelljwithinanareawouldhavetheform
whereisthecellsize.Tofitsuchamodelwewouldspecifyanextralevel1explanatoryvariableequalto
forthej-thcell,withvarianceparameteratlevel1whichwasallowedtobenegative(seechapter3).Moregenerally,wecanfitamodelwithanextrabinomialparametertogetherwithafurthertermsuchasabovetogivethefollowinglevel1variancestructure(omittingsubscripts)
Wedonot,ofcourse,knowthetruevalueoforsothatateachiterationweuseestimatesbaseduponthecurrentvaluesoftheparameters.Becauseweareusingonlythemeanandvarianceofthebinomialdistributiontocarryouttheestimation,theestimationisknownas‘quasilikelihood’(seeappendix5.1).
Anotherwayofmodellingsuchextrabinomialvariation,whichhascertainadvantages,istoinserta‘pseudolevel’abovelevel1.Thus,forindividualssampledwithinhouseholds,level1wouldbethatoftheindividualandwewouldspecifylevel2asthatoftheindividualsalsotogiveexactly1level1unitperlevel2unit.Wespecifybinomialvariationatlevel1andatlevel2wecannowfitfurtherrandomcoefficients.Forexample,ifwefitarandomcoefficientfortheexplanatoryvariablewithavariancewhichcanbeallowedtobenegativethisisequivalenttospecifyinganextralevel1variable
asabove.Intheaboveexamplewhereindividualsareclassifiedbygenderandsocialclasswecancreatealevel2unitcoincidingwitheachlevel1unit,fitbinomialvariationatlevel1andaddlevel2variationwhichisafunctionofgenderandsocialclass,forexampleanadditivefunctionwith4parameters(seechapter3).Wemaywishtomodelthebetween-areavariationofthecellproportionsintermsofasimplevarianceterm,ratherasinverselyproportionalto.Inthiscasewewouldchooseasimpledummyvariablestructureratherthanexplanatoryvariablesproportionalto
.This‘pseudolevel’procedureisrathersimilartothewayinwhichametaanalysiswithknownlevel1variationismodelled(chapter3).
Inchapter5wemadethedistinctionbetweenmodelswherethecurrentlevel2residualestimateswereaddedtothelinearcomponentofthenonlinearfunctionwhenformingtheTaylorexpansioninordertoworkwithalinearisedmodel,andthosecaseswheretheywerenot.Theformerisreferredtoaspredictivequasilikelihood(PQL)andthelattermarginalquasilikelihood(MQL).InmanyapplicationstheMQLprocedurewilltendtounderestimatethevaluesofboththefixedandrandomparameters,especiallywhereissmall.InadditionwepointedoutthatgreateraccuracyistobeexpectedifthesecondorderapproximationisusedratherthanthefirstorderbaseduponthefirsttermintheTaylorexpansion.Also,whenthesamplesizeissmalltheunbiased(RIGLS,REML)procedureshouldbeused.Appendix7.1givesexpressionsfortheseconddifferentialsrequiredforthesecondorderprocedure..Toillustratethedifferencetable7.2presentstheresultsofsimulatingthefollowingmodelwheretheresponseisbinary(0,1).Theexampleassumesonemoderateandonelargelevel2variance.
Thereare50level2unitswith20level1unitsineachlevel2unit.Thefollowingresultsarebasedupon400simulationsoftheabovemodelforeachvariancevalue.
Table7.2Meanvaluesof400simulations.Empiricalstandarderrorinfirstbracket;meanofestimatedstandarderrorsinsecondbracket(IGLS).
True
True
Parameter
MQLfirstorder
PQLsecondorder
MQLfirstorder
PQLsecondorder
0.386(0.115)(0.130)
0.480(0.157)(0.152)
0.672(0.157)(0.188)
0.964(0.278)(0.255)
0.448(0.126)(0.129)
0.499(0.139)(0.138)
0.420(0.145)(0.149)
0.500(0.171)(0.172)
0.934(0.154)(0.147)
1.018(0.168)(0.154)
0.875(0.147)(0.145)
1.017(0.171)(0.158)
Here,thedenominatoris1.0inallcases.ItisclearthattheMQLfirstordermodelunderestimatesalltheparametervalues,whereasthesecondorderPQLmodelproducesestimatesclosetothetruevalues.TheestimatesgivenarebaseduponIGLS.Ineverycaseconvergencewasachievedinlessthan10iterations.VerysimilarestimatesforthefixedcoefficientsareobtainedusingRIGLS,andforthelevel2variancesthePQLestimatesbecome0.498and0.996respectively,whichareevenclosertothetruevalues.Inaddition,theaveragesofthestandarderrorsgivenbybothmodelsarereasonablyclosetothosecalculatedempiricallyfromthereplications.Ifwecalculate95%confidenceintervalsfortheparametersinthesecondorderPQLmodelusingtheestimatedstandarderrorsandassumingNormalitythenforthevariancewefindthatabout91%oftheintervalsincludethetruevalueandforandabout95%doso.Hence,inferencesaboutthetruevalueswouldnotbetoomisleading.TheresultsofTable7.2arebaseduponabalanceddatasetwithequalnumbersoflevel1unitswithineachlevel2unit.Further,limited,simulationssuggestthatevenwherethedataareveryunbalanced,forexamplewithsomelevel2unitscontainingonlyasinglelevel1unit,thePQLsecondorderestimatesremainclosetothetruevalues.Theseestimatesappeartohavegoodpropertiesevenwithaverageobservedprobabilitiesassmallas0.1oraslargeas0.9andalevel2varianceof1.0forthesamplestructureofthisexample.
Moregenerally,whentheaverageobservedprobabilityisverysmall(orverylarge),ifmanyofthelevel2unitshavefewlevel1unitsandthereareveryfewlevel2unitswithlargenumbersoflevel1units,wewilloftenfindthatwheretheresponseisbinary,therewillbemanylevel2unitswheretheresponsesareallzero.Insuchacaseconvergenceoftenmaynotbepossibleandevenwhereestimatesareobtained,ingeneraltheywillnotbeunbiased.Thisproblemcanbeavoidedbyhavingasufficientnumberoflargelevel2unitswherethereisadequateresponseheterogeneity,andinsuchcaseswecanobtainsatisfactoryestimatesevenwheretheaverageprobabilitiesareverysmallorlarge.FurtherworkonthisissueisreportedbyGol
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