多水平统计模型 第6章

#Chapter6RepeatedmeasuresdataModelsforrepeatedmeasuresWhenmeasurementsarerepeatedonthesamesubjects,forexamplestudentsoranimals,a2-levelhierarchyisestablishedwithmeasurementrepetitionsoroccasionsaslevel1unitsandsubjectsaslevel2units.Suchdataareoftenreferredtoas‘longitudinal'asopposedto‘cross-sectional'whereeachsubjectismeasuredonlyonce.Thus,wemayhaverepeatedmeasuresofbodyweightongrowinganimalsorchildren,repeatedtestscoresonstudentsorrepeatedinterviewswithsurveyrespondents.Itisimportanttodistinguishtwoclassesofmodelswhichuserepeatedmeasurementsonthesamesubjects.Inone,earliermeasurementsaretreatedascovariatesratherthanresponses.Thiswasdonefortheeducationaldataanalysedinchapters2and3,andwilloftenbemoreappropriatewhenthereareasmallnumberofdiscreteoccasionsandwheredifferentmeasuresareusedateachone.Intheother,usuallyreferredtoas‘repeatedmeasures'models,allthemeasurementsaretreatedasresponses,anditisthisclassofmodelsweshalldiscusshere.Adetaileddescriptionofthedistinctionbetweentheformer'conditional'modelsandthelatter'unconditional'modelscanbefoundinGoldstein(1979)andPlewis(1985).Wemayalsohaverepetitionathigherlevelsofadatahierarchy.Forexample,wemayhaveannualexaminationdataonsuccessivecohortsof16-year-oldstudentsinasampleofschools.Inthiscasetheschoolisthelevel3unit,yearisthelevel2unitandstudentthelevel1unit.Wemayevenhaveacombinationofrepetitionsatdifferentlevels:inthepreviousexample,withthestudentsthemselvesbeingmeasuredonsuccessiveoccasionsduringtheyearswhentheytaketheirexamination.Weshallalsolookatanexamplewherethereareresponsesatbothlevel1andlevel2,thatisspecifictotheoccasionandtothesubject.Itisworthpointingoutthatinrepeatedmeasuresmodelstypicallymostofthevariationisatlevel2,sothattheproperspecificationofamultilevelmodelforthedataisofparticularimportance.Thelinkwiththemultivariatedatamodelsofchapter4isalsoapparentwheretheoccasionsarefixed.Forexample,wemayhavemeasurementsontheheightofasampleofchildrenatages11.0,12.0,13.0and14.0years.Wecanregardthisashavingamultivariateresponsevectorof4responsesforeachchild,andperformanequivalentanalysis,forexamplerelatingthemeasurementstoapolynomialfunctionofage.Thismultivariateapproachhastraditionallybeenusedwithrepeatedmeasuresdata(GrizzleandAllen,1969).Itcannot,however,dealwithdatawithanarbitraryspacingornumberofoccasionsandweshallnotconsideritfurther.Inallthemodelsconsideredsofarwehaveassumedthatthelevel1residualsareuncorrelated.Forsomekindsofrepeatedmeasuresdata,however,thisassumptionwillnotbereasonable,andweshallinvestigatemodelswhichallowaserialcorrelationstructurefortheseresiduals.Wedealonlywithcontinuousresponsevariablesinthischapter.Weshalldiscussrepeatedmeasuresmodelsfordiscreteresponsedatainchapter7.A2-levelrepeatedmeasuresmodelConsideradatasetconsistingofrepeatedmeasurementsoftheheightsofarandomsampleofchildren.Wecanwriteasimplemodely—P+卩x+e(6.1)ij0j1jijijThismodelassumesthatheight(Y)islinearlyrelatedtoage(X)witheachsubjecthavingtheirowninterceptandslopesothatE(卩)二卩，E(卩)二卩0j01j1var(卩)=C2,var(卩)=C2,cov(卩,卩)=a,var(e)=C20ju01ju10j1ju01ijeThereisnorestrictiononthenumberorspacingofages,sothatwecanfitasinglemodeltosubjectswhomayhaveoneorseveralmeasurements.Wecanclearlyextend(6.1)toincludefurtherexplanatoryvariables,measuredeitherattheoccasionlevel,suchastimeofyearorstateofhealth,oratthesubjectlevelsuchasbirthweightorgender.Wecanalsoextendthebasiclinearfunctionin(6.1)toincludehigherordertermsandwecanfurthermodelthelevel1residualsothatthelevel1varianceisafunctionofage.Weexploredbrieflyanonlinearmodelforgrowthmeasurementsinchapter5.Suchmodelshaveanimportantroleincertainkindsofgrowthmodelling,especiallywheregrowthapproachesanasymptoteasintheapproachtoadultstatusinanimals.Inthefollowingsectionsweshallexploretheuseofpolynomialmodelswhichhaveamoregeneralapplicabilityandformanyapplicationsaremoreflexible(seeGoldstein,1979forafurtherdiscussion).Weintroduceexamplesofincreasingcomplexity,andincludingsomenonlinearmodelsforlevel1variationusingtheresultsofchapter5.ApolynomialmodelexampleforadolescentgrowthandthepredictionofadultheightOurfirstexamplecombinesthebasic2-levelrepeatedmeasuresmodelwithamultivariatemodeltoshowhowageneralgrowthpredictionmodelcanbeconstructed.Thedataconsistof436measurementsoftheheightsof110boysbetweentheagesof11and16yearstogetherwithmeasurementsoftheirheightasadultsandestimatesoftheirboneagesateachheightmeasurementbaseduponwristradiographs.AdetaileddescriptioncanbefoundinGoldstein(1989b).Wefirstwritedownthethreebasiccomponentsofthemodel,startingwithasimplerepeatedmeasuresmodelforheightusinga5-thdegreepolynomial.y(1)二艺B(1)xh+工u(1)xh+e⑴ijhijhjijij(6.2)h=0h=0wherethelevel1termemayhaveacomplexstructure,forexampleadecreasingvariancewithijincreasingage.Themeasureofboneageisalreadystandardisedsincetheaverageboneageforboysofagivenchronologicalageisequaltothisageforthepopulation.Thuswemodelboneageusinganoverallconstanttodetectanyaveragedepartureforthisgrouptogetherwithbetween-individualandwithin-individualvariation.y(2)=B(2)+£u(2)xh+e(2)ij0hjijij(6.3)Foradultheightwehaveasimplemodelwithanoverallmeanandlevel2variation.Ifwehadmorethanoneadultmeasurementonindividualswewouldbeabletoestimatealsothelevel1variationamongadultheightmeasurements;ineffectmeasurementerrors.(6.4)y(3)=B(3)+u(3)j00j(6.4)Wenowcombinetheseintoasinglemodelusingthefollowingindicators

8⑴=1,ijifgrowthperiodmeasurement,0otherwise8⑵=1,ijifboneagemeasurement,08⑴=1,ijifgrowthperiodmeasurement,0otherwise8⑵=1,ijifboneagemeasurement,0otherwise8⑶=1,ifadultheightmeasurement,0otherwiseij=8(1)(1)xh+u(1)xh+e(1))+8⑵(卩⑵+乙u(2)Xh+e(2))ijhijhjijijij0hjijijh=0h=0h=0(6.5)+8(3)(卩⑶+U⑶)j00jAtlevel1thesimplestmodel,whichweshallassume,isthattheresidualsforboneageandheightareindependent,althoughdependenciescouldbecreated,forexampleifthemodelwasincorrectlyspecifiedatlevel2.Thus,level1variationisspecifiedintermsoftwovarianceterms.Althoughthemodelisstrictlyamultivariatemodel,becausethelevel1randomvariablesareindependentitisunnecessarytospecifya'dummy'level1withnorandomvariationasinchapter4.If,however,weallowcorrelationbetweenheightandboneagethenwewillneedtospecifythemodelwithnovariationatlevel1,thevariancesandcovariancebetweenboneageandheightatlevel2andthebetween-individualvariationatlevel3.Table6.1showsthefixedandrandomparametersforthismodel,omittingtheestimatesforthebetween-individualvariationinthequadraticandcubiccoefficientsofthepolynomialgrowthcurve.Weseethatthereisalargecorrelationbetweenadultheightandheightandsmallcorrelationsbetweentheadultheightandtheheightgrowthandtheboneagecoefficients.Thisimpliesthattheheightandboneagemeasurementscanbeusedtomakepredictionsofadultheight.Infactthesepredictedvaluesaresimplytheestimatedresidualsforadultheight.Foranewindividual,withinformationavailableatoneormoreagesonheightorboneage,wesimplyestimatetheadultheightresidualusingthemodelparameters.Table6.2showstheestimatedstandarderrorsassociatedwithpredictionsmadeonthebasisofvaryingamountsofinformation.Itisclearthatthemaingaininefficiencycomeswiththeuseofheightwithasmallergainfromtheadditionofboneage.Table6.1Height(cm)foradolescentgrowth,boneage,andadultheightforasampleofboys.Agemeasuredabout13.0years.Level2variancesandcovariancesshown;correlationsinbrackets.ParameterEstimate(s.e.)FixedAdultHeight174.40.25(0.50)174.40.25(0.50)Height:InterceptAge2Height:InterceptAge2Age3Age4Age5BoneAge:Intercept153.06.91(0.20)0.43(0.09)-0.14(0.03)-0.03(0.01)0.03(0.03)0.21(0.09)0.03(0.03)

RandomLevel2AdultHeightHeightinterceptAgeBoneAgeIntcpt.AdultHeight62.5Heightintercept49.5(0.85)54.5Age1.11(0.09)1.14(0.09)2.5BoneAgeIntcpt.0.57(0.08)3.00(0.44)0.02(0.01)0.85Level1varianceHeight0.89Boneage0.18Themethodcanbeusedforanymeasurements,eithertobepredictedoraspredictors.Inparticular,covariatessuchasfamilysizeorsocialbackgroundcanbeincludedtoimprovetheprediction.Wecanalsopredictothereventsofinterest,suchastheestimatedageatmaximumgrowthvelocity.Fig6.2Standarderrorsforheightpredictionsforspecifiedcombinationsofheightandboneagemeasurements.BoneagemeasuresNone11.0BoneagemeasuresNone11.011.012.0None11.011.012.04.34..Heightmeasures(age)6.4Modellinganautocorrelationstructureatlevel1.Sofarwehaveassumedthatthelevel1residualsareindependent.Inmanysituations,however,suchanassumptionwouldbefalse.Forgrowthmeasurementsthespecificationoflevel2variationservestomodelaseparatecurveforeachindividual,butthebetween-individualvariationwilltypicallyinvolveonlyafewparameters,asinthepreviousexample.Thusifmeasurementsonanindividualareobtainedveryclosetogetherintime,theywilltendtohavesimilardeparturesfromthatindividual'sunderlyinggrowthcurve.Thatis,therewillbe'autocorrelation'betweenthelevel1residuals.Examplesarisefromotherareas,suchaseconomics,wheremeasurementsoneachunit,forexampleanenterpriseoreconomicsystem,exhibitanautocorrelationstructureandwheretheparametersoftheseparatetimeseriesvaryacrossunitsatlevel2.AdetaileddiscussionofmultileveltimeseriesmodelsisgivenbyGoldsteinetal(1994).Theydiscussboththediscretetimecase,wherethemeasurementsaremadeatthesamesetofequalintervalsforalllevel2units,andthecontinuoustimecasewherethetimeintervalscanvary.Weshalldevelopthecontinuoustimemodelheresinceitisbothmoregeneralandflexible.Tosimplifythepresentation,weshalldropthelevel1and2subscriptsandwriteageneralmodelforthelevel1residualsasfollowscov(ee)=b2f(s)(6.6)tt一se

Thus,thecovariancebetweentwomeasurementsdependsonthetimedifferencebetweenthemeasurements.Thefunctionf(s)isconvenientlydescribedbyanegativeexponentialreflectingthecommonassumptionthatwithincreasingtimedifferencethecovariancetendstoafixedvalue,ac2,eandtypicallythisisassumedtobezerof(s)=a+exp(-g(卩,z,s))(6.7)wherepisavectorofparametersforexplanatoryvariablesz.SomechoicesforgaregiveninTable6.3.WecanapplythemethodsdescribedinAppendix5.1toobtainmaximumlikelihoodestimatesforthesemodels,bywritingtheexpansionf(s,卩，z)={1+工卩zg(H)}f(H)-工卩zg(H)f(H)(6.8)k,tkttk,t+1kttkksothatthemodelfortherandomparametersislinear.FulldetailsaregivenbyGoldsteinetal(1994).6.5AgrowthmodelwithautocorrelatedresidualsThedataforthisexampleconsistofasampleof26boyseachmeasuredonnineoccasionsbetweentheagesof11and14years(HarrisonandBrush,(1990).Themeasurementsweretakenapproximately3monthsapart.Table6.4showstheestimatesfromamodelwhichassumesindependentlevel1residualswithaconstantvariance.Themodelalsoincludesacosinetermtomodeltheseasonalvariationingrowthwithtimemeasuredfromthebeginningoftheyear.Iftheseasonalcomponenthasamplitudeaandphase7wecanwriteacos(t+7)=acos(t)-asin(t)12Inthepresentcasethesecondcoefficientisestimatedtobeveryclosetozeroandissettozerointhefollowingmodel.Thiscomponentresultsinanaveragegrowthdifferencebetweensummerandwinterestimatedtobeabout0.5cm.Wenowfitintable6.5themodelwithg=pswhichisthecontinuoustimeversionofthefirst0orderautoregressivemodel.Thefixedpartandlevel2estimatesarelittlechanged.Theautocorrelationparameterimpliesthatthecorrelationbetweenresiduals3months(0.25years)apartis0.19.Table6.3Somechoicesforthecovariancefunctiongforlevel1residuals.Forequalintervalsthisisafirstorderautoregressiveseries.varianceisaquadraticfunctionoftime.varianceisaquadraticfunctionoftime.g=Ps+P(t+t)+P(t2+t2)0112212Fortimepointst1,t2thisimpliesthatthePPsifnoreplicate0PifreplicateForreplicatedmeasurementsthisgivesanestimateofmeasurementreliabilityexp(-P1).

g=g=(卩0+卩1Z1j+P2'JSPs+Ps-1,s〉001Allowsaflexiblefunctionalform,wherethetimeintervalsarenotclosetozero.0,s=0ParameterEstimate(s.e.)FixedIntercept148.9age6.19(0.35)age22.17(0.46)age30.39(0.16)age4-1.55(0.44)cos(time)-0.24(0.07)Randomlevel2InterceptageIntercept61.6(17.1)age8.0(0.61)2.8(0.7)age21.4(0.22)0.9(0.67)level1b20.20(0.02)Table6.4Heightasafourthdegreepolynomialonage,measuredabout13.0years.Standarderrorsinbrackets;correlationsinbracketsforcovarianceterms.age20.7(0.2)ParameterEstimate(s.e.)FixedIntercept148.9age6.19(0.35)age22.16(0.45)age30.39(0.17)age4-1.55(0.43)cos(time)-0.24(0.07)Randomlevel2InterceptIntercept61.5(17.1)age7.9(0.61)age21.5(0.25)level1b20.23(0.04)矿6.90(2.07)Table6.5Heightasafourthdegreepolynomialonage,measuredabout13.0years.Standarderrorsinbrackets;correlationsinbracketsforcovarianceterms.Autocorrelationstructurefittedforlevel1residuals.ageage22.7(0.7)0.9(0.68)0.6(0.2)MultivariaterepeatedmeasuresmodelsWehavealreadydiscussedthebivariaterepeatedmeasuresmodelwherethelevel1residualsforthetworesponsesareindependent.Inthegeneralmultivariatecasewherecorrelationsatlevel1areallowed,wecanfitafullmultivariatemodelbyaddingafurtherlowestlevelasdescribedinchapter4.Fortheautocorrelationmodelthiswillinvolveextendingthemodelstoincludecrosscorrelations.Forexamplefortworesponsevariableswiththemodeloftable6.5wewouldwriteg=ggexp(-Ps)e1e212Thespecialcaseofarepeatedmeasuresmodelwheresomeoralloccasionsarefixedisofinterest.Wehavealreadydealtwithoneexampleofthiswhereadultheightistreatedseparatelyformtheothergrowthmeasurements.Thesameapproachcouldbeusedwith,forexample,birthweightorlengthatbirth.Insomestudies,allindividualsmaybemeasuredatthesameinitialoccasionandwecanchoosetotreatthisasacovariateratherthanasaresponse.Thismightbeappropriatewhereindividualsweredividedintogroupsfordifferenttreatmentsfollowinginitialmeasurements.ScalingacrosstimeForsomekindsofdata,forexampleeducationalachievementscores,differentmeasurementsmaybetakenovertimeonthesameindividualssothatsomeformofstandardisationmaybeneededbeforetheycanbemodelledusingthemethodsofthischapter.Itiscommoninsuchcasestostandardisethemeasurementssothatateachmeasuringoccasiontheyhavethesamepopulationdistribution.Ifthisisdonethenweshouldnotexpectanytrendineitherthemeanorvarianceovertime,althoughtherewillstill,ingeneral,bebetween-individualvariation.Analternativestandardisationprocedureistoconvertscorestoageequivalents;thatistoassigntoeachscoretheageforwhichthatscoreisthepopulationmeanormedian.Wherescoreschangesmoothlywithagethishastheattractionofprovidingareadilyinterpretablescale.Plewis(1993)usesavariantofthisinwhichthecoefficientofvariationateachageisalsofixedtoaconstantvalue.Ingeneral,differentstandardisationsmaybeexpectedtoleadtodifferentinferences.Thechoiceofstandardisationisineffectachoiceabouttheappropriatescalealongwhichmeasuremen