多水平统计模型 第5章.docx

多水平统计模型 第5章.docx

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多水平统计模型第5章

Chapter5

Nonlinearmultilevelmodels

5.1Nonlinearmodels

ThemodelsofChapters1-4arelinearinthesensethattheresponseisalinearfunctionoftheparametersinthefixedpartandtheelementsof

arelinearfunctionsoftheparametersintherandompart.Inmanyapplications,however,itisappropriatetoconsidermodelswherethefixedorrandompartsofthemodel,orboth,containnonlinearfunctions.Forexample,inthestudyofgrowth,JenssandBayley（1937）proposedthefollowingfunctiontodescribethegrowthinheightofyoungchildren

（5.1）

where

istheageofthej-thchildatthei-thmeasurementoccasion.Generalisedlinearmodels（McCullaghandNelder,1989）areaspecialcaseofnonlinearmodelswheretheresponseisanonlinearfunctionofafixedpartlinearpredictor.Modelsfordiscretedata,suchascountsorproportionsfallintothiscategoryandweshalldevotechapter7tostudyingthese.Forexample,a2-levelloglinearmodelcanbewritten

（5.2）

where

isassumedtypicallytohaveaPoissondistribution,inthiscaseacrosslevel1units.Notehere,thatinthemultilevelextensionofthestandardsinglelevelmodel,thelinearpredictorcontainsrandomvariablesdefinedatlevel2orabove.

Inthischapterweconsiderageneralnonlinearmodel.Laterchapterswillusetheresultsforparticularapplications.

5.2Nonlinearfunctionsoflinearcomponents

ThefollowingresultsareanextensionofthosepresentedbyGoldstein（1991）andappendix5.1givesdetails.Wheretherandomvariablesarenotpartofthenonlinearfunction,theproceduregivesmaximumlikelihoodestimates（seeappendix5.1）.Inthecasewherethelevel1variationisnonNormaltheprocedurecanberegardedasageneralisationofquasilikelihoodestimation（McCullaghandNelder,1989）andsuchmodelsarediscussedinchapter7.

Restrictingattentiontoa2-levelstructurewecanwriteafairlygeneralmodelasfollows

（5.3）

wherethefunction

isnonlinearandwherethe

indicatesthatadditionalnonlinearfunctionscanbeincluded,involvingfurtherfixedpartexplanatoryvariables

orrandompartexplanatoryvariablesatlevels1and2,respectively

.ThemodelisfirstlinearisedbyasuitableTaylorseriesexpansionandthisleadstoconsiderationofalinearmodelwheretheexplanatoryvariablesin

aretransformedusingfirstandsecondderivativesofthenonlinearfunction.Notethatthelinearcomponentof（5.3）istreatedinthestandardway,andthattherandomvariablesatagivenlevelinthelinearandnonlinearcomponentsmaybecorrelated.

Considerthenonlinearfunction

.Appendix5.1showsthatwecanwritethisasthesumofafixedpartcomponentandarandompart.TheTaylorexpansionfortherandompartuptoasecondorderapproximationfortheij-thunitisasfollows

（5.4）

Thefirsttermontherighthandsideisthefixedpartvalueof

atthecurrent（（t+1）-th）iterationoftheIGLSorRIGLSalgorithm,thatisignoringtherandompart.Theothertwotermsinvolvethefirstandseconddifferentialsofthenonlinearfunctionevaluatedatthecurrentvaluesfromthepreviousiteration.Wehave

（5.5）

Wewritetheexpansionforthefixedpartvalueas

（5.6）

where

arethecurrentandpreviousiterationvaluesofthefixedpartcoefficients.

Wecanchoose

tobeeitherthecurrentvalueofthefixedpartpredictor,thatis

orwecanaddthecurrentestimatedresidualstoobtainanimprovedapproximationtothenonlinearcomponentforeachunit.Theformerisreferredtoasa'marginal'（quasilikelihood）modelandthelatterasa'penalised'or'predictive'（quasilikelihood）model（seeBreslowandClayton,1993,forafurtherdiscussion）.Wecanalsochoosewhetherornottoincludethetermin（5.4）involvingthesecondderivativeandwewouldexpectitsinclusioningeneraltoimprovetheestimates.Itsinclusiondefinesafurtheroffsetforthefixedpartandonefortherandompart（seeappendix5.1）.Weshallillustratetheeffectofthesechoicesintheexamplesgiveninchapter7.FurtherdetailsoftheestimationprocedurearegiveninAppendix5.1.Inpracticegeneralmodelssuchas（5.1）mayposeconsiderableestimationproblems.Wenoticethatthesameexplanatoryvariablesoccurinthelinearandnonlinearcomponentsandthiscanleadtoinstabilityandfailuretoconverge.Furtherworkinthisareaisrequired.

Table5.1givesexpressionsforthefirstandseconddifferentialsforsomecommonlyusednonlinearmodels.

Table5.1Differentialsforsomecommonnonlinearmodels.

Model

Function

Firstdifferential

Seconddifferential

loglinear

logit

log-log

inverse

5.3Estimatingpopulationmeans

Considertheexpectedvalueoftheresponseforagivensetofcovariatevalues.Becauseofthenonlinearitythisisnotingeneralequaltothepredictedvaluewhentherandomvariablesinthenonlinearfunctionarezero.Forexample,ifwewritethevariancecomponentsmodel（5.2）

andassumingNormalityfor

weobtain

Where

isthedensityfunctionoftheNormaldistribution.Zegeretal（1988）considerthisissueandproposea‘populationaverage’modelfordirectlyobtainingpopulationpredictedvaluesbyeliminatingrandomvariablesfromthenonlinearcomponent.Ingeneral,however,thisapproachislessefficientwhenthefullmodelwithrandomvariableswithinthenonlinearfunctionisthecorrectmodel.Thepopulationpredictedvalues,conditionaloncovariates,canbeobtainedifrequired,asabove,bytakingexpectationsoverthepopulation.Anapproximationtothiscanbeobtainedfromthesecondordertermsin（5.1.4）withhigherordertermsintroducedifnecessarytoobtainabetterapproximation.Alternativelywemaygeneratealargenumberofsimulatedsetsofvaluesfortherandomvariablesandforeachsetevaluatetheresponsefunctiontoobtainanestimateofthefullpopulationdistribution.

5.4Nonlinearfunctionsforvariancesandcovariances

Wesawinchapter3howwecouldmodelcomplexfunctionsofthelevel1variance.Aswiththelinearcomponentofthemodel,therearecaseswherewemaywishtomodelvariancesorcovariancesasnonlinearfunctions.Inprinciplewecandothisatanylevelbutwerestrictourattentiontolevel1andtothevarianceonly.Inchapter6wegiveanexamplewherethecovariancesaremodelledinthisway.

Supposethatthelevel1variancedecreaseswithincreasingvaluesofanexplanatoryvariablesuchthatitapproachesafixedvalueasymptotically.Wecouldthenmodelthisfora2-levelmodel,say,asfollows

where

areparameterstobeestimated.Suchamodelalsoguaranteesthatthelevel1varianceispositivewhichisnotthecasewithlinearmodels,suchasthosebasedonpolynomials.TheestimationprocedureisanalogoustothatdescribedaboveanddetailsaregiveninAppendix5.1.

5.5Examplesofnonlineargrowthandnonlinearlevel1variance

Wegivefirstanexampleofamodelwithanonlinearfunctionforthelinearcomponentandwethenconsiderthecaseofanonlinearlevel1variancefunction.

Weuseanexamplefromchildgrowth,consistingof577repeatedmeasurementsofheighton197FrenchCanadianboysagedfrom5to10years（Demirjianetal,1982）withbetween3and7measurementseach.Thisisa2-levelstructurewithmeasurementoccasionsnestedwithinchildren.WefitthefollowingversionoftheJenss-Bayleycurvetoillustratetheprocedure

（5.7）

sothatthefixedpartisaninterceptplusanonlinearcomponentandtherandompartvarianceatlevel2ispartofthenonlinearcomponent.Theresultsaregivenintable5.2,usingthefirstorderapproximationwithpredictionbaseduponthefixedpartonly.Weshallcomparetheperformanceofthedifferentapproximationsinchapter7.

Thelevel1varianceissmallandoftheorderofthemeasurementerrorofheightmeasurements.Thestartingvaluesforthismodelneedtobechosenwithcare,andinthepresentcasethemodelwasruntoconvergencewithoutthelinearintercept

whichwasthenaddedwithastartingvalueof100.Bock（1992）usesanEMalgorithmtofitanonlinear2-levelmodeltogrowthdatafromage2yearstoadulthoodusingamixtureofthreelogisticcurves.

ThesecondexampleusestheJSPdatasetwherewestudiedthelevel1varianceinchapter3.WewillfitmodelBofTable3.1withanonlinearfunctionofthelevel1varianceinsteadofthelevel1varianceasaquadraticfunctionofthe8-year-score.Thislevel1variancefortheij-thlevel1unitis

andtable5.3showsthemodelestimates.

TheestimatesarealmostidenticaltothoseofmodelBoftable3.1asisthelikelihoodvalue.

Figure5.1showsthepredictedlevel1varianceforthismodelandmodelBofTable3.1.

Fig5.1Level1varianceasafunctionof8-yearMathsscore

Table5.2Nonlinearmodelestimateswithfirstorderfixedpartprediction.Ageismeasuredabout8.0years.

Fixedcoefficient

Estimate（s.e.）

Intercept（linear）

90.3

Intercept（nonlinear）

3.58

Age

0.15（0.10）

Agesquared

-0.016（0.02）

Agecubed

0.002（0.004）

Nonlinearmodellevel2covariancematrix（s.e.）

Intercept

Age

Intercept

0.025（0.003）

Age

-0.0027（0.0003）

0.00036（0.00005）

Level1variance=0.25

Inthesedatathenonlinearfunctiongivesverysimilarresultstothequadraticone.Itisclear,however,thatwherethevarianceasymptoticallyapproachesaconstantvalue,forextremevaluesofanexplanatoryvariable,alinearorevenquadraticapproximationmaybeexpectedtofail.Inthepresentcasealinearfunctiondoespredictanegativelevel1variancewithintherangeofthedata.Anexamplewhereanonlinearfunctionisnecessar