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1、Chapter5Nonlinearmultilevelmodels5.1NonlinearmodelsThemodelsofChapters1-4arelinearinthesensethattheresponseisalinearfunctionoftheparametersinthefixedpartandtheelementsofarelinearfunctionsoftheparametersintherandompart.Inmanyapplications,however,itisappropriatetoconsidermo
2、delswherethefixedorrandompartsofthemodel,orboth,containnonlinearfunctions.Forexample,inthestudyofgrowth,JenssandBayley(1937)proposedthefollowingfunctiontodescribethegrowthinheightofyoungchildren(5.1)whereistheageofthej-thchildatthei-thmeasurementoccasion.Generalisedlinear
3、models(McCullaghandNelder,1989)areaspecialcaseofnonlinearmodelswheretheresponseisanonlinearfunctionofafixedpartlinearpredictor.Modelsfordiscretedata,suchascountsorproportionsfallintothiscategoryandweshalldevotechapter7tostudyingthese.Forexample,a2-levelloglinearmodelcanbe
4、written(5.2)whereisassumedtypicallytohaveaPoissondistribution,inthiscaseacrosslevel1units.Notehere,thatinthemultilevelextensionofthestandardsinglelevelmodel,thelinearpredictorcontainsrandomvariablesdefinedatlevel2orabove.Inthischapterweconsiderageneralnonlinearmodel.Later
5、chapterswillusetheresultsforparticularapplications.5.2NonlinearfunctionsoflinearcomponentsThefollowingresultsareanextensionofthosepresentedbyGoldstein(1991)andappendix5.1givesdetails.Wheretherandomvariablesarenotpartofthenonlinearfunction,theproceduregivesmaximumlikelihoo
6、destimates(seeappendix5.1).Inthecasewherethelevel1variationisnonNormaltheprocedurecanberegardedasageneralisationofquasilikelihoodestimation(McCullaghandNelder,1989)andsuchmodelsarediscussedinchapter7.Restrictingattentiontoa2-levelstructurewecanwriteafairlygeneralmodelasfo
7、llows(5.3)10wherethefunctionisnonlinearandwheretheindicatesthatadditionalnonlinearfunctionscanbeincluded,involvingfurtherfixedpartexplanatoryvariablesorrandompartexplanatoryvariablesatlevels1and2,respectively.ThemodelisfirstlinearisedbyasuitableTaylorseriesexpansionandthi
8、sleadstoconsiderationofalinearmodelwheretheexplanatoryvariablesinaretransformedusingfirstandseco
