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1、CHAPTER10GeneralizedEstimatingEquations(GEEs)10.1IntroductionInpreviouschapters,wehavemostlyfocusedonlikelihoodmethodsforesti-mationandinference.Thelikelihoodmethodsarebasedondistributionalas-sumptionsforthedata.Forexample,giventherandomeffects,inGLMMsweassumethattheresponsesinth
2、emodelsfollowdistributionsintheexponen-tialfamily,andinLMEandNLMEmodelsweassumethattheresponsesinthemodelsfollownormaldistributions.Ifthedistributionalassumptionshold,thelikelihoodmethodsareveryattractivesincetheMLEsareasymptoticallymostefcientandasymptoticallynormalundersomereg
3、ularityconditions(seeChapter12).Inpractice,however,thedistributionalassumptionsforregressionmodelsmaynothold,especiallyforGLMsorGLMMs.Forexample,inabinomialGLMoraPoissonGLMthevarianceiscompletelydeterminedbythemean,butthismaynotbeconsistentwiththeobserveddata.Iftheobservedvariati
4、oninthedataislargerthanthevariancedeterminedbythedistributionassumedinthemodel,wehaveanover-dispersionproblemandthedistributionalassumptionsforthemodeldonothold.Thus,inthiscasethelikelihoodmethod,whichisbasedonthedistributionalassumptions,maynotperformwell.Whentheobservedvariatio
5、ninthedataisinconsistentwiththetheoreticalvariationdeterminedbytheassumedmodel,oneapproachistointroducead-ditionaldispersionparameterstoaccountfortheextravariationinthedata.Inthiscase,onecanstillobtainparameterestimatesbysolvingtheresult-ing“score”equations,butthecorresponding“li
6、kelihood”isnolongeratruelikelihoodbutratheraquasi-likelihood,andthese“score”equationsarecalledgeneralizedestimatingequations(GEEs).Forlongitudinalorclustereddata,LiangandZeger(1986)introducedtheideaofusingaworkingcorrelationma-trixintheGEEs.Thatis,GEEskeepthesameassumptionsaboutt
7、hemeanand333334MIXEDEFFECTSMODELSFORCOMPLEXDATAcovariancestructuresasinquasi-likelihoodmethodsbutintroducetheworkingcovariancematrixwhichmaydependonfewernuisanceparameterstosimplifythecorrelationstructure.Inquasi-likelihoodandGEEmethods,weassumeameanstructureandavariance-covarian
8、cestructurebasedontheobserveddatawithout
