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1、CHAPTER9RobustMixedEffectsModels9.1IntroductionInpreviouschapters,wehavemostlyfocusedonparametricmixedeffectsmodels.Inthesemodels,therandomeffectsaretypicallyassumedtofollow(multivariate)normaldistributionsandthewithinindividual(cluster)randomerrorsareassumedtofollowp
2、arametricdistributionsintheexponentialfam-ily.Forexample,inLMEandNLMEmodelsweassumethatboththerandomeffectsandtheresponses(orthewithinindividualerrors)follownormaldis-tributions,andinGLMMsweassumethattherandomeffectsfollownormaldistributionsandtheresponses(orthewithin
3、individualerrors)followparamet-ricdistributionsintheexponentialfamily.Likelihoodinferenceisthenbasedontheassumeddistributions.Inpractice,however,theassumedparametricdistributionsmaynothold.More-over,outliersmaybepresent.Sometimes,outliersmaynotbeeasilydetected,especia
4、llyformulti-dimensionaldata.Likelihoodinferenceforstandard(para-metric)mixedeffectsmodelsaretypicallysensitivetotheassumedparametricdistributionsandtooutliers.Forexample,afewoutliersinadatasetcanhavealargeinuenceontheparameterestimationandinference.Therefore,likeli-h
5、oodinferenceignoringoutliersisunreliableandmisleading.Inabroadsense,outliersmaybeviewedasincompletelyobserveddata.Some-timesoutliersmaybedatameasuredwitherrors.Iftheoutlierscanbedetectedorscreenedmanuallyinadvance,onemayremovetheseoutliersintheanalysisorchecktheirsour
6、cessincesometimesoutliersmaycontainvaluableinforma-tion.Formulti-dimensional(ormultivariate)data,however,itmaybedifculttodetectoutliersinadvance.Therefore,robustmethodswhicharelesssensitivetooutliersarehighlyvaluable.Robustmethodsusuallyoutperformclassicalmethodsinth
7、epresenceofoutliers.Goodrobustestimatescanbereasonablyefcientwhentheassumeddistributionsholdandmoreefcientthanstandardestimateswhentheassumeddistributionsdonothold.Anoutliercanbedenedasanobservationwhichappearstobeinconsistent293294MIXEDEFFECTSMODELSFORCOMPLEXDATAw
8、iththeremainderofthedata.Forlongitudinaldata,therearetwotypesofoutliers:outlierscanoccuratindividuallevel,sometimescallede-
