mixed effects models with measurement errors

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1、CHAPTER5MixedEffectsModelswithMeasurementErrors5.1IntroductionInSection3.6ofChapter3,webrieydescribedcovariatemeasurementerrorproblemsinregressionmodelsandbrieyreviewedseveralgeneralapproachestoaddressmeasurementerrors.Itisknownthatthenaivemethodwhichig-norescovari

2、atemeasurementerrorsinregressionmodelsmayproducebiasedandmisleadingresults.Tovisuallyillustratethis,Figure5.1showshowtheestimatesofregressioncoefcientsmaybebiasedifmeasurementerrorsinacovariateisnotaddressed,basedonanarticialdatasetgeneratedfromthesimplelinearregre

3、ssiony=0+1x+e.Toformallyaddressmeasurementerrors,wetypicallyneedtohavevalidationdataorreplicationdata(Carrolletal.2006).Inpractice,thesevalidationdataorreplicationdatamaynotbeavailable.However,forlongitudinalstudiestherepeatedmeasurementswithineachindividualmaybeview

4、edasreplicationdata,whichallowsustopartiallyaddressmeasurementerrorsintime-dependentcovariates.Toaddresscovariatemeasurementerrors,weoftenassumeanerrorstructureorameasurementerrormodel.Letzijbetheobservedcovariatevalueforindividualiattimetij(orreplicationj),withpossi

5、blemeasurementerrors,andletzbethecorrespondingunobservedtruecovariatevalue.AsdiscussedijinSection3.6ofChapter3,thefollowingtwomeasurementerrormodelsarecommonlyused(Carrolletal.2006):theclassicalmeasurementerrormodelassumesthatz=z+e;E(ejz)=0;(5.1)ijijijijijandtheBe

6、rksonmeasurementerrormodel(Berkson1950)assumesthatzij=zij+eij;E(eijjzij)=0;(5.2)whereeijisthemeasurementerrorforindividualiattimetij(orreplicationj).177178MIXEDEFFECTSMODELSFORCOMPLEXDATA7654y32withoutmeasurementerrorwithmeasurementerror10012345xFigure5.1Theeffectof

7、covariatemeasurementerroronttedleastsquarelinesforsimplelinearregressionmodely=0+1x+e.Thesolidlineistheleastsquarelineforxwithoutmeasurementerror,whilethedottedlineistheleastsquarelineforxwithmeasurementerror.Intheclassicalmeasurementerrormodel(5.1),thevariabilityof

8、theob-servedcovariateszislargerthanthevariabilityofthetruecovariatesz,ijiji.e.,var(z)var(z),whileintheBerksonmeasurementerrormod