數(shù)學(xué)建模之統(tǒng)計-第六章

DistinguishingBetweenRandomandFixed:Variables,Effects,andCoefficientsTheterms“random”and“fixed”areusedfrequentlyinthehierarchicallinearmodelingliterature.Thedistinctionisadifficultonetobeginwithand esmoreconfusingbecausethetermsareusedtorefertodifferentcircumstances.Herearesomesummarycommentsthatmayhelp.RandomandFixedA“fixedvariable”isonethatisassumedtobemeasuredwithouterror.Itisalsoassumedthatthevaluesofafixedvariableinonestudyarethesameasthevaluesofthefixedvariableinanotherstudy.“Randomvariables”areassumedtobevaluesthataredrawnfromalargerpopulationofvaluesandthuswillrepresentthem.Youcanthinkofthevaluesofrandomvariablesasrepresentingarandomsampleofallpossiblevaluesofthatvariable.Thus,weexpecttogeneralizetheresultsobtainedwitharandomvariabletoallotherpossiblevaluesofthatrandomvariable.MostofthetimeinANOVAandregression ysisweassumetheindependentvariablesarefixed.隨量：是取自一個大總體的變量，作為總體的代表，即統(tǒng)計學(xué)意義上的樣本。因此，可以將隨量中得到的結(jié)論延伸至整個總體。RandomandFixedTheterms“random”and“fixed”areusedinthecontextofANOVAandregressionmodels,andrefertoacertaintypeofstatisticalmodel.Almostalways,researchersusefixedeffectsregressionorANOVAandtheyarerarelyfacedwithasituationinvolvingrandomeffectsyses.AfixedeffectsANOVAreferstoassumptionsabouttheindependentvariableandtheerrordistributionforthevariable.Anexperimentaldesignistheeasiestexampleforillustratingtheprincipal.Usually,theresearcherisinterestedinonlygeneralizingtheresultstoexperimentalvaluesusedinthestudy.Forinstance,adrugstudyusing0mg,5mg,or10mgofanexperimentaldrug.ThisiswhenafixedeffectsANOVAwouldbeappropriate.Inthiscase,theextrapolationistootherstudiesortreatmentsthatmightusethesamevaluesofthedrug(i.e.,0mg,5mg,and10mg).However,iftheresearcherwantstomakeinferencestheparticularvaluesoftheindependentvariableusedinthestudy,arandomeffectsmodelisused.Acommonexamplewouldbetheuseofpublicartworksrepresentinglow,moderate,andhigh ness(e.g.,statueofawarherovs.apivotinggeometricdesign).Theresearcherwouldliketomakeinferencesthethreepiecesused,sotheartpiecesareconceptualizedaspiecesrandomlydrawnfromalargeruniverseofpossiblepiecesandtheinferencesaremadetoalargeruniverseofartworkandrangeof nessvalues.Suchageneralizationismoreofaninferentialleap,and,consequently,therandomeffectsmodelislesspowerful.Randomeffectsmodelsaresometimesreferredtoas“ModelII”or“variancecomponentmodels.”ysesusingbothfixedandrandomeffectsarecalled“mixedmodels.”FixedandRandomCoefficientsinMultilevelTherandomvs.fixeddistinctionforvariablesandeffectsisimportantinmultilevelregression.Inmultilevelregressionmodels,bothlevel-1andlevel-2predictorsareassumedtobefixed.However,level-1interceptsandslopesaretypicallyassumedtovaryrandomlyacrossgroups.Becauseoftheassumptionsabouttheirerrordistributions,wecalltheirvariances,oo=var(U0j)and11=var(U1j),“randomcoefficients.”Thismeansthatwecanthinkabout0jand1jasakintotherandomvariablesIdescribedabove.Insteadofattemptingtogeneralizetheparticularvaluesoftheindependentvariable,weareattemptingtogeneralizetheparticulargroupsinthestudy.Forinstance,wemayhave50companies,butwewishtogeneralizetoalargeruniverseofcompanieswhenweexaminethemeans(intercepts)orthex-yrelationship(slopes).InHLM,variancesfortheinterceptsandslopesareestimatedbydefault.Thatis,whenyouconstructamodel,U0jandU1jareestimatedbydefault.However,theresearcherhasthechoiceofsettingU0jandU1jtobezero.Bysettingthemtozero,wearetestingamodelinwhichwe0jand1jdonotvaryrandomlyacrossgroups.Infact,theirvarianceisassumedtobezero,sotheyareassumedtobeconstantor“nonvarying”acrossgroups.Forexample,fixed,nonvaryinginterceptswouldimplythegroupaverageforthedependentvariableisassumedtobeequalineachgroup.Althoughwetypicallyrefertothisconstraintas“fixingtheintercepts”or“fixingtheslopes,”thetermissomewhatlooselyapplied,becausewearereallyassumingtheyarefixedandnonvarying.InterceptModelandRandomEffectsInthesimplestHLMmodel,therearenopredictors.Wehaveonelevel-1equationandonelevel-2Yij0jwhereYisthedependentvariable,0jistheintercept,andRijistheresidualorerror.0jcanbethoughtofasthemeanofeachgroup.Thelevel-2equationalsohasnopredictorsinit’ssimplest0j00U0where0jisthedependentvariable,00isthelevel-2intercept,andU0jisthelevel-2error.Inthisequation,00representsthegrandmeanorthemeanoftheintercepts.U0jrepresentsthedeviationofeaeanfromthegrandmean.Whentheaveragedeviationislarge,therearelargegroupItcanbeshownthatthisisreallytheANOVAmodel.ANOVAexaminesthedeviationsofgroupmeansfromthegrandmean.Becauseweassumethatthegroupmeans,representedby0jand,thus,theirdeviationsarerandomlyvarying,thismodelisequivalenttotherandomeffectsANOVAmodel.WecanrewriteourtwoHLMequationsasasingleequation,ifluginthelevel-2termsintolevel-1Yij00U0jInthisequation,00representsthegrandmean,theU0jtermrepresentsthedeviationsfromthemean(ormeandifferencesortreatmenteffect,ifyouwouldlike),andRijrepresentsthewithin-grouperrororvariation.InmoremathematicallyorientedANOVAtextbooks,youwillseeanANOVAmodelwrittensomethinglike:YijjIfthemodelisanrandomeffectsANOVA,jisassumedtohavearandomdistribution.Inmultilevelregression,becauseweassumeU0jtovaryrandomly,thesimpleHLMmodelwithnolevel-1orlevel-2predictorsisequivalenttotherandomeffectsANOVAmodel.Whenweaddpredictorstothelevel-1equation,theyarecovariatesandthemodel esarandomeffectsANCOVAinwhichthemeansareadjustedforthecovariate.SummaryRandomvs.UseinMultilevelRandomvariable:(1)isassumedtobemeasuredwithmeasurementerror.Thescoresareafunctionofatruescoreandrandomerror;(2)thevaluescomefromandareintendedtogeneralizetoamuchlargerpopulationofpossiblevalueswithacertainprobabilitydistribution(e.g.,normaldistribution);(3)thenumberofvaluesinthestudyissmallrelativetothevaluesofthevariableasitappearsinthepopulationitisdrawnfrom.Fixedvariable:(1)assumedtobemeasuredwithoutmeasurementerror;(2)desiredgeneralizationtopopulationorotherstudiesistothesamevalues;(3)thevariableusedinthestudycontainsallormostofthevariable’svaluesinthepopulation.Randomvariable:representingindividualswithdifferinglevelsofmanipulatedinanexperiment,censusFixedvariable:gender,race,orinterventionvs.controlvariablesinMLRassumedtobeItisimportanttodistinguishbetweenavariablethatvaryingandavariablethatisrandom.Afixedvariablecanhavedifferentvalues,itisnotnecessarily(equal)acrossRandomeffect:(1)differentstatisticalmodelofregressionorANOVAmodelwhichassumesthatanindependentvariableisrandom;(2)generallyusedifthelevelsoftheindependentvariablearethoughttobeasmallsubsetofthepossiblevalueswhichonewishestogeneralizeto;(3)willprobablyproducelargerstandarderrors(lesspowerful).Fixedeffect:(1)statisticalmodeltypicallyusedinregressionandANOVAassumingindependentvariableisfixed;(2)generalizationoftheresultsapplytosimilarvaluesofindependentvariableinthepopulationorinotherstudies;(3)willprobablyproducesmallerstandarderrors(morepowerful).Randomeffect:randomeffectsANOVA,randomeffectsregressionFixedeffect:fixedeffectsANOVA,fixedeffectsregressionInterceptonlymodelsinMLRareequivalenttorandomeffectsANOVAorRandomcoefficient:termappliesonlytoysesinwhichintercepts,slopes,andvariancescanbeassumedtoberandom.MLRysesmosttypicallyassumerandomcoefficients.Onecanconceptualizeth