Introduction to Regression Analysis Chapter 4

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1、Chapter4ISimultaneousinferenceIRegressionthroughtheoriginIInverseprediction(leftasreading)IEectsofmeasurementerrors(leftasreading)IChoiceofXlevels1/9Condenceintervals(CIs)andsimultaneousinferenceIFora(1)100%CIforoneparameterofinterest,e.g.95%CIfor1,

2、wecansay:IBeforeobservingthedata:Prob(Event1295%CI)=0.95IAfterobservingthedata:Weare95%condentthat1isinits95%CIIFormultipleparameters,eachwithitsown(1)100%CIforoneparameterofinterest,e.g.a95%CIfor0anda95%CIfor1:IBeforeobservingthedata:Prob(fEvent0295

3、%CIgANDfEvent1295%CIg)=?IAfterobservingthedata:Weare?%condentthat0isinits95%CI,AND1isinits95%CIIWhenconstructingcondenceintervalsformultipleparameters:IEachparameterhasitsowncondenceinterval,withitsownindividualcondencelevel/coecient,e.g.1ICombin

4、edcondencelevel=familycondencecoecientIGoalinsimultaneousinference:Controlthefamilycondencecoecient(bycontrollingtheindividualcondencecoecients)2/9BonferronijointcondenceintervalsIBonferroni:Whenconstructing2condenceintervals,wecanuse(11)and(

5、12)condencelevelsforindividualintervals,with1+2=,toguaranteeanoverallcondencecoecientof(1-)(proofonboard)IE.g.wecanuse95%condenceintervalsfor0and1,suchthatIbeforeobservingthedata:theprobabilityoftheeventthatatleastoneCIdoesnotcontaintheir"happeni

6、ngis0.10Iafterobservingthedata:weare90%condentthateachCIcontainsitsIExtendBonferroni:Whenconstructingmcondenceintervals,wecanuse(1k)Pindividualcondencelevels,withk=,toguaranteeanoverallcondencelevelof(1-)IBonferronigivesconservativebounds:thefamil

7、ycondencecoecientisATLEAST13/9SimultaneouspredictionforthemeanresponseIGoal:predictthemeanE(Yh)atmlevelsofX:fXh;h=1;:::;mgITwoapproaches:IBonferroniprocedure:use(1-=m)asindividualcondencecoecientforeachXh:Y^hB
sfY^hg;B=t(1;n2)2mIWorking-Hotell

8、ingprocedure(condencebandfromCh2):pY^hW
sfY^hg;W=2F(1;2;n2)IBothapproachesgiveconservativebounds,sochoosethetightestoneIInterpretationofa95%familycondencecoecient?IA95%familycondencecoecientformmeansmeansthatweare95%condenttha