linear regression and anova

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1、CHAPTER11LinearRegressionandANOVAIntroductionInstatistics,modelingiswherewegetdowntobusiness.Modelsquantifytherelation-shipsbetweenourvariables.Modelsletusmakepredictions.Asimplelinearregressionisthemostbasicmodel.It’sjusttwovariablesandismodeledasalinearrelationshipwithaner

2、rorterm:yi=β0+β1xi+εiWearegiventhedataforxandy.Ourmissionistofitthemodel,whichwillgiveusthebestestimatesforβ0andβ1(Recipe11.1).Thatgeneralizesnaturallytomultiplelinearregression,wherewehavemultiplevari-ablesontherighthandsideoftherelationship(Recipe11.2):yi=β0+β1ui+β2vi+β3wi

3、+εiStatisticianscallu,v,andwthepredictorsandytheresponse.Obviously,themodelisusefulonlyifthereisafairlylinearrelationshipbetweenthepredictorsandtheresponse,butthatrequirementismuchlessrestrictivethanyoumightthink.Recipe11.11dis-cussestransformingyourvariablesintoa(more)linea

4、rrelationshipsothatyoucanusethewell-developedmachineryoflinearregression.ThebeautyofRisthatanyonecanbuildtheselinearmodels.Themodelsarebuiltbyafunction,lm,whichreturnsamodelobject.Fromthemodelobject,wegetthecoef-ficients(βi)andregressionstatistics.It’seasy.ThehorrorofRisthat

5、anyonecanbuildthesemodels.Nothingrequiresyoutocheckthatthemodelisreasonable,muchlessstatisticallysignificant.Beforeyoublindlybe-lieveamodel,checkit.Mostoftheinformationyouneedisintheregressionsummary(Recipe11.4):Isthemodelstatisticallysignificant?ChecktheFstatisticatthebotto

6、mofthesummary.267Arethecoefficientssignificant?Checkthecoefficient’ststatisticsandp-valuesinthesummary,orchecktheirconfidenceintervals(Recipe11.13).Isthemodeluseful?ChecktheR2nearthebottomofthesummary.Doesthemodelfitthedatawell?Plottheresidualsandchecktheregressiondiagnostic

7、s(Recipes11.14and11.15).Doesthedatasatisfytheassumptionsbehindlinearregression?Checkwhetherthediagnosticsconfirmthatalinearmodelisreasonableforyourdata(Recipe11.15).ANOVAAnalysisofvariance(ANOVA)isapowerfulstatisticaltechnique.First-yeargraduatestudentsinstatisticsaretaughtA

8、NOVAalmostimmediatelybecauseofitsimportance,boththeoreticalandpractical.Iam