(1.34)-PowerPoint Ch 13商務(wù)統(tǒng)計課件

Chapter13

MultipleRegressionMultipleRegressionModelLeastSquaresMethodMultipleCoefficientofDeterminationModelAssumptionsTestingforSignificanceUsingtheEstimatedRegressionEquation forEstimationandPredictionCategoricalIndependentVariables Theequationthatdescribeshowthedependentvariableyisrelatedtotheindependentvariablesx1,x2,...xpandanerrortermis:MultipleRegressionModely=b0+b1x1+b2x2+

...+bpxp+ewhere: b0,b1,b2,...,bparetheparameters,and eisarandomvariablecalledtheerrortermMultipleRegressionModel Theequationthatdescribeshowthemeanvalueofyisrelatedtox1,x2,...xpis:MultipleRegressionEquationE(y)=

0+

1x1+

2x2+...+

pxpMultipleRegressionEquation Asimplerandomsampleisusedtocomputesamplestatisticsb0,b1,b2,...,bpthatareusedasthepointestimatorsoftheparametersb0,b1,b2,...,bp.EstimatedMultipleRegressionEquation^y=b0+b1x1+b2x2+...+bpxpEstimatedMultipleRegressionEquationEstimationProcessMultipleRegressionModelE(y)=

0+

1x1+

2x2+...+

pxp+eMultipleRegressionEquationE(y)=

0+

1x1+

2x2+...+

pxp

Unknownparametersareb0,b1,b2,...,bpSampleData:x1x2...xpy........

EstimatedMultipleRegressionEquation

Samplestatisticsareb0,b1,b2,...,bpb0,b1,b2,...,bpprovideestimatesofb0,b1,b2,...,bpLeastSquaresMethodLeastSquaresCriterionComputationofCoefficientValuesTheformulasfortheregressioncoefficientsb0,b1,b2,...bpinvolvetheuseofmatrixalgebra.Wewillrelyoncomputersoftwarepackagestoperformthecalculations. Theyearsofexperience,scoreontheaptitudetesttest,andcorrespondingannualsalary($1000s)forasampleof20programmersisshownonthenextslide.Example:ProgrammerSalarySurveyMultipleRegressionModelAsoftwarefirmcollecteddataforasampleof20computerprogrammers.Asuggestionwasmadethatregressionanalysiscouldbeusedtodetermineifsalarywasrelatedtotheyearsofexperienceandthescoreonthefirm’sprogrammeraptitudetest.47158100166921056846337810086828684758083918873758174877994708924.043.023.734.335.838.022.223.130.033.038.026.636.231.629.034.030.133.928.230.0Exper.(Yrs.)TestScoreTestScoreExper.(Yrs.)Salary($000s)Salary($000s)MultipleRegressionModel Supposewebelievethatsalary(y)isrelatedtotheyearsofexperience(x1)andthescoreontheprogrammeraptitudetest(x2)bythefollowingregressionmodel: MultipleRegressionModelwhere

y=annualsalary($000) x1=yearsofexperience

x2=scoreonprogrammeraptitudetesty=

0+

1x1+

2x2+

SolvingfortheEstimatesof

0,

1,

2

InputDataLeastSquaresOutputx1

x2

y47824710043......38930ComputerPackageforSolvingMultipleRegressionProblemsb0=b1=b2=R2=etc.Excel’sRegressionEquationOutputNote:ColumnsF-Iarenotshown.SolvingfortheEstimatesof

0,

1,

2EstimatedRegressionEquationSALARY=3.174+1.404(EXPER)+0.251(SCORE)Note:Predictedsalarywillbeinthousandsofdollars.InterpretingtheCoefficients Inmultipleregressionanalysis,weinterpreteachregressioncoefficientasfollows:birepresentsanestimateofthechangeinycorrespondingtoa1-unitincreaseinxiwhenallotherindependentvariablesareheldconstant. Salaryisexpectedtoincreaseby$1,404for eachadditionalyearofexperience(whenthevariable

scoreonprogrammerattitudetestisheldconstant).b1=1.404InterpretingtheCoefficients Salaryisexpectedtoincreaseby$251foreach additionalpointscoredontheprogrammeraptitude test(whenthevariableyearsofexperienceisheld constant).b2=0.251InterpretingtheCoefficientsMultipleCoefficientofDeterminationRelationshipAmongSST,SSR,SSEwhere:

SST=totalsumofsquares

SSR=sumofsquaresduetoregression

SSE=sumofsquaresduetoerrorSST=SSR+SSE=+Excel’sANOVAOutputMultipleCoefficientofDeterminationSSRSSTMultipleCoefficientofDeterminationR2=500.3285/599.7855=.83418R2=SSR/SSTAdjustedMultipleCoefficientofDeterminationThevarianceof

,denotedby

2,isthesameforallvaluesoftheindependentvariables.Theerror

isanormallydistributedrandomvariablereflectingthedeviationbetweentheyvalueandtheexpectedvalueofygivenby

0+

1x1+

2x2+..+

pxp.AssumptionsAbouttheErrorTerm

Theerror

isarandomvariablewithmeanofzero.Thevaluesof

areindependent.Insimplelinearregression,theFandttestsprovidethesameconclusion.TestingforSignificanceInmultipleregression,theFandttestshavedifferentpurposes.TestingforSignificance:FTestTheFtestisreferredtoasthetestforoverall

significance.TheFtestisusedtodeterminewhetherasignificantrelationshipexistsbetweenthedependentvariableandthesetofalltheindependentvariables.Aseparatettestisconductedforeachoftheindependentvariablesinthemodel.IftheFtestshowsanoverallsignificance,thettestisusedtodeterminewhethereachoftheindividualindependentvariablesissignificant.TestingforSignificance:tTestWerefertoeachofthesettestsasatestforindividual

significance.TestingforSignificance:FTestHypothesesRejectionRuleTestStatisticsH0:

1=

2=...=

p=0Ha:Oneormoreoftheparametersisnotequaltozero.F=MSR/MSERejectH0ifp-value<

aorifF>F

,whereF

isbasedonanFdistributionwithpd.f.inthenumeratorandn-p-1d.f.inthedenominator.FTestforOverallSignificanceHypothesesH0:

1=

2=0Ha:Oneorbothoftheparametersisnotequaltozero.RejectionRuleFor

=.05andd.f.=2,17;F.05=3.59RejectH0ifp-value<.05orF

>3.59Excel’sANOVAOutputFTestforOverallSignificancep-valueusedtotestforoverallsignificanceFTestforOverallSignificanceTestStatisticsF=MSR/MSE=250.16/5.85=42.76Conclusionp-value<.05,sowecanrejectH0.(Also,F=42.76>3.59)TestingforSignificance:tTestHypothesesRejectionRuleTestStatisticsRejectH0ifp-value<

aorift

<-t

ort

>

t

wheret

isbasedonatdistributionwithn-p-1degreesoffreedom.tTestforSignificanceofIndividualParametersHypothesesRejectionRuleFor

=.05andd.f.=17,t.025=2.11RejectH0ifp-value<.05,orift

<-2.11ort

>2.11Excel’sRegressionEquationOutputNote:ColumnsF-Iarenotshown.tTestforSignificanceofIndividualParameterststatisticandp-valueusedtotestfortheindividualsignificanceof“Experience”Excel’sRegressionEquationOutputNote:ColumnsF-Iarenotshown.tTestforSignificanceofIndividualParameterststatisticandp-valueusedtotestfortheindividualsignificanceof“TestScore”tTestforSignificanceofIndividualParametersTestStatisticsConclusionsRejectboth

H0:

1=0andH0:

2=0.Bothindependentvariablesaresignificant.TestingforSignificance:MulticollinearityThetermmulticollinearityreferstothecorrelationamongtheindependentvariables.Whentheindependentvariablesarehighlycorrelated(say,|r|>.7),itisnotpossibletodeterminetheseparateeffectofanyparticularindependentvariableonthedependentvariable.TestingforSignificance:MulticollinearityEveryattemptshouldbemadetoavoidincludingindependentvariablesthatarehighlycorrelated.Iftheestimatedregressionequationistobeusedonlyforpredictivepurposes,multicollinearityisusuallynotaseriousproblem.UsingtheEstimatedRegressionEquation

forEstimationandPredictionTheproceduresforestimatingthemeanvalueofyandpredictinganindividualvalueofyinmultipleregressionaresimilartothoseinsimpleregression.Wesubstitutethegivenvaluesofx1,x2,...,xpintotheestimatedregressionequationandusethecorrespondingvalueofyasthepointestimate.UsingtheEstimatedRegressionEquation

forEstimationandPredictionSoftwarepackagesformultipleregressionwilloftenprovidetheseintervalestimates.Theformulasrequiredtodevelopintervalestimatesforthemeanvalueofy

andforanindividualvalueofyarebeyondthescopeofthetextbook.^Inmanysituationswemustworkwithcategorical

independentvariables

suchasgender(male,female),methodofpayment(cash,check,creditcard),etc.Forexample,x2mightrepresentgenderwherex2=0indicatesmaleandx2=1indicatesfemale.CategoricalIndependentVariablesInthiscase,x2iscalledadummyorindicatorvariable. Theyearsofexperience,thescoreontheprogrammeraptitudetest,whethertheindividualhasarelevantgraduatedegree,andtheannualsalary($000)foreachofthesampled20programmersareshownonthenextslide.CategoricalIndependentVariablesExample:ProgrammerSalarySurvey Asanextensionoftheprobleminvolvingthecomputerprogrammersalarysurvey,supposethatmanagementalsobelievesthattheannualsalaryisrelatedtowhethertheindividualhasagraduatedegreeincomputerscienceorinformationsystems.47158100166921056846337810086828684758083918873758174877994708924.043.023.734.335.838.022.223.130.033.038.026.636.231.629.034.030.133.928.230.0Exper.(Yrs.)TestScoreTestScoreExper.(Yrs.)Salary($000s)Salary($000s)Degr.NoYesNoYesYesYesNoNoNoYesDegr.YesNoYesNoNoYesNoYesNoNoCategoricalIndependentVariablesEstimatedRegressionEquation^where:

y=annualsalary($1000)

x1=yearsofexperience

x2=scoreonprogrammerapt