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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
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