商務(wù)統(tǒng)計(jì)學(xué)第13章 出版社：中國人民大學(xué)出版社

Chapter13MultipleRegressionBusinessStatistics,AFirstCourse4thEdition1商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社LearningObjectivesInthischapter,youlearn:

HowtodevelopamultipleregressionmodelHowtointerprettheregressioncoefficientsHowtodeterminewhichindependentvariablestoincludeintheregressionmodelHowtousecategoricalvariablesinaregressionmodelHowtousequadratictermsinaregressionmodel2商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社TheMultipleRegressionModelIdea:Examinethelinearrelationshipbetween1dependent(Y)&2ormoreindependentvariables(Xi)MultipleRegressionModelwithkIndependentVariables:Y-interceptPopulationslopesRandomError3商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社MultipleRegressionEquationThecoefficientsofthemultipleregressionmodelareestimatedusingsampledataEstimated(orpredicted)valueofYEstimatedslopecoefficientsMultipleregressionequationwithkindependentvariables:EstimatedinterceptInthischapterwewillalwaysuseExceltoobtaintheregressionslopecoefficientsandotherregressionsummarymeasures.4商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社TwovariablemodelYX1X2SlopeforvariableX1SlopeforvariableX2MultipleRegressionEquation(continued)5商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社Example:

2IndependentVariablesAdistributoroffrozendesertpieswantstoevaluatefactorsthoughttoinfluencedemandDependentvariable:Piesales(unitsperweek)Independentvariables:Price(in$)

Advertising($100’s)Dataarecollectedfor15weeks6商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社PieSalesExampleSales=b0+b1(Price) +b2(Advertising)WeekPieSalesPrice($)Advertising($100s)13505.503.324607.503.333508.003.044308.004.553506.803.063807.504.074304.503.084706.403.794507.003.5104905.004.0113407.203.5123007.903.2134405.904.0144505.003.5153007.002.7Multipleregressionequation:7商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社MultipleRegressionOutputRegressionStatisticsMultipleR0.72213RSquare0.52148AdjustedRSquare0.44172StandardError47.46341Observations15ANOVA

dfSSMSFSignificanceFRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333

CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.708888商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社TheMultipleRegressionEquationb1=-24.975:saleswilldecrease,onaverage,by24.975piesperweekforeach$1increaseinsellingprice,netoftheeffectsofchangesduetoadvertisingb2=74.131:saleswillincrease,onaverage,by74.131piesperweekforeach$100increaseinadvertising,netoftheeffectsofchangesduetopricewhere SalesisinnumberofpiesperweekPriceisin$Advertisingisin$100’s.9商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社UsingTheEquationtoMakePredictionsPredictsalesforaweekinwhichthesellingpriceis$5.50andadvertisingis$350:Predictedsalesis428.62piesNotethatAdvertisingisin$100’s,so$350meansthatX2=3.510商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社PredictionsinExcelusingPHStatPHStat|regression|multipleregression…Checkthe“confidenceandpredictionintervalestimates”box11商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社InputvaluesPredictionsinPHStat(continued)

PredictedYvalue<ConfidenceintervalforthemeanYvalue,giventheseX’s<PredictionintervalforanindividualYvalue,giventheseX’s<12商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社Coefficientof

MultipleDeterminationReportstheproportionoftotalvariationinYexplainedbyallXvariablestakentogether13商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社RegressionStatisticsMultipleR0.72213RSquare0.52148AdjustedRSquare0.44172StandardError47.46341Observations15ANOVA

dfSSMSFSignificanceFRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333

CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.7088852.1%ofthevariationinpiesalesisexplainedbythevariationinpriceandadvertisingMultipleCoefficientof

Determination(continued)14商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社Adjustedr2r2neverdecreaseswhenanewXvariableisaddedtothemodelThiscanbeadisadvantagewhencomparingmodelsWhatistheneteffectofaddinganewvariable?WeloseadegreeoffreedomwhenanewXvariableisaddedDidthenewXvariableaddenoughexplanatorypowertooffsetthelossofonedegreeoffreedom?15商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社ShowstheproportionofvariationinYexplainedbyallXvariablesadjustedforthenumberofX

variablesused

(wheren=samplesize,k=numberofindependentvariables)PenalizeexcessiveuseofunimportantindependentvariablesSmallerthanr2UsefulincomparingamongmodelsAdjustedr2(continued)16商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社RegressionStatisticsMultipleR0.72213RSquare0.52148AdjustedRSquare0.44172StandardError47.46341Observations15ANOVA

dfSSMSFSignificanceFRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333

CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.7088844.2%ofthevariationinpiesalesisexplainedbythevariationinpriceandadvertising,takingintoaccountthesamplesizeandnumberofindependentvariables(continued)Adjustedr217商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社IstheModelSignificant?FTestforOverallSignificanceoftheModelShowsifthereisalinearrelationshipbetweenalloftheXvariablesconsideredtogetherandYUseF-teststatisticHypotheses:H0:β1=β2=…=βk=0(nolinearrelationship)H1:atleastoneβi≠0(atleastoneindependent variableaffectsY)

18商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社FTestforOverallSignificanceTeststatistic:

whereFhas (numerator)=kand (denominator)=(n–k-1)

degreesoffreedom

19商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社RegressionStatisticsMultipleR0.72213RSquare0.52148AdjustedRSquare0.44172StandardError47.46341Observations15ANOVA

dfSSMSFSignificanceFRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333

CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888(continued)FTestforOverallSignificanceWith2and12degreesoffreedomP-valuefortheFTest20商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社H0:β1=β2=0H1:β1andβ2notbothzero=.05df1=2df2=12TestStatistic:Decision:Conclusion:SinceFteststatisticisintherejectionregion(p-value<.05),rejectH0ThereisevidencethatatleastoneindependentvariableaffectsY0

=.05F.05=3.885RejectH0DonotrejectH0CriticalValue:F=3.885FTestforOverallSignificance(continued)F21商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社TwovariablemodelYX1X2YiYi<x2ix1i

Thebestfitequation,Y,isfoundbyminimizingthesumofsquarederrors,e2<SampleobservationResidualsinMultipleRegressionResidual=ei=(Yi–Yi)<22商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社MultipleRegressionAssumptionsAssumptions:TheerrorsarenormallydistributedErrorshaveaconstantvarianceThemodelerrorsareindependentei=(Yi–Yi)<Errors(residuals)fromtheregressionmodel:23商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社ResidualPlotsUsed

inMultipleRegressionTheseresidualplotsareusedinmultipleregression:Residualsvs.YiResidualsvs.X1iResidualsvs.X2iResidualsvs.time(iftimeseriesdata)<Usetheresidualplotstocheckforviolationsofregressionassumptions24商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社AreIndividualVariablesSignificant?UsettestsofindividualvariableslopesShowsifthereisalinearrelationshipbetweenthevariableXjandYHypotheses:H0:βj=0(nolinearrelationship)H1:βj≠0(linearrelationshipdoesexist betweenXjandY)25商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社AreIndividualVariablesSignificant?H0:βj=0(nolinearrelationship)H1:βj≠0(linearrelationshipdoesexist betweenxjandy)TestStatistic: (df=n–k–1)(continued)26商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社RegressionStatisticsMultipleR0.72213RSquare0.52148AdjustedRSquare0.44172StandardError47.46341Observations15ANOVA

dfSSMSFSignificanceFRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333

CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888t-valueforPriceist=-2.306,withp-value.0398t-valueforAdvertisingist=2.855,withp-value.0145(continued)AreIndividualVariablesSignificant?27商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社d.f.=15-2-1=12=.05t/2=2.1788InferencesabouttheSlope:

t

TestExampleH0:βi=0H1:βi

0Theteststatisticforeachvariablefallsintherejectionregion(p-values<.05)ThereisevidencethatbothPriceandAdvertisingaffectpiesalesat=.05FromExceloutput:RejectH0foreachvariable

CoefficientsStandardErrortStatP-valuePrice-24.9750910.83213-2.305650.03979Advertising74.1309625.967322.854780.01449Decision:Conclusion:RejectH0RejectH0a/2=.025-tα/2DonotrejectH00tα/2a/2=.025-2.17882.178828商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社ConfidenceIntervalEstimate

fortheSlopeConfidenceintervalforthepopulationslopeβjExample:Forma95%confidenceintervalfortheeffectofchangesinprice(X1)onpiesales:-24.975±(2.1788)(10.832)Sotheintervalis(-48.576,-1.374)(Thisintervaldoesnotcontainzero,sopricehasasignificanteffectonsales)

CoefficientsStandardErrorIntercept306.52619114.25389Price-24.9750910.83213Advertising74.1309625.96732wherethas(n–k–1)d.f.Here,thas(15–2–1)=12d.f.29商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社ConfidenceIntervalEstimate

fortheSlopeConfidenceintervalforthepopulationslopeβiExample:Exceloutputalsoreportstheseintervalendpoints:Weeklysalesareestimatedtobereducedbybetween1.37to48.58piesforeachincreaseof$1inthesellingprice

CoefficientsStandardError…Lower95%Upper95%Intercept306.52619114.25389…57.58835555.46404Price-24.9750910.83213…-48.57626-1.37392Advertising74.1309625.96732…17.55303130.70888(continued)30商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社UsingDummyVariablesAdummyvariableisacategoricalexplanatoryvariablewithtwolevels:yesorno,onoroff,maleorfemalecodedas0or1RegressioninterceptsaredifferentifthevariableissignificantAssumesequalslopesforothervariables31商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社Dummy-VariableExampleLet:Y=piesalesX1=priceX2=holiday(X2=1ifaholidayoccurredduringtheweek) (X2=0iftherewasnoholidaythatweek)32商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社SameslopeDummy-VariableExample(continued)X1(Price)Y(sales)b0+b2b0

HolidayNoHolidayDifferentinterceptHoliday(X2=1)NoHoliday(X2=0)IfH0:β2=0isrejected,then“Holiday”hasasignificanteffectonpiesales33商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社Sales:numberofpiessoldperweekPrice:piepricein$Holiday:InterpretingtheDummyVariableCoefficientExample:1Ifaholidayoccurredduringtheweek0Ifnoholidayoccurredb2=15:onaverage,saleswere15piesgreaterinweekswithaholidaythaninweekswithoutaholiday,giventhesameprice34商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社InteractionBetweenIndependentVariablesHypothesizesinteractionbetweenpairsofXvariablesResponsetooneXvariablemayvaryatdifferentlevelsofanotherXvariableContainscross-productterm

35商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社EffectofInteractionGiven:

Withoutinteractionterm,effectofX1onYismeasuredbyβ1Withinteractionterm,effectofX1onYismeasuredbyβ1+β3X2EffectchangesasX2changes36商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社X2=1:Y=1+2X1+3(1)+4X1(1)=4+6X1

X2=0:Y=1+2X1+3(0)+4X1(0)=1+2X1

InteractionExampleSlopesaredifferentiftheeffectofX1onYdependsonX2valueX148120010.51.5Y=1+2X1+3X2+4X1X2

SupposeX2isadummyvariableandtheestimatedregressionequationis37商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社TherelationshipbetweenthedependentvariableandanindependentvariablemaynotbelinearCanreviewthescatterdiagramtocheckfornon-linearrelationshipsExample:QuadraticmodelThesecondindependentvariableisthesquareofthefirstvariableNonlinearRelationships38商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社QuadraticRegressionModelwhere:

β0=Yintercept

β1=regressioncoefficientforlineareffectofXonY

β2=regressioncoefficientforquadraticeffectonY

εi=randomerrorinYforobservationiModelform:39商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社LinearfitdoesnotgiverandomresidualsLinearvs.NonlinearFitNonlinearfitgivesrandomresidualsXresidualsXYXresidualsYX40商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社QuadraticRegressionModelQuadraticmodelsmaybeconsideredwhenthescatterdiagramtakesononeofthefollowingshapes:X1YX1X1YYYβ1<0β1>0β1<0β1>0β1=thecoefficientofthelineartermβ2=thecoefficientofthesquaredtermX1β2>0β2>0β2<0β2<041商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社TestingtheOverall

QuadraticModelTestforOverallRelationship H0:β1=β2=0(nooverallrelationshipbetweenXandY) H1:

β1and/orβ2≠0(thereisarelationshipbetweenXandY)F-teststatistic=Estimatethequadraticmodeltoobtaintheregressionequation:

42商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社TestingforSignificance:QuadraticEffectTestingtheQuadraticEffectComparequadraticregressionequation

withthelinearregressionequation43商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社TestingforSignificance:QuadraticEffectTestingtheQuadraticEffectConsiderthequadraticregressionequation

Hypotheses(Thequadratictermdoesnotimprovethemodel)(Thequadratictermimprovesthemodel)H0:β2=0H1:β2

0(continued)44商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社TestingforSignificance:QuadraticEffectTestingtheQuadraticEffect

Hypotheses(Thequadratictermdoesnotimprovethemodel)(Thequadratictermimprovesthemodel)TheteststatisticisH0:β2=0H1:β2

0(continued)where:b2=squaredtermslopecoefficient

β2=hypothesizedslope(zero)Sb=standarderroroftheslope245商務(wù)統(tǒng)計(jì)學(xué)第13章出版社：中國人民大學(xué)出版社TestingforSignificance:QuadraticEffectTestingtheQuadraticEffect

Comparer2fromsimpleregressionto adjustedr2fromthequadraticmodelI