利用Excel进行统计分析-Chapter14-Introduction to Multiple Regression.ppt

Chap14 1 StatisticsforManagersUsingMicrosoft Excel5thEdition Chapter14IntroductiontoMultipleRegression Chap14 2 LearningObjectives Inthischapter youlearn HowtodevelopamultipleregressionmodelHowtointerprettheregressioncoefficientsHowtodeterminewhichindependentvariablestoincludeintheregressionmodelHowtodeterminewhichindependentvariablesaremostimportantinpredictingadependentvariableHowtousecategoricalvariablesinaregressionmodel Chap14 3 TheMultipleRegressionModel Idea Examinethelinearrelationshipbetween1dependent Y 2ormoreindependentvariables Xi MultipleRegressionModelwithkIndependentVariables Y intercept Populationslopes RandomError Chap14 4 MultipleRegressionEquation Thecoefficientsofthemultipleregressionmodelareestimatedusingsampledata Estimated orpredicted valueofY Estimatedslopecoefficients Multipleregressionequationwithkindependentvariables Estimatedintercept InthischapterwewillalwaysuseExceltoobtaintheregressionslopecoefficientsandotherregressionsummarymeasures Chap14 5 MultipleRegressionEquation Examplewithtwoindependentvariables Y X1 X2 SlopeforvariableX1 SlopeforvariableX2 Chap14 6 MultipleRegressionEquation2VariableExample AdistributoroffrozendessertpieswantstoevaluatefactorsthoughttoinfluencedemandDependentvariable Piesales unitsperweek Independentvariables Price in Advertising 100 s Dataarecollectedfor15weeks Chap14 7 MultipleRegressionEquation2VariableExample Sales b0 b1 Price b2 Advertising Sales b0 b1X1 b2X2WhereX1 PriceX2 Advertising Multipleregressionequation Chap14 8 MultipleRegressionEquation2VariableExample Excel Chap14 9 MultipleRegressionEquation2VariableExample b1 24 975 saleswilldecrease onaverage by24 975piesperweekforeach 1increaseinsellingprice netoftheeffectsofchangesduetoadvertising b2 74 131 saleswillincrease onaverage by74 131piesperweekforeach 100increaseinadvertising netoftheeffectsofchangesduetoprice whereSalesisinnumberofpiesperweekPriceisin Advertisingisin 100 s Chap14 10 MultipleRegressionEquation2VariableExample Predictsalesforaweekinwhichthesellingpriceis 5 50andadvertisingis 350 Predictedsalesis428 62pies NotethatAdvertisingisin 100 s so 350meansthatX2 3 5 Chap14 11 CoefficientofMultipleDetermination ReportstheproportionoftotalvariationinYexplainedbyallXvariablestakentogether Chap14 12 CoefficientofMultipleDetermination Excel 52 1 ofthevariationinpiesalesisexplainedbythevariationinpriceandadvertising Chap14 13 Adjustedr2 r2neverdecreaseswhenanewXvariableisaddedtothemodelThiscanbeadisadvantagewhencomparingmodelsWhatistheneteffectofaddinganewvariable WeloseadegreeoffreedomwhenanewXvariableisaddedDidthenewXvariableaddenoughindependentpowertooffsetthelossofonedegreeoffreedom Chap14 14 Adjustedr2 ShowstheproportionofvariationinYexplainedbyallXvariablesadjustedforthenumberofXvariablesused wheren samplesize k numberofindependentvariables PenalizesexcessiveuseofunimportantindependentvariablesSmallerthanr2Usefulincomparingmodels Chap14 15 Adjustedr2 44 2 ofthevariationinpiesalesisexplainedbythevariationinpriceandadvertising takingintoaccountthesamplesizeandnumberofindependentvariables Chap14 16 F TestforOverallSignificance F TestforOverallSignificanceoftheModelShowsifthereisalinearrelationshipbetweenalloftheXvariablesconsideredtogetherandYUseFteststatisticHypotheses H0 1 2 k 0 nolinearrelationship H1 atleastone i 0 atleastoneindependentvariableaffectsY Chap14 17 F TestforOverallSignificance Teststatistic whereFhas numerator kand denominator n k 1 degreesoffreedom Chap14 18 F TestforOverallSignificance P valuefortheF Test Chap14 19 F TestforOverallSignificance H0 1 2 0H1 1and 2notbothzero 05df1 2df2 12 TestStatistic Decision Conclusion SinceFteststatisticisintherejectionregion p value 05 rejectH0 ThereisevidencethatatleastoneindependentvariableaffectsY 0 05 F 05 3 89 RejectH0 DonotrejectH0 CriticalValue F 3 885 F Chap14 20 ResidualsinMultipleRegression Twovariablemodel Y X1 X2 Yi Yi x2i x1i Thebestfittinglinearregressionequation Y isfoundbyminimizingthesumofsquarederrors e2 Sampleobservation Residual ei Yi Yi Chap14 21 MultipleRegressionAssumptions Assumptions TheerrorsareindependentTheerrorsarenormallydistributedErrorshaveanequalvariance ei Yi Yi Errors residuals fromtheregressionmodel Chap14 22 MultipleRegressionAssumptions Theseresidualplotsareusedinmultipleregression Residualsvs YiResidualsvs X1iResidualsvs X2iResidualsvs time iftimeseriesdata Usetheresidualplotstocheckforviolationsofregressionassumptions Chap14 23 IndividualVariablesTestsofHypothesis Uset testsofindividualvariableslopesShowsifthereisalinearrelationshipbetweenthevariableXiandYHypotheses H0 i 0 nolinearrelationship H1 i 0 linearrelationshipdoesexistbetweenXiandY Chap14 24 IndividualVariablesTestsofHypothesis H0 j 0 nolinearrelationship H1 j 0 linearrelationshipdoesexistbetweenXiandY TestStatistic df n k 1 Chap14 25 IndividualVariablesTestsofHypothesis t valueforPriceist 2 306 withp value 0398t valueforAdvertisingist 2 855 withp value 0145 Chap14 26 IndividualVariablesTestsofHypothesis d f 15 2 1 12 05t 2 2 1788 H0 j 0H1 j 0 Theteststatisticforeachvariablefallsintherejectionregion p values 05 ThereisevidencethatbothPriceandAdvertisingaffectpiesalesat 05 RejectH0foreachvariable Decision Conclusion RejectH0 RejectH0 a 2 025 t 2 DonotrejectH0 0 t 2 a 2 025 Chap14 27 ConfidenceIntervalEstimatefortheSlope Confidenceintervalforthepopulationslope i Example Forma95 confidenceintervalfortheeffectofchangesinprice X1 onpiesales holdingconstanttheeffectsofadvertising 24 975 2 1788 10 832 Sotheintervalis 48 576 1 374 wherethas n k 1 d f Here thas 15 2 1 12d f Chap14 28 ConfidenceIntervalEstimatefortheSlope Confidenceintervalforthepopulationslope i Example Exceloutputalsoreportstheseintervalendpoints Weeklysalesareestimatedtobereducedbybetween1 37to48 58piesforeachincreaseof 1inthesellingprice holdingconstanttheeffectsofadvertising Chap14 29 TestingPortionsoftheMultipleRegressionModel ContributionofaSingleIndependentVariableXjSSR Xj allvariablesexceptXj SSR allvariables SSR allvariablesexceptXj MeasuresthecontributionofXjinexplainingthetotalvariationinY SST Chap14 30 TestingPortionsoftheMultipleRegressionModel MeasuresthecontributionofX1inexplainingSST FromANOVAsectionofregressionfor FromANOVAsectionofregressionfor ContributionofaSingleIndependentVariableXj assumingallothervariablesarealreadyincluded considerherea3 variablemodel SSR X1 X2andX3 SSR allvariables SSR X2andX3 Chap14 31 ThePartialF TestStatistic Considerthehypothesistest H0 variableXjdoesnotsignificantlyimprovethemodelafterallothervariablesareincludedH1 variableXjsignificantlyimprovesthemodelafterallothervariablesareincludedTestusingtheF teststatistic with1andn k 1d f Chap14 32 TestingPortionsofModel Example Testatthe 05leveltodeterminewhetherthepricevariablesignificantlyimprovesthemodelgiventhatadvertisingisincluded Example Frozendessertpies Chap14 33 TestingPortionsofModel Example H0 X1 price doesnotimprovethemodelwithX2 advertising includedH1 X1doesimprovemodel 05 df 1and12FcriticalValue 4 75 ForX1andX2 ForX2only Chap14 34 TestingPortionsofModel Example Conclusion RejectH0 addingX1doesimprovemodel ForX1andX2 ForX2only Chap14 35 RelationshipBetweenTestStatistics ThepartialFteststatisticdevelopedinthissectionandthetteststatisticarebothusedtodeterminethecontributionofanindependentvariabletoamultipleregressionmodel Thehypothesistestsassociatedwiththesetwostatisticsalwaysresultinthesamedecision thatis thep valuesareidentical Wherea degreesoffreedom Chap14 36 CoefficientofPartialDeterminationforkVariableModel MeasurestheproportionofvariationinthedependentvariablethatisexplainedbyXjwhilecontrollingfor holdingconstant theotherindependentvariables Chap14 37 UsingDummyVariables Adummyvariableisacategoricalindependentvariablewithtwolevels yesorno onoroff maleorfemalecodedas0or1AssumesequalslopesforothervariablesIfmorethantwolevels thenumberofdummyvariablesneededis numberoflevels 1 Chap14 38 DummyVariableExample Let Y piesalesX1 priceX2 holiday X2 1ifaholidayoccurredduringtheweek X2 0iftherewasnoholidaythatweek Chap14 39 DummyVariableExample Sameslope X1 Price Y sales b0 b2 b0 HolidayNoHoliday Differentintercept Holiday X2 1 NoHoliday X2 0 IfH0 2 0isrejected then Holiday hasasignificanteffectonpiesales Chap14 40 DummyVariableExample Sales numberofpiessoldperweekPrice piepricein Holiday 1Ifaholidayoccurredduringtheweek 0Ifnoholidayoccurred b2 15 onaverage saleswere15piesgreaterinweekswithaholidaythaninweekswithoutaholiday giventhesameprice Chap14 41 DummyVariableModelMoreThanTwoLevels ThenumberofdummyvariablesisonelessthanthenumberoflevelsExample Y houseprice X1 squarefeetIfstyleofthehouseisalsothoughttomatterStyle ranch splitlevel colonialThreelevels sotwodummyvariablesareneeded Chap14 42 DummyVariableModelMoreThanTwoLevels Example Let colonial bethedefaultcategory andletX2andX3beusedfortheothertwocategories Y housepriceX1 squarefeetX2 1ifranch 0otherwiseX3 1ifsplitlevel 0otherwiseThemultipleregressionequationis Chap14 43 DummyVariableModelMoreThanTwoLevels Withthesamesquarefeet aranchwillhaveanestimatedaveragepriceof23 53thousanddollarsmorethanacolonial Withthesamesquarefeet asplit levelwillhaveanestimatedaveragepriceof18 84thousanddollarsmorethanacolonial Considertheregressionequation Foracolonial X2 X3 0 Foraranch X2 1 X3 0 Forasplitlevel X2 0 X3 1 Chap14 44 InteractionBetweenIndependentVariables HypothesizesinteractionbetweenpairsofXvariablesResponsetooneXvariablemayvaryatdifferentlevelsofanotherXvariableContainsatwo waycrossproductterm Chap14 45 EffectofInteraction Given Withoutinteractionterm effectofX1onYismeasuredby 1Withinteractionterm effectofX1onYismeasuredby 1 3X2EffectchangesasX2changes Chap14 46 InteractionExample X2 1 Y 1 2X1 3 1 4X1 1 4 6X1 X2 0