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MultipleRegressionDr.AndyFieldSlide2AimsUnderstandWhenToUseMultipleRegression.Understandthemultipleregressionequationandwhatthebetasrepresent.UnderstandDifferentMethodsofRegressionHierarchicalStepwiseForcedEntryUnderstandHowtodoaMultipleRegressiononPASW/SPSSUnderstandhowtoInterpretmultipleregression.UnderstandtheAssumptionsofMultipleRegressionandhowtotestthemSlide3WhatisMultipleRegression?LinearRegressionisamodeltopredictthevalueofonevariablefromanother.MultipleRegressionisanaturalextensionofthismodel:Weuseittopredictvaluesofanoutcomefromseveralpredictors.Itisahypotheticalmodeloftherelationshipbetweenseveralvariables.Regression:AnExampleArecordcompanybosswasinterestedinpredictingrecordsalesfromadvertising.Data200differentalbumreleasesOutcomevariable:Sales(CDsandDownloads)intheweekafterreleasePredictorvariablesTheamount(in￡s)spentpromotingtherecordbeforerelease(seelastlecture)Numberofplaysontheradio(newvariable)Slide5TheModelwithOnePredictorSlide6MultipleRegressionasanEquationWithmultipleregressiontherelationshipisdescribedusingavariationoftheequationofastraightline.Slide7b0b0

istheintercept.TheinterceptisthevalueoftheYvariablewhenallXs=0.ThisisthepointatwhichtheregressionplanecrossestheY-axis(vertical).Slide8BetaValuesb1

istheregressioncoefficientforvariable1.b2

istheregressioncoefficientforvariable2.bn

istheregressioncoefficientfornthvariable.Slide9TheModelwithTwoPredictorsbAdvertsbairplayb0Slide10MethodsofRegressionHierarchical:Experimenterdecidestheorderinwhichvariablesareenteredintothemodel.ForcedEntry:Allpredictorsareenteredsimultaneously.Stepwise:Predictorsareselectedusingtheirsemi-partialcorrelationwiththeoutcome.Slide12HierarchicalRegressionKnownpredictors(basedonpastresearch)areenteredintotheregressionmodelfirst.Newpredictorsarethenenteredinaseparatestep/block.Experimentermakesthedecisions.Slide13HierarchicalRegressionItisthebestmethod:Basedontheorytesting.Youcanseetheuniquepredictiveinfluenceofanewvariableontheoutcomebecauseknownpredictorsareheldconstantinthemodel.BadPoint:Reliesontheexperimenterknowingwhatthey’redoing!Slide14ForcedEntryRegressionAllvariablesareenteredintothemodelsimultaneously.Theresultsobtaineddependonthevariablesenteredintothemodel.Itisimportant,therefore,tohavegoodtheoreticalreasonsforincludingaparticularvariable.Slide15StepwiseRegressionIVariablesareenteredintothemodelbasedonmathematicalcriteria.Computerselectsvariablesinsteps.Step1SPSSlooksforthepredictorthatcanexplainthemostvarianceintheoutcomevariable.ExamPerformanceRevisionTimeVarianceaccountedforbyRevisionTime(33.1%)PreviousExamVarianceexplained(1.7%)DifficultyVarianceexplained(1.3%)Slide18StepwiseRegressionIIStep2:Havingselectedthe1stpredictor,asecondoneischosenfromtheremainingpredictors.Thesemi-partialcorrelationisusedasacriterionforselection.Slide19Semi-PartialCorrelationPartialcorrelation:measurestherelationshipbetweentwovariables,controllingfortheeffectthatathirdvariablehasonthemboth.Asemi-partialcorrelation:Measurestherelationshipbetweentwovariablescontrollingfortheeffectthatathirdvariablehasononlyoneoftheothers.Slide20ExamAnxietyExamAnxietyRevisionRevisionPartial

CorrelationSemi-PartialCorrelationSlide21Semi-PartialCorrelationinRegressionThesemi-partialcorrelationMeasurestherelationshipbetweenapredictorandtheoutcome,controllingfortherelationshipbetweenthatpredictorandanyothersalreadyinthemodel.Itmeasurestheuniquecontributionofapredictortoexplainingthevarianceoftheoutcome.Slide22Slide23ProblemswithStepwiseMethods

Relyonamathematicalcriterion.VariableselectionmaydependupononlyslightdifferencesintheSemi-partialcorrelation.Theseslightnumericaldifferencescanleadtomajortheoreticaldifferences.Shouldbeusedonlyforexploration

Slide24DoingMultipleRegressionSlide25DoingMultipleRegressionRegressionStatisticsRegressionDiagnosticsSlide28Output:ModelSummarySlide29RandR2

RThecorrelationbetweentheobservedvaluesoftheoutcome,andthevaluespredictedbythemodel.R2Yheproportionofvarianceaccountedforbythemodel.Adj.R2AnestimateofR2inthepopulation(shrinkage).Slide30Output:ANOVASlide31AnalysisofVariance:ANOVATheF-testlooksatwhetherthevarianceexplainedbythemodel(SSM)issignificantlygreaterthantheerrorwithinthemodel(SSR).Ittellsuswhetherusingtheregressionmodelissignificantlybetteratpredictingvaluesoftheoutcomethanusingthemean.Slide32Output:betasSlide33HowtoInterpretBetaValues

Betavalues:thechangeintheoutcomeassociatedwithaunitchangeinthepredictor.Standardisedbetavalues:tellusthesamebutexpressedasstandarddeviations.Slide34BetaValuesb1=0.087.So,asadvertisingincreasesby￡1,recordsalesincreaseby0.087units.b2=3589.So,eachtime(perweek)asongisplayedonradio1itssalesincreaseby3589units.Slide35ConstructingaModelSlide36StandardisedBetaValues

1=0.523Asadvertisingincreasesby1standarddeviation,recordsalesincreaseby0.523ofastandarddeviation.

2=0.546Whenthenumberofplaysonradioincreasesby1s.d.itssalesincreaseby0.546standarddeviations.Slide37InterpretingStandardisedBetasAsadvertisingincreasesby￡485,655,recordsalesincreaseby0.52380,699=42,206.Ifthenumberofplaysonradio1perweekincreasesby12,recordsalesincreaseby0.54680,699=44,062.ReportingtheModelSlide39HowwelldoestheModelfitthedata?

Therearetwowaystoassesstheaccuracyofthemodelinthesample:ResidualStatisticsStandardizedResidualsInfluentialcasesCook’sdistanceSlide40StandardizedResiduals

Inanaveragesample,95%ofstandardizedresidualsshouldliebetween2.99%ofstandardizedresidualsshouldliebetween2.5.OutliersAnycaseforwhichtheabsolutevalueofthestandardizedresidualis3ormore,islikelytobeanoutlier.Slide41Cook’sDistanceMeasurestheinfluenceofasinglecaseonthemodelasawhole.Weisberg(1982):Absolutevaluesgreaterthan1maybecauseforconcern.Slide42Generalization

Whenwerunregression,wehopetobeabletogeneralizethesamplemodeltotheentirepopulation.Todothis,severalassumptionsmustbemet.Violatingtheseassumptionsstopsusgeneralizingconclusionstoourtargetpopulation.Slide43StraightforwardAssumptions

VariableType:OutcomemustbecontinuousPredictorscanbecontinuousordichotomous.Non-ZeroVariance:Predictorsmustnothavezerovariance.Linearity:Therelationshipwemodelis,inreality,linear.Independence:Allvaluesoftheoutcomeshouldcomefromadifferentperson.Slide44TheMoreTrickyAssumptionsNoMulticollinear

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