簡單線性回歸與相關(guān)TheSimpleLinearRegressionandCorrelationPurposeofRegressionandCorrelationAnalysis

回歸與相關(guān)分析的目的

RegressionAnalysisisUsedPrimarilyfor Prediction（回歸主要用于預(yù)測）

Astatisticalmodelusedtopredictthevaluesofa dependentorresponsevariablebasedonvaluesof atleastoneindependentorexplanatoryvariable

CorrelationAnalysisisUsedtoMeasure StrengthoftheAssociationBetween NumericalVariables（度量關(guān)系密切程度）TheScatterDiagram

散點(diǎn)圖Plotofall(Xi,

Yi)pairsTypesofRegressionModelsPositiveLinearRelationshipNegativeLinearRelationshipRelationshipNOTLinearNoRelationshipSimpleLinearRegressionModel

簡單線性回歸模型

YinterceptSlopeTheStraightLinethatBestFittheDataRelationshipBetweenVariablesIsaLinearFunctionRandomErrorDependent(Response)VariableIndependent(Explanatory)Variable

i=RandomErrorYXPopulation

LinearRegressionModelObservedValueObservedValue

YXiX

01YXiii

01SampleLinearRegressionModel

簡單線性相關(guān)模型

Yi

=PredictedValueofYforobservationiXi

=ValueofXforobservation

ib0

=SampleY-interceptusedasestimateof thepopulation

0b1

=SampleSlopeusedasestimateofthe population

1SimpleLinearRegressionEquation:ExampleYouwishtoexaminetherelationshipbetweenthesquarefootageofproducestoresanditsannualsales.Sampledatafor7storeswereobtained.Findtheequationofthestraightlinethatfitsthedatabest

AnnualStore Square Sales Feet ($000)11,726 3,68121,542 3,3953 2,816 6,6534 5,555 9,5435 1,292 3,3186 2,208 5,5637 1,313 3,760 ScatterDiagramExampleEquationfortheBestStraightLine

FromExcelPrintout:GraphoftheBestStraightLineYi=1636.415+1.487Xi

InterpretingtheResultsYi=1636.415+1.487XiTheslopeof1.487meansforeachincreaseofoneunitinX,theYisestimatedtoincrease1.487units.Foreachincreaseof1squarefootinthesizeofthestore,themodelpredictsthattheexpectedannualsalesareestimatedtoincreaseby$1487.

MeasuresofVariation:

TheSumofSquares變異平方和

SST

=

TotalSumofSquares

measuresthevariationofthe

Yi

valuesaroundtheir meanYSSR=RegressionSum

ofSquaresexplainedvariationattributabletotherelationship betweenXandYSSE

=

ErrorSumofSquares

variationattributabletofactorsotherthanthe relationshipbetweenXandY_MeasuresofVariation:TheSumofSquaresXiYi=b0+b1XiYXYSST=

(Yi

-

Y)2SSE=

(Yi

-

Yi)2

SSR=

(Yi-

Y)2

___

MeasuresofVariation TheSumofSquares:ExampleExcelOutputforProduceStoresSSRSSESSTTheCoefficientofDetermination

決定系數(shù)----用回歸解釋相關(guān)的橋梁SSRregressionsumofsquaresSSTtotalsumofsquaresr2==

MeasurestheproportionofvariationthatisexplainedbytheindependentvariableXintheregressionmodelCoefficientsofDetermination(r2)andCorrelation(r)

r2=1,r2=1,r2=.8,r2=0,YYi=b0+b1XiX^YYi=b0+b1XiX^YYi=b0+b1XiX^YYi=b0+b1XiX^r=+1r=-1r=+0.9r=0StandardErrorofEstimate

標(biāo)準(zhǔn)誤的估計(jì)

=ThestandarddeviationofthevariationofobservationsaroundtheregressionlineMeasuresofVariation:ExampleExcelOutputforProduceStores

r2=.94Syx94%ofthevariationinannualsalescanbeexplainedbythevariabilityinthesizeofthestoreasmeasuredbysquarefootageLinearRegressionAssumptions1. NormalityYValuesAreNormallyDistributedForEachXProbabilityDistributionofErrorisNormal2. Homoscedasticity(ConstantVariance)3. IndependenceofErrorsForLinearModelsVariationofErrorsAroundtheRegressionLineX1X2XYf(e)yvaluesarenormallydistributedaroundtheregressionline.Foreachxvalue,thespreadsorvariancearoundtheregressionlineisthesame.RegressionLineResidualAnalysis

殘差分析PurposesExamineLinearityEvaluateviolationsofassumptionsGraphicalAnalysisofResidualsPlotresidualsVs.

XivaluesDifferencebetweenactual

Yi

&predicted

YiStudentizedresiduals:Allowsconsiderationforthemagnitudeoftheresiduals

ResidualAnalysisforLinearityNotLinearLinear

XeeXResidualAnalysisforHomoscedasticityHeteroscedasticity

HomoscedasticityUsingStandardizedResidualsSRXSRXResidualAnalysis:

ComputerOutputExampleProduceStoresExcelOutputTheDurbin-WatsonStatisticUsedwhendataiscollectedovertimetodetect autocorrelation(Residualsinonetimeperiod arerelatedtoresidualsinanotherperiod)MeasuresViolationofindependenceassumptionShouldbecloseto2.Ifnot,examinethemodelforautocorrelation.ResidualAnalysisforIndependence獨(dú)立性殘差分析NotIndependentIndependent

XSRXSRInferencesabouttheSlope:tTest（回歸系數(shù)的檢驗(yàn)）tTestforaPopulationSlope IsaLinearRelationshipBetweenX&Y?TestStatistic:anddf=n-2NullandAlternativeHypotheses

H0:

1=0 (NoLinearRelationship) H1:

1

0 (LinearRelationship)WhereExample:ProduceStoresDatafor7Stores:RegressionModelObtained:Theslopeofthismodelis

1.487.

Istherealinearrelationshipbetweenthesquarefootageofastoreanditsannualsales?

AnnualStore Square Sales Feet ($000)11,726 3,68121,542 3,3953 2,816 6,6534 5,555 9,5435 1,292 3,3186 2,208 5,5637 1,313 3,760 Yi=1636.415+1.487Xi H0:

1=0 H1:

1

0

.05df

7-2=7CriticalValue(s):TestStatistic:Decision:Conclusion:Thereisevidenceofarelationship.t02.5706-2.5706.025RejectReject.025FromExcelPrintoutRejectH0InferencesabouttheSlope:tTestExampleInferencesabouttheSlope:ConfidenceIntervalExampleConfidenceIntervalEstimateoftheSlopeb1

tn-2

ExcelPrintoutforProduceStoresAt95%levelofConfidenceTheconfidenceIntervalfortheslopeis(1.062,1.911).Doesnotinclude0.Conclusion:

Thereisasignificantlinearrelationshipbetweenannualsalesandthesizeofthestore.EstimationofPredictedValuesConfidenceIntervalEstimate for

XYTheMeanofYgivenaparticularXitvaluefromtablewithdf=n-2StandarderroroftheestimateSizeofintervalvaryaccordingtodistanceawayfrommean,X.EstimationofPredictedValues區(qū)間預(yù)測ConfidenceIntervalEstimateforIndividualResponseYi

ataParticularXiAdditionofthis1increasedwidthofintervalfromthatforthemeanYIntervalEstimatesforDifferentValuesofXXYXConfidenceIntervalforaindividualYiAGivenXConfidenceIntervalforthemeanofYYi=b0+b1Xi

_Example:ProduceStoresYi=1636.415+1.487XiDatafor7Stores:RegressionModelObtained:Predicttheannualsalesforastorewith2000squarefeet.

AnnualStore Square Sales Feet ($000)11,726 3,68121,542 3,3953 2,816 6,6534 5,555 9,5435 1,292 3,3186 2,208 5,5637 1,313 3,760 EstimationofPredictedValues:ExampleConfidenceIntervalEstimate forIndividualYFindthe95%confidenceintervalfortheaverageannualsalesforstoresof2,000squarefeetPredictedSalesYi=1636.415+1.487Xi=4610.45($000)

X=2350.29SYX=611.75

tn-2=t5=2.5706=4610.45

980.97ConfidenceintervalformeanYEstimationofPredictedValues:ExampleConfidenceIntervalEstimate for

XYFindthe95%confidenceintervalforannualsalesofoneparticularstoresof2,000squarefeetPredictedSalesYi=1636.415+1.487Xi=4610.45($000)

X=2350.29SYX=611.75

tn-2=t5=2.5706=4610.45

1853.45ConfidenceintervalforindividualYCorre