第十七-章相關(guān)分析與回歸分析

ChapterSeventeen

CorrelationAndRegression

第十七章相關(guān)分析與回歸分析

學(xué)習(xí)目標(biāo)討論積矩相關(guān)系數(shù)、偏相關(guān)和部分相關(guān)的概念，并說明這些相關(guān)關(guān)系如何為回歸分析建立基礎(chǔ)。解釋二元回歸的特點和方法，描述其一般模型、參數(shù)估計、標(biāo)準(zhǔn)化回歸系數(shù)、顯著性檢驗、預(yù)測準(zhǔn)確性、殘差分析和模型交叉檢驗。解釋多元回歸分析的特點和方法，尤其是逐步回歸、含虛擬變量的回歸以及回歸中的方差和協(xié)方差分析。描述多元回歸分析中用的特殊方法，尤其是逐步回歸、含虛擬變量的回歸以及回歸中的方差和協(xié)方差分析。探討非定量相關(guān)及其測量指標(biāo)學(xué)習(xí)內(nèi)容積矩相關(guān)系數(shù)偏相關(guān)非定量相關(guān)回歸分析二元回歸統(tǒng)計與二元回歸分析的關(guān)系進行二元回歸分析多元回歸

與二元回歸分析相關(guān)的統(tǒng)計量

進行多元回歸分析逐步回歸多重共線性預(yù)測的相對重要性交叉驗證回歸與虛擬變量方差分析與回歸分析總結(jié)ProductMomentCorrelation

積矩相關(guān)系數(shù)積矩相關(guān)系數(shù)r是最常用的概括兩個定量（定距或定比尺度）變量X與Y的關(guān)系強度的統(tǒng)計量它是一個決定X與Y是否存在線性關(guān)系的指標(biāo)。由于這個指標(biāo)最早由KarlPearson提出的，因此也被稱為Pearson相關(guān)系數(shù)。

它同時也叫簡單相關(guān)系數(shù)、雙變量相關(guān)系數(shù)或者相關(guān)系數(shù)。ProductMomentCorrelation

積矩相關(guān)系數(shù)對于n個觀測值的樣本，變量為X和Y，積矩相關(guān)系數(shù)r計算為r=(Xi-X)(Yi-Y)Si=1n(Xi-X)2Si=1n(Yi-Y)2Si=1nDivisionofthenumeratoranddenominatorby(n-1)givesr=(Xi-X)(Yi-Y)n-1Si=1n(Xi-X)2n-1Si=1n(Yi-Y)2n-1Si=1n=COVxySxSyProductMomentCorrelation

積矩相關(guān)系數(shù)r在-1.0和+1.0之間變化。不論兩個變量各自的測量單位是什么，相關(guān)系數(shù)都是不變的。

ExplainingAttitudeTowardtheCityofResidence

研究對居住城市的態(tài)度Table17.1調(diào)查對象編號對城市的態(tài)度居住年限天氣的重要性161032912113812443415101211646175878224911188109910111017812225ProductMomentCorrelation

積矩相關(guān)系數(shù)相關(guān)系數(shù)的計算如下： =(10+12+12+4+12+6+8+2+18+9+17+2)/12 =9.333

XY =(6+9+8+3+10+4+5+2+11+9+10+2)/12 =6.583(Xi-X)(Yi-Y)Si=1n =(10-9.33)(6-6.58)+(12-9.33)(9-6.58) +(12-9.33)(8-6.58)+(4-9.33)(3-6.58) +(12-9.33)(10-6.58)+(6-9.33)(4-6.58) +(8-9.33)(5-6.58)+(2-9.33)(2-6.58) +(18-9.33)(11-6.58)+(9-9.33)(9-6.58) +(17-9.33)(10-6.58)+(2-9.33)(2-6.58) =-0.3886+6.4614+3.7914+19.0814 +9.1314+8.5914+2.1014+33.5714 +38.3214-0.7986+26.2314+33.5714 =179.6668ProductMomentCorrelation

積矩相關(guān)系數(shù)DecompositionoftheTotalVariation

總變差分解r2

=

Explained

variationTotal

variation

=

SSxSSy

=

Total

variation

-

Error

variationTotal

variation=

SSy

-

SSerrorSSy

DecompositionoftheTotalVariation

總方差分解

DecompositionoftheTotalVariation

總變差分解r=0時的非線性關(guān)系PartialCorrelation偏相關(guān)偏相關(guān)系數(shù)是用于測量在控制或調(diào)整了一個或多個變量的基礎(chǔ)上，兩個變量之間的關(guān)系計算偏相關(guān)系數(shù)是需要考慮其“階數(shù)”，這

“階數(shù)”說明有多少個變量被控制或調(diào)整簡單相關(guān)系數(shù)r是零階的，因為在測量兩個變量之間關(guān)系時不需要控制額外變量的作用。

PartialCorrelation偏相關(guān)

PartCorrelationCoefficient

部分相關(guān)系數(shù)部分相關(guān)系數(shù)代表從X中去除其他自變量線性影響后，Y和X之間的相關(guān)性。ry(x.z)部分相關(guān)系數(shù)計算如下：通常認(rèn)為偏相關(guān)系數(shù)比部分相關(guān)系數(shù)重要。ry(x.z)

=

rxy

-

ryzrxz1

-

rxz2NonmetricCorrelation非定量相關(guān)

rs

t

rt

t

rs

rRegressionAnalysis回歸分析Regression

analysis

examinesassociativerelationshipsbetweenametricdependentvariableandoneormoreindependentvariablesinthefollowingways:回歸分析是分析定量因變量與一個或多個自變量之間相關(guān)關(guān)系的有效且易用的方法，可以用于以下幾方面；Determinewhethertheindependentvariablesexplainasignificantvariationinthedependentvariable:whetherarelationshipexists.確定自變量是否能夠解釋因變量的重要變差，即二者之間是否存在關(guān)系。Determinehowmuchofthevariationinthedependentvariablecanbeexplainedbytheindependentvariables:strengthoftherelationship.確定因變量中有多大比例的變差可以有自變量來解釋，即關(guān)系的強度有多大。RegressionAnalysis回歸分析Determinethestructureorformoftherelationship:themathematicalequationrelatingtheindependentanddependentvariables.確定二者關(guān)系的形式，即與自變量和因變量有關(guān)的數(shù)學(xué)方程式。Predictthevaluesofthedependentvariable.預(yù)測因變量的值。Controlforotherindependentvariableswhenevaluatingthecontributionsofaspecificvariableorsetofvariables.在評估特定變量貢獻時，控制其他變量的作用。Regressionanalysisisconcernedwiththenatureanddegreeofassociationbetweenvariablesanddoesnotimplyorassumeanycausality.盡管自變量可能解釋一部分因變量的變差，但這并不表示必然存在因果關(guān)系StatisticsAssociatedwithBivariate

RegressionAnalysis與二元回歸分析相關(guān)的統(tǒng)計量Bivariateregressionmodel.ThebasicregressionequationisYi=+Xi

+ei,whereY=dependentorcriterionvariable,X=independentorpredictorvariable,=interceptoftheline,=slopeoftheline,andeiistheerrortermassociatedwiththeithobservation.二元回歸模型，基本的回歸等式為Yi=+Xi

+ei,其中Yi

是因變量或標(biāo)準(zhǔn)變量，Xi為自變量或預(yù)測變量，為直線截距，為直線斜率，ei為第i個觀測值的誤差。Coefficientofdetermination.Thestrengthofassociationismeasuredbythecoefficientofdetermination,r2.Itvariesbetween0and1andsignifiestheproportionofthetotalvariationinYthatisaccountedforbythevariationinX.可決系數(shù)變量之間聯(lián)系的強度由可決系數(shù)r2

類測量，其值在0和1之間變化，表表示Y的總變差中能被X變差解釋的比例。Estimatedorpredictedvalue.TheestimatedorpredictedvalueofYiisi

=a+bx,whereiisthepredictedvalueofYi,andaandbareestimatorsof

and,respectively.

估計值或預(yù)測值：Yi的估計值或預(yù)測值為=a+bx，為Yi預(yù)測值，a

和

b

分別為和的估計值。

b0

b1

b0

b1

b0

b1StatisticsAssociatedwithBivariate

RegressionAnalysis與二元回歸分析相關(guān)的統(tǒng)計量Regressioncoefficient.Theestimatedparameterbisusuallyreferredtoasthenon-standardizedregressioncoefficient.回歸系數(shù)。估計的參數(shù)b通常是指非標(biāo)準(zhǔn)化回歸系數(shù)。Scattergram.Ascatterdiagram,orscattergram,isaplotofthevaluesoftwovariablesforallthecasesorobservations.散點圖。散點圖是根據(jù)兩個變量的所有觀測值繪制的圖。Standarderrorofestimate.Thisstatistic,SEE,isthestandarddeviationoftheactualYvaluesfromthepredictedvalues.估計標(biāo)準(zhǔn)誤。SEE表示Y的實際值與預(yù)測值之間的標(biāo)準(zhǔn)差Standarderror.Thestandarddeviationofb,SEb,iscalledthestandarderror.標(biāo)準(zhǔn)誤。B的標(biāo)準(zhǔn)差Seb被稱作標(biāo)準(zhǔn)誤。YStatisticsAssociatedwithBivariate

RegressionAnalysis與二元回歸分析相關(guān)的統(tǒng)計量Standardizedregressioncoefficient.Alsotermedthebetacoefficientorbetaweight,thisistheslopeobtainedbytheregressionofYonXwhenthedataarestandardized.標(biāo)準(zhǔn)化回歸系數(shù)。也被稱作beta系數(shù)或beta權(quán)數(shù)，是X與Y均為標(biāo)準(zhǔn)化數(shù)據(jù)時的斜率。Sumofsquarederrors.Thedistancesofallthepointsfromtheregressionlinearesquaredandaddedtogethertoarriveatthesumofsquarederrors,whichisameasureoftotalerror,誤差平方和。將所有偏離回歸擬合線的點的距離的平方和加總就得到誤差平方和，值總誤差的測量指標(biāo)，記作tstatistic.Atstatisticwithn-2degreesoffreedomcanbeusedtotestthenullhypothesisthatnolinearrelationshipexistsbetweenXandY,orH0:β=0,wheret=b/SEbT統(tǒng)計量。自由度為n-2的t統(tǒng)計量可用于檢驗X與Y不存在線性關(guān)系的零假設(shè)。

ejS2ConductingBivariateRegressionAnalysis

PlottheScatterDiagram

二元回歸分析散點圖Ascatterdiagram,orscattergram,isaplotofthevaluesoftwovariablesforallthecasesorobservations.

散點圖就是根據(jù)兩個變量的所有觀測值繪制的圖表Themostcommonlyusedtechniqueforfittingastraightlinetoascattergramistheleast-squaresprocedure.Infittingtheline,theleast-squaresprocedureminimizesthesumofsquarederrors,用一條直線對散點圖進行擬合的最常用方法為最小二乘法.為找到最佳擬合線，最小二乘法可以令誤差平方和最小。

ejS2ConductingBivariateRegressionAnalysis進行二元回歸分析法Fig.17.2圖PlottheScatterDiagram繪制散點圖FormulatetheGeneralModel建立二元回歸模型EstimatetheParameters估計參數(shù)EstimateStandardizedRegressionCoefficients估計標(biāo)準(zhǔn)化回歸系數(shù)TestforSignificance顯著性檢驗DeterminetheStrengthandSignificanceofAssociation確定相關(guān)關(guān)系的強度和顯著性CheckPredictionAccuracy檢查預(yù)測準(zhǔn)確度ExaminetheResiduals殘差檢驗

Cross-ValidatetheModel模型交叉檢驗ConductingBivariateRegressionAnalysis

FormulatetheBivariateRegressionModel分析二元回歸模型Inthebivariateregressionmodel,thegeneralformofastraightlineis:Y

=X

b0+

b1whereY=dependentorcriterionvariable因變量或標(biāo)準(zhǔn)變量X=independentorpredictorvariable自變量或預(yù)測變量

=interceptoftheline直線的截距

b0

b1=slopeoftheline直線的斜率

Theregressionprocedureaddsanerrortermtoaccountfortheprobabilisticorstochasticnatureoftherelationship:在回歸分析中需要加上誤差項，以便考察變量之間關(guān)系的隨機性Yi

=

b0+

b1

Xi+eiwhereeiistheerrortermassociatedwiththeithobservation.式中ei為第I個觀察值相關(guān)的誤差項PlotofAttitudewithDurationFig.17.34.52.256.7511.25913.593615.7518DurationofResidenceAttitudeWhichStraightLineIsBest?Fig.17.49

6

3

2.25

4.5

6.75

9

11.25

13.5

15.75

18

Line1

Line2

Line3

Line4

BivariateRegression二元回歸

Fig.17.5X2X1X3X5X4YJeJeJYJXYβ0+β1XConductingBivariateRegressionAnalysis

EstimatetheParameters二元回歸參數(shù)分析 areunknownandareestimatedfromthesampleobservationsusingtheequation在大多數(shù)情況下，和是未知的，需要根據(jù)等式從樣本觀測值中估計

whereiistheestimatedorpredictedvalueofYi,andaandbareestimatorsofInmostcases,

b0and

b1Yi=a+bxiYand,respectively.

b=COVxySx2=(Xi-X)(Yi-Y)Si=1n(Xi-X)Si=1n2=XiYi-nXYSi=1nXi2-nX2Si=1n

b0

b1b0ConductingBivariateRegressionAnalysis

EstimatetheParameters二元回歸參數(shù)分析Theintercept,a,maythenbecalculatedusing:截距a則可以計算如下 a=ForthedatainTable17.1,theestimationofparametersmaybe

illustratedasfollows:

=(10)(6)+(12)(9)+(12)(8)+(4)(3)+(12)(10)+(6)(4) +(8)(5)+(2)(2)+(18)(11)+(9)(9)+(17)(10)+(2)(2) =917

Xi2 =102+122+122+42+122+62 +82+22+182+92+172+22 =1350-bYXS12iS=112=i1XiYiConductingBivariateRegressionAnalysis

EstimatetheParameters二元回歸參數(shù)分析Itmayberecalledfromearliercalculationsofthesimplecorrelationthat:前面我們講過簡單相關(guān)系數(shù)的計算為： =9.333 =6.583

Givenn=12,bcanbecalculatedas:

=0.5897

a=XYb

=

917

-

(12)

(9.333)

(

6.583)1350

-

(12)

(9.333)2Y-bX

=6.583-(0.5897)(9.333) =1.0793ConductingBivariateRegressionAnalysis

EstimatetheStandardizedRegressionCoefficient估計標(biāo)準(zhǔn)化回歸系數(shù)Standardizationistheprocessbywhichtherawdataaretransformedintonewvariablesthathaveameanof0andavarianceof1(Chapter14).標(biāo)準(zhǔn)化就是將原始數(shù)據(jù)轉(zhuǎn)換為均值為0，方差為1的新變量的過程（見14章）Whenthedataarestandardized,theinterceptassumesavalueof0.數(shù)據(jù)進行標(biāo)準(zhǔn)化后，截距取值為0Thetermbetacoefficientorbetaweight

isusedtodenotethestandardizedregressioncoefficient.Bata系數(shù)被用來表示標(biāo)準(zhǔn)化回歸系數(shù)。 Byx=Bxy

=rxy

Thereisasimplerelationshipbetweenthestandardizedandnon-standardizedregressioncoefficients:標(biāo)準(zhǔn)化和非標(biāo)準(zhǔn)化回歸系數(shù)的關(guān)系可以簡單表示如下：

Byx=byx(Sx/Sy)ConductingBivariateRegressionAnalysis

TestforSignificance二元回歸顯著性檢驗ThestatisticalsignificanceofthelinearrelationshipbetweenXandYmaybetestedbyexaminingthehypotheses:對于X和Y之間的線性關(guān)系的統(tǒng)計顯著性可以通過以下假設(shè)進行檢驗Atstatisticwithn-2degreesoffreedomcanbeused,where通常采用雙尾檢驗，對此要采用自由度為n-2的t統(tǒng)計量SEbdenotesthestandarddeviationofbandiscalledthestandarderror.Seb表示b的標(biāo)準(zhǔn)差，被稱為標(biāo)準(zhǔn)誤。

H0:b1=0H1:b110t

=

bSEbConductingBivariateRegressionAnalysis

TestforSignificance二元回歸顯著性檢驗Usingacomputerprogram,theregressionofattitudeondurationofresidence,usingthedatashowninTable17.1,yieldedtheresultsshowninTable17.2.Theintercept,a,equals1.0793,andtheslope,b,equals0.5897.Therefore,theestimatedequationis:用計算機程序，根據(jù)表17-1的數(shù)據(jù)，可以建立對城市態(tài)度與居住年限的回歸方程，其中截距a=1.0793,斜率b=0.5897，估計的方程式為：Attitude()=1.0793+0.5897(Durationofresidence)Thestandarderror,orstandarddeviationofbisestimatedas0.07008,andthevalueofthetstatisticast=0.5897/0.0700=8.414,withn-2=10degreesoffreedom.B的標(biāo)準(zhǔn)誤或標(biāo)準(zhǔn)差為0.07008，t=0.5897/0.0700自由度為n-2=10FromTable4intheStatisticalAppendix,weseethatthecriticalvalueoftwith10degreesoffreedomand=0.05is2.228foratwo-tailedtest.Sincethecalculatedvalueoftislargerthanthecriticalvalue,thenullhypothesisisrejected.從附錄統(tǒng)計表4中，我們可以找到自由度為10，a=0.05時雙尾檢驗t的臨界值為2.228，由于t的計算值大于臨界值，零假設(shè)被拒絕。

aYConductingBivariateRegressionAnalysis

DeterminetheStrengthandSignificanceofAssociation確定相關(guān)關(guān)系的強度和顯著性Thetotalvariation,SSy,maybedecomposedintothevariationaccountedforbytheregressionline,SSreg,andtheerrororresidualvariation,SSerrororSSres,asfollows:總變差SSy可以分解為回歸變差,SSreg和殘差SSerror或Ssres，即SSy=SSreg+SSreswhere

S

S

y

=

(

Y

i

-

Y

)

2

n

S

i

=1

S

S

r

e

g

=

(

Y

i

-

Y

)

2

S

S

r

e

s

=(

Y

i

-

Y

i

)

2

n

S

i

=1n

S

i

=1DecompositionoftheTotal

VariationinBivariateRegression

二元回歸中的總變差分解Fig.17.6X2X1X3X5X4YXTotalVariationSSyResidualVariation殘余變差SSresExplainedVariation解釋變差SSregYConductingBivariateRegressionAnalysis

DeterminetheStrengthandSignificanceofAssociation確定相關(guān)關(guān)系的強度和顯著性Toillustratethecalculationsofr2,letusconsideragaintheeffectofattitudetowardthecityonthedurationofresidence.Itmayberecalledfromearliercalculationsofthesimplecorrelationcoefficientthat:為說明r2

計算，我們?nèi)稳灰跃幼∧晗迣Τ鞘袘B(tài)度的影響為例。在此之前我們曾經(jīng)計算過簡單相關(guān)系數(shù)

=120.9168SSy=(Yi-Y)2Si=1n

r

2

=

S

S

r

e

g

S

S

y

=

S

S

y

-

S

S

r

e

s

S

S

y

Thestrengthofassociationmaythenbecalculatedasfollows:變量之間聯(lián)系的強度計算如下；ConductingBivariateRegressionAnalysis

DeterminetheStrengthandSignificanceofAssociation確定相關(guān)關(guān)系的強度和顯著性Thepredictedvalues()canbecalculatedusingtheregression預(yù)測值可以通過回歸方程來計算equation:Attitude()=1.0793+0.5897(Durationofresidence)ForthefirstobservationinTable17.1,thisvalueis:()=1.0793+0.5897x10=6.9763.Foreachsuccessiveobservation,thepredictedvaluesare,inorder,8.1557,8.1557,3.4381,8.1557,4.6175,5.7969,2.2587,11.6939,6.3866,11.1042,and2.2587.對以后各項觀測值，預(yù)測值依次為8.1557,8.1557,3.4381,8.1557,4.6175,5.7969,2.2587,11.6939,6.3866,11.1042,和2.2587YYYConductingBivariateRegressionAnalysis

DeterminetheStrengthandSignificanceofAssociation確定相關(guān)關(guān)系的強度和顯著性Therefore,

=(6.9763-6.5833)2+(8.1557-6.5833)2 +(8.1557-6.5833)2+(3.4381-6.5833)2 +(8.1557-6.5833)2+(4.6175-6.5833)2 +(5.7969-6.5833)2+(2.2587-6.5833)2 +(11.6939-6.5833)2+(6.3866-6.5833)2 +(11.1042-6.5833)2+(2.2587-6.5833)2 =0.1544+2.4724+2.4724+9.8922+2.4724 +3.8643+0.6184+18.7021+26.1182 +0.0387+20.4385+18.7021

=105.9524SSreg=(Yi-Y)2Si=1nConductingBivariateRegressionAnalysis

DeterminetheStrengthandSignificanceofAssociation確定相關(guān)關(guān)系的強度和顯著性 =(6-6.9763)2+(9-8.1557)2+(8-8.1557)2

+(3-3.4381)2+(10-8.1557)2+(4-4.6175)2 +(5-5.7969)2+(2-2.2587)2+(11-11.6939)2 +(9-6.3866)2+(10-11.1042)2+(2-2.2587)2

=14.9644ItcanbeseenthatSSy=SSreg+SSres.Furthermore,

r2 =SSreg/SSy =105.9524/120.9168 =0.8762SSres=(Yi-Yi)2Si=1nConductingBivariateRegressionAnalysis

DeterminetheStrengthandSignificanceofAssociation

確定相關(guān)關(guān)系的強度和顯著性Another,equivalenttestforexaminingthesignificanceofthelinearrelationshipbetweenXandY(significanceofb)isthetestforthesignificanceofthecoefficientofdetermination.Thehypothesesinthiscaseare:

另外一個考察X與Y之間線性關(guān)系顯著性（b的顯著性）的等價檢驗，是可決系數(shù)顯著性檢驗。該檢驗的假設(shè)為：

H0:R2pop=0

H1:R2pop>0ConductingBivariateRegressionAnalysis

DeterminetheStrengthandSignificanceofAssociation

確定相關(guān)關(guān)系的強度和顯著性TheappropriateteststatisticistheFstatistic:適當(dāng)?shù)慕y(tǒng)計檢驗量為F統(tǒng)計量

whichhasanFdistributionwith1andn-2degreesoffreedom.TheFtestisageneralizedformofthettest(seeChapter15).Ifarandomvariableistdistributedwithndegreesoffreedom,thent2isFdistributedwith1andndegreesoffreedom.Hence,theFtestfortestingthesignificanceofthecoefficientofdeterminationisequivalenttotestingthefollowinghypotheses:它服從F分布，自由度為1和n-2。F檢驗是t檢驗的一般形式，如果隨機變量服從自由度為n的t分布，那么t2就服從自由度為1和n的F分布。因此檢驗可決系數(shù)顯著性的F檢驗與以下假設(shè)意義相同： orF

=

SSregSSres/(n-2)

H0:b1=0H1:b110

H0:r=0H1:r10ConductingBivariateRegressionAnalysis

DeterminetheStrengthandSignificanceofAssociation

確定相關(guān)關(guān)系的強度和顯著性FromTable17.2,itcanbeseenthat:

r2=105.9522/(105.9522+14.9644)

=0.8762

Whichisthesameasthevaluecalculatedearlier.ThevalueoftheFstatisticis:

F=105.9522/(14.9644/10)=70.8027

with1and10degreesoffreedom.ThecalculatedFstatisticexceedsthecriticalvalueof4.96determinedfromTable5intheStatisticalAppendix.Therefore,therelationshipissignificantat=0.05,corroboratingtheresultsofthettest.自由度為1和10.計算出的F統(tǒng)計量超過了根據(jù)附錄統(tǒng)計表5查到的臨界值4.96，因此，變量之間的關(guān)系在a=0.05的對平下顯著，證實了t檢驗的結(jié)果

aBivariateRegression

二元回歸Table17.2MultipleR 0.93608R2 0.87624AdjustedR2 0.86387StandardError 1.22329

ANALYSISOFVARIANCE

df SumofSquares MeanSquareRegression 1 105.95222 105.95222Residual 10 14.964441.49644F=70.80266 SignificanceofF=0.0000VARIABLESINTHEEQUATIONVariable b SEb Beta(?) TSignificance ofTDuration 0.58972 0.070080.93608 8.414 0.0000(Constant) 1.07932 0.74335 1.452 0.1772BivariateRegression

二元回歸Table17.2多元R

0.93608R2 0.87624調(diào)整的

R2 0.86387標(biāo)準(zhǔn)誤

1.22329

方差分析

自由度

平方和

均方回歸方程 1 105.95222 105.95222殘差

10 14.964441.49644F=70.80266

F

的顯著性

=0.0000等式中的變量變量

b SEb Beta(?) TSignificanceofT

居住年限 0.58972 0.070080.93608 8.414 0.0000(常數(shù)項 1.07932 0.74335 1.452 0.1772ConductingBivariateRegressionAnalysis

CheckPredictionAccuracy檢查預(yù)測準(zhǔn)確度Toestimatetheaccuracyofpredictedvalues,,itisusefultocalculatethestandarderrorofestimate,SEE.為估計預(yù)測值的準(zhǔn)確性，有必要計算估計的標(biāo)準(zhǔn)誤SEE，這個統(tǒng)計量表示Y的實際值與預(yù)測值之間的標(biāo)準(zhǔn)差。

orormoregenerally,iftherearekindependentvariables,如果有K個自變量，一般形式為

ForthedatagiveninTable17.2,theSEEisestimatedasfollows:

=1.22329Y2(12)?--=?=nSEEniiiYY2-=nSEESSres1--=knSEESSresSEE

=

14.9644/(12-2)Assumptions假設(shè)Theerrortermisnormallydistributed.ForeachfixedvalueofX,thedistributionofYisnormal.誤差項呈正態(tài)分布，對于每個X的固定值，Y為正態(tài)分布ThemeansofallthesenormaldistributionsofY,givenX,lieonastraightlinewithslopeb.給定X，所有正態(tài)分布的Y的均值位于一條斜率為b的直線上Themeanoftheerrortermis0.誤差項的均值為0Thevarianceoftheerrortermisconstant.ThisvariancedoesnotdependonthevaluesassumedbyX.誤差項的方差固定，方差不隨X值變化Theerrortermsareuncorrelated.Inotherwords,theobservationshavebeendrawnindependently.誤差項是不相關(guān)的，即觀測值是相互獨立的。MultipleRegression多元回歸Thegeneralformofthemultipleregressionmodelisasfollows:多元回歸模型的一般形式如下：whichisestimatedbythefollowingequation:該模型通過以下公式盡享估算 =a+b1X1+b2X2+b3X3+...+bkXk

Asbefore,thecoefficientarepresentstheintercept,buttheb'sarenowthepartialregressioncoefficients.如前所述，系數(shù)a代表的是截距，但b現(xiàn)在是偏回歸系數(shù)。Y

Y=b0+b1X1+b2X2+b3X3+...+bkXk+eeStatisticsAssociatedwithMultipleRegression與多元回歸有關(guān)的統(tǒng)計量AdjustedR2.R2,coefficientofmultipledetermination,isadjustedforthenumberofindependentvariablesandthesamplesizetoaccountforthediminishingreturns.Afterthefirstfewvariables,theadditionalindependentvariablesdonotmakemuchcontribution.

調(diào)整的。將多元可決系數(shù)根據(jù)自變量和樣本規(guī)模進行調(diào)整，除了前幾個自變量，其他自變量對因變量的影響不大。Coefficientofmultipledetermination.Thestrengthofassociationinmultipleregressionismeasuredbythesquareofthemultiplecorrelationcoefficient,R2,whichisalsocalledthecoefficientofmultipledetermination.多元可決系數(shù)。多元回歸中變量之間關(guān)系的強度由多元相關(guān)系數(shù)的平方R2來測量Ftest.TheFtestisusedtotestthenullhypothesisthatthecoefficientofmultipledeterminationinthepopulation,R2pop,iszero.Thisisequivalenttotestingthenullhypothesis.TheteststatistichasanFdistributionwithkand(n-k-1)degreesoffreedom.F檢驗。F檢驗用于檢驗樣本總體多元可決系數(shù)R2pop為0的假設(shè)。這與檢驗零假設(shè)是等價的。檢驗統(tǒng)計量服從F分布，自由度為k和（n-k-1）.StatisticsAssociatedwithMultipleRegression與多元回歸有關(guān)的統(tǒng)計量PartialFtest.Thesignificanceofapartialregressioncoefficient,,ofXimaybetestedusinganincrementalFstatistic.TheincrementalFstatisticisbasedontheincrementintheexplainedsumofsquaresresultingfromtheadditionoftheindependentvariableXitotheregressionequationafteralltheotherindependentvariableshavebeenincluded.偏F檢驗。對Xi的偏回歸系數(shù)進行顯著性檢驗可以應(yīng)用遞增F統(tǒng)計量。遞增F統(tǒng)計量取決于在所有其他自變量都包括在模型中的情況下，向回歸方程引入新自變量時可解釋平方和的增量。

Partialregressioncoefficient.Thepartialregressioncoefficient,b1,denotesthechangeinthepredictedvalue,,perunitchangeinX1whentheotherindependentvariables,X2toXk,areheldconstant.偏回歸系數(shù)。偏回歸系數(shù)b1表示在X2到Xk均固定不變時，改變一單位X1引起的預(yù)測值的變化。Y

biConductingMultipleRegressionAnalysis

PartialRegressionCoefficients偏回歸系數(shù)

Tounderstandthemeaningofapartialregressioncoefficient,letusconsideracaseinwhichtherearetwoindependentvariables,sothat:

為便于理解偏回歸系數(shù)的意義，我們假設(shè)有兩個自變量，所以有如下公式

=a+b1X1+b2X2First,notethattherelativemagnitudeofthepartialregressioncoefficientofanindependentvariableis,ingeneral,differentfromthatofitsbivariateregressioncoefficient.首先，注意一個自變量的偏回歸系數(shù)的相對重要性在總體上不如其二元回歸系數(shù)。Theinterpretationofthepartialregressioncoefficient,b1,isthatitrepresentstheexpectedchangeinYwhenX1ischangedbyoneunitbutX2isheldconstantorotherwisecontrolled.Likewise,b2representstheexpectedchangein

YforaunitchangeinX2,whenX1isheldconstant.Thus,callingb1andb2partialregressioncoefficientsisappropriate.偏回歸系數(shù)b1代表的意義是，X2在不變或受到控制的前提下，X1變化一個單位會使Y產(chǎn)生的預(yù)期變化。同樣b2代表的意義是，X1在不變或受到控制的前提下，X2變化一個單位會引起Y產(chǎn)生的預(yù)期變化。YConductingMultipleRegressionAnalysis

PartialRegressionCoefficients偏回歸系數(shù)ItcanalsobeseenthatthecombinedeffectsofX1andX2onYareadditive.Inotherwords,ifX1andX2areeachchangedbyoneunit,theexpectedchangeinYwouldbe(b1+b2).。X2,和

X1對Y的聯(lián)合作用是累加的。即如果都改變一個單位，Y的預(yù)期變化就是（b1+b2

）SupposeonewastoremovetheeffectofX2fromX1.ThiscouldbedonebyrunningaregressionofX1onX2.Inotherwords,onewouldestimatetheequation1=a+bX2andcalculatetheresidualXr=(X1-1).Thepartialregressioncoefficient,

b1,isthenequaltothebivariateregressioncoefficient,br,obtainedfromtheequation=a+brXr.假設(shè)我們希望從X1中X2去除的影響，可以用X2對X1進行回歸，也就是估計方程=a+bX2

，并計算殘差Xr=(X1-1)，因此，偏回歸系數(shù)br與方程Y=a+brXr.中的二元回歸系數(shù)相等。XXYConductingMultipleRegressionAnalysis

PartialRegressionCoefficients偏回歸系數(shù)Extensiontothecaseofkvariablesisstraightforward.Thepartialregressioncoefficient,b1,representstheexpectedchangeinYwhenX1ischangedbyoneunitandX2throughXkareheldconstant.Itcanalsobeinterpretedasthebivariateregressioncoefficient,b,fortheregressionofYontheresidualsofX1,whentheeffectofX2throughXkhasbeenremovedfromX1.

以上方程可以直接擴展到K個變量的情況。偏回歸系數(shù)b1道標(biāo)X2到Xk固定時，X1 變化一單位引起Y的預(yù)期變化。它也可以解釋為去除X2到Xk對X1的影響后，Y對X1殘差回歸的二元回歸系數(shù)。Therelationshipofthestandardizedtothenon-standardizedcoefficientsremainsthesameasbefore:

標(biāo)準(zhǔn)化與非標(biāo)準(zhǔn)化系數(shù)之間的關(guān)系為：

B1=b1(Sx1/Sy) Bk=bk(Sxk/Sy)Theestimatedregressionequationis:估計出的回歸方程為；

()=0.33732+0.48108X1+0.28865X2orAttitude=0.33732+0.48108(Duration)+0.28865(Importance)YMultipleRegressionTable17.3MultipleR 0.97210R2 0.94498AdjustedR2 0.93276StandardError 0.85974

ANALYSISOFVARIANCE

df SumofSquares MeanSquareRegression 2 114.26425 57.13213

Residual 9 6.65241 0.73916

F=77.29364 SignificanceofF=0.0000VARIABLESINTHEEQUATIONVariable b SEb Beta(?) TSignificance ofTIMPORTANCE 0.28865 0.086080.31382 3.353 0.0085

DURATION 0.48108 0.058950.76363 8.160 0.0000

(Constant) 0.33732 0.56736 0.595 0.5668

多元回歸Table17.3多元

R

0.97210R2 0.94498調(diào)整的

R2 0.93276標(biāo)準(zhǔn)誤 0.85974

ANALYSISOFVARIANCE

df SumofSquares MeanSquare回歸方程 2 114.26425 57.13213

殘差 9 6.65241 0.73916

F=77.29364 SignificanceofF=0.0000VARIABLESINTHEEQUATION變量

b SEb Beta(?) TSignificance ofT天氣重要性 0.28865 0.086080.31382 3.353 0.0085

居住年限 0.48108 0.058950.76363 8.160 0.0000

(常數(shù)項

0.33732 0.56736 0.595 0.5668

ConductingMultipleRegressionAnalysis

StrengthofAssociation聯(lián)系的強度SSy=SSreg+SSreswhereSSreg=(Yi-Y)2Si=1nSSy=(Yi-Y)2Si=1nSSres=(Yi-Yi)2Si=1nConductingMultipleRegressionAnalysis

StrengthofAssociation聯(lián)系的強度Thestrengthofassociationismeasuredbythesquareofthemultiplecorrelationcoefficient,R2,whichisalsocalledthecoefficientofmultipledetermination. 變量之間聯(lián)系的強度可以用多元相關(guān)系數(shù)的平方R2,來測量，也稱多元可決系數(shù)。R2

=

SSregSSyR2isadjustedforthenumberofindependentvariablesandthesamplesizebyusingthefollowingformula:

R2可以根據(jù)自變量的數(shù)量和樣本規(guī)模按照如下公式調(diào)整AdjustedR2

=R2

-

k(1

-

R2)n