教育應用統(tǒng)計方法第四章-回歸分析

第四章回歸分析多元回歸方法：在眾多的相關的變量中，根據(jù)問題的要求，考察其中一個或幾個變量與其余變量的依賴關系。多元回歸問題：如果只考察某一個變量（常稱為響應變量，因變量，指標）與其余多個變量（自變量或因素）的相互依賴關系。多因變量的多元回歸問題（多對多回歸）3/26/20241應用統(tǒng)計方法第四章例如：若某公司管理人員要預測來年該公司的銷售額y時，研究認為影響銷售額的因素不只是廣告宣傳費x1,還有個人可支配收入x2,價格x3,研究與發(fā)展費用x4,各種投資x5,銷售費用x6.3/26/20242應用統(tǒng)計方法第四章多元線性回歸回歸變量的選擇與逐步回歸?？苫癁槎嘣€性回歸的問題3/26/20243應用統(tǒng)計方法第四章第一節(jié)多元線性回歸3/26/20244應用統(tǒng)計方法第四章3/26/20245應用統(tǒng)計方法第四章一、多元線性回歸模型的基本假定解釋變量x1,x2,…,xm是確定性變量，不是隨機變量，而且解釋變量之間互不相關隨機誤差項具有零均值和同方差

隨機誤差項在不同樣本點之間是相互獨立的，不存在序列相關

3/26/20246應用統(tǒng)計方法第四章隨機誤差項與解釋變量之間不相關隨機誤差項服從零均值，同方差的正態(tài)分布

3/26/20247應用統(tǒng)計方法第四章二、建立回歸方程設令即3/26/20248應用統(tǒng)計方法第四章3/26/20249應用統(tǒng)計方法第四章3/26/202410應用統(tǒng)計方法第四章3/26/202411應用統(tǒng)計方法第四章3/26/202412應用統(tǒng)計方法第四章3/26/202413應用統(tǒng)計方法第四章3/26/202414應用統(tǒng)計方法第四章3/26/202415應用統(tǒng)計方法第四章例2中，方差分析表為：y3/26/202416應用統(tǒng)計方法第四章3/26/202417應用統(tǒng)計方法第四章3/26/202418應用統(tǒng)計方法第四章3/26/202419應用統(tǒng)計方法第四章3/26/202420應用統(tǒng)計方法第四章3/26/202421應用統(tǒng)計方法第四章3/26/202422應用統(tǒng)計方法第四章3/26/202423應用統(tǒng)計方法第四章3/26/202424應用統(tǒng)計方法第四章3/26/202425應用統(tǒng)計方法第四章3/26/202426應用統(tǒng)計方法第四章3/26/202427應用統(tǒng)計方法第四章3/26/202428應用統(tǒng)計方法第四章3/26/202429應用統(tǒng)計方法第四章datad411;inputx1-x4y;cards;72666078.5129155274.31156820104.3113184787.675263395.91155922109.2371176102.7131224472.5254182293.12147426115.9140233483.81166912113.31068812109.4；procregdata=d411;modely=x1-x4;run;quit;3/26/202430應用統(tǒng)計方法第四章datad411;inputx1-x4y;cards;72666078.5129155274.31156820104.3113184787.675263395.91155922109.2371176102.7131224472.5254182293.12147426115.9140233483.81166912113.31068812109.4;procregdata=d411;modely=x1-x4/selection=stepwise

sle=0.10sls=0.10;run;quit;3/26/202431應用統(tǒng)計方法第四章

TheSASSystem13:43Wednesday,March10,20087TheREGProcedureModel:MODEL1DependentVariable:yAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel42667.89944666.97486111.48<.0001Error847.863645.98295CorrectedTotal122715.76308RootMSE2.44601R-Square0.9824DependentMean95.42308AdjR-Sq0.9736

Coeff

Var2.56333ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept162.4053770.070960.890.3991x111.551100.744772.080.0708x210.510170.723790.700.5009x310.101910.754710.140.8959x41-0.144060.70905-0.200.84413/26/202432應用統(tǒng)計方法第四章3/26/202433應用統(tǒng)計方法第四章3/26/202434應用統(tǒng)計方法第四章datad411;inputx1-x4y;cards;72666078.5129155274.31156820104.3113184787.675263395.91155922109.2371176102.7131224472.5254182293.12147426115.9140233483.81166912113.31068812109.4;procregdata=d411;modely=x1x2;run;quit;3/26/202435應用統(tǒng)計方法第四章

TheSASSystem13:43Wednesday,March10,200811TheREGProcedureModel:MODEL1DependentVariable:yAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel22657.858591328.92930229.50<.0001Error1057.904485.79045CorrectedTotal122715.76308RootMSE2.40634R-Square0.9787DependentMean95.42308AdjR-Sq0.9744

Coeff

Var2.52175ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept152.577352.2861723.00<.0001x111.468310.1213012.10<.0001x210.662250.0458514.44<.0001擬合的很好，x1,x2對y的影響顯著3/26/202436應用統(tǒng)計方法第四章

AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel11831.896161831.8961622.800.0006Error11883.8669280.35154CorrectedTotal122715.76308ParameterStandardVariableEstimateErrorTypeIISSFValuePr>FIntercept117.567935.2622140108499.16<.0001x4-0.738160.154601831.8961622.800.0006Boundsonconditionnumber:1,1------------------------------------------------------------------------------------------------------StepwiseSelection:Step2Variablex1Entered:R-Square=0.9725andC(p)=5.4959AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel22641.000961320.50048176.63<.0001Error1074.762117.47621CorrectedTotal122715.763083/26/202437應用統(tǒng)計方法第四章

StepwiseSelection:Step3Variablex2Entered:R-Square=0.9823andC(p)=3.0182AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel32667.79035889.26345166.83<.000Error947.972735.33030CorrectedTotal122715.763083/26/202438應用統(tǒng)計方法第四章

StepwiseSelection:Step4Variablex4Removed:R-Square=0.9787andC(p)=2.6782AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel22657.858591328.92930229.50<.0001Error1057.904485.79045CorrectedTotal122715.76308ParameterStandardVariableEstimateErrorTypeIISSFValuePr>FIntercept52.577352.286173062.60416528.91<.0001x11.468310.12130848.43186146.52<.0001x20.662250.045851207.78227208.58<.0001Boundsonconditionnumber:1.0551,4.22053/26/202439應用統(tǒng)計方法第四章

Allvariablesleftinthemodelaresignificantatthe0.1000level.Noothervariablemetthe0.1000significancelevelforentryintothemodel.SummaryofStepwiseSelectionVariableVariableNumberPartialModelStepEnteredRemovedVarsInR-SquareR-Square

C(p)FValuePr>F1x410.67450.6745138.73122.800.00062x120.29790.97255.4959108.22<.00013x230.00990.98233.01825.030.05174x420.00370.97872.67821.860.20543/26/202440應用統(tǒng)計方法第四章三.回歸變量的選擇與逐步回歸（1）enter:強迫進入法（2）stepwise:逐步選擇法（3）remove:強迫消除法（4）backward:向后剔除法（5）forward:向前引入法3/26/202441應用統(tǒng)計方法第四章3/26/202442應用統(tǒng)計方法第四章3/26/202443應用統(tǒng)計方法第四章3/26/202444應用統(tǒng)計方法第四章3/26/202445應用統(tǒng)計方法第四章3/26/202446應用統(tǒng)計方法第四章3/26/202447應用統(tǒng)計方法第四章3/26/202448應用統(tǒng)計方法第四章3/26/202449應用統(tǒng)計方法第四章3/26/202450應用統(tǒng)計方法第四章3/26/202451應用統(tǒng)計方法第四章3/26/202452應用統(tǒng)計方法第四章3/26/202453應用統(tǒng)計方法第四章datad411;inputx1-x4y;cards;72666078.5129155274.31156820104.3113184787.675263395.91155922109.2371176102.7131224472.5254182293.12147426115.9140233483.81166912113.31068812109.4;procregdata=d411;modely=x1-x4/selection=rsquare

badjrsqcpaic

mse

sbc;run;quit;3/26/202454應用統(tǒng)計方法第四章

TheREGProcedureModel:MODEL1DependentVariable:yR-SquareSelectionMethodNumberinAdjustedModelR-SquareR-Square

C(p)AICMSESBC10.67450.6450138.730858.851680.3515459.9815410.66630.6359142.486459.178082.3942160.3078910.53390.4916202.548863.5195115.0624364.6493710.28590.2210315.154369.0674176.3091370.19730-----------------------------------------------------------------------------------------20.97870.97442.678225.42005.7904527.1148420.97250.96705.495928.74177.4762130.4365520.93530.922322.373139.852617.5738041.5474320.84700.816462.437751.037141.5442752.7319920.68010.6161138.225960.629386.8880162.3241720.54820.4578198.094765.1167122.7072166.81153-----------------------------------------------------------------------------------------30.98230.97643.018224.97395.3303027.2336830.98230.97643.041325.01125.3456227.2709930.98130.97503.496825.72765.6484627.9873530.97280.96387.337530.57598.2016232.83568-----------------------------------------------------------------------------------------40.98240.97365.000026.94435.9829529.769033/26/202455應用統(tǒng)計方法第四章

Numberin--------------------------ParameterEstimates--------------------------ModelR-SquareInterceptx1x2x3x410.6745117.56793...-0.7381610.666357.42368.0.78912..10.533981.479341.86875...10.2859110.20266..-1.25578.------------------------------------------------------------------------------------------------20.978752.577351.468310.66225..20.9725103.097381.43996..-0.6139520.9353131.28241..-1.19985-0.7246020.847072.07467.0.73133-1.00839.20.680194.16007.0.31090.-0.4569420.548272.348992.31247.0.49447.------------------------------------------------------------------------------------------------30.982371.648311.451940.41611.-0.2365430.982348.193631.695890.656910.25002.30.9813111.684411.05185.-0.41004-0.6428030.9728203.64196.-0.92342-1.44797-1.55704------------------------------------------------------------------------------------------------40.982462.405371.551100.510170.10191-0.144063/26/202456應用統(tǒng)計方法第四章3/26/202457應用統(tǒng)計方法第四章3/26/202458應用統(tǒng)計方法第四章3/26/202459應用統(tǒng)計方法第四章3/26/202460應用統(tǒng)計方法第四章3/26/202461應用統(tǒng)計方法第四章3/26/202462應用統(tǒng)計方法第四章datad431;inputyearx1-x5y1y2;cards;19490.90.80.146.630.241.477.3119501.02.10.157.070.461.257.4219512.96.30.337.601.022.0511.1319525.04.40.7812.881.612.4916.0819538.213.31.1815.861.633.1622.86195413.116.81.5618.791.933.8729.52195523.817.82.1114.632.314.5034.54195634.827.83.0919.793.326.0941.22195735.422.13.5816.504.446.7847.54195847.032.27.3126.227.1810.7360.00195962.633.29.6128.008.7717.6578.00196068.055.612.8527.569.8926.8496.20196135.324.46.7610.955.5824.2052.37196231.317.95.0810.156.0320.0837.77196335.224.85.5414.237.1819.2840.07196445.337.87.1420.388.8022.8950.36196549.578.811.2026.5610.4528.9465.33196659.7101.615.8933.1812.5139.0583.64196747.874.910.8623.9011.4239.0968.16196817.740.25.1017.569.0326.8141.64196936.073.313.1427.208.0537.1967.30197062.0138.625.5436.2810.3054.09103.57197197.0247.031.3141.5314.1877.39135.80197295.2270.028.7940.2415.1984.02118.101973118.4233.528.0338.2015.7788.39119.62197499.9205.026.5031.5412.2986.32112.391975151.0288.038.6146.8717.36107.94144.411976108.0262.231.4638.6215.10102.76130.661977162.5358.646.2152.4820.48118.84175.101978238.2454.855.8655.9626.40139.30214.44;procprint;run;procregdata=d431;modely1y2=x1-x5;

mtestx3,x4,x5;run;quit;3/26/202463應用統(tǒng)計方法第四章

TheSASSystem07:49Sunday,March21,20084TheREGProcedureModel:MODEL1MultivariateTest1MultivariateStatisticsandFApproximationsS=2M=0N=10.5StatisticValueFValueNumDFDenDFPr>F

Wilks'Lambda0.1739086010.72646<.0001

Pillai'sTrace1.089531229.57648<.0001

Hotelling-LawleyTrace3.2353293712.16628.955<.0001Roy'sGreatestRoot2.6674367221.34324<.0001NOTE:FStatisticforRoy'sGreatestRootisanupperbound.NOTE:FStatisticforWilks'Lambdaisexact.3/26/202464應用統(tǒng)計方法第四章datad431;inputyearx1-x5y1y2;cards;19490.90.80.146.630.241.477.3119501.02.10.157.070.461.257.4219512.96.30.337.601.022.0511.1319525.04.40.7812.881.612.4916.0819538.213.31.1815.861.633.1622.86195413.116.81.5618.791.933.8729.52195523.817.82.1114.632.314.5034.54195634.827.83.0919.793.326.0941.22195735.422.13.5816.504.446.7847.54195847.032.27.3126.227.1810.7360.00195962.633.29.6128.008.7717.6578.00196068.055.612.8527.569.8926.8496.20196135.324.46.7610.955.5824.2052.37196231.317.95.0810.156.0320.0837.77196335.224.85.5414.237.1819.2840.07196445.337.87.1420.388.8022.8950.36196549.578.811.2026.5610.4528.9465.33196659.7101.615.8933.1812.5139.0583.64196747.874.910.8623.9011.4239.0968.16196817.740.25.1017.569.0326.8141.64196936.073.313.1427.208.0537.1967.30197062.0138.625.5436.2810.3054.09103.57197197.0247.031.3141.5314.1877.39135.80197295.2270.028.7940.2415.1984.02118.101973118.4233.528.0338.2015.7788.39119.62197499.9205.026.5031.5412.2986.32112.391975151.0288.038.6146.8717.36107.94144.411976108.0262.231.4638.6215.10102.76130.661977162.5358.646.2152.4820.48118.84175.101978238.2454.855.8655.9626.40139.30214.44;procprint;run;procregdata=d431;modely1y2=x1-x5/selection=stepwise

sle=0.05sls=0.05;run;procregdata=d431;modely1y2=x3-x5;run;quit;3/26/202465應用統(tǒng)計方法第四章

TheREGProcedureModel:MODEL1DependentVariable:y1AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel34648415495291.53<.0001Error261381.8921853.14970CorrectedTotal2947865RootMSE7.29038R-Square0.9711DependentMean40.11533AdjR-Sq0.9678

Coeff

Var18.17356ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept18.499454.650241.830.0791x312.841280.342488.30<.0001x41-0.849540.34357-2.470.0203x511.347640.703051.920.06633/26/202466應用統(tǒng)計方法第四章

TheREGProcedureModel:MODEL1DependentVariable:y2AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel37635225451425.62<.0001Error261554.7201059.79693CorrectedTotal2977907RootMSE7.73285R-Square0.9800DependentMean73.75167AdjR-Sq0.9777

Coeff

Var10.48498ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept15.293114.932461.070.2931x311.725330.363264.75<.0001x411.005290.364422.760.0105x511.973050.745722.650.01363/26/202467應用統(tǒng)計方法第四章回歸方程的殘差分析殘差序列的正態(tài)性分析殘差序列的隨機性分析殘差序列的獨立性分析奇異值診斷異方差診斷

返回3/26/202468應用統(tǒng)計方法第四章殘差序列的正態(tài)性分析：通過繪制標準化殘差序列的帶正態(tài)曲線的直方圖或累計概率圖來分析，確定殘差是否接近正態(tài)返回3/26/202469應用統(tǒng)計方法第四章殘差序列的隨機性分析：可以繪制殘差序列和對應的預測值序列的散點圖。如果殘差序列是隨機的，那么殘差序列應與預測值序列無關，殘差序列點將隨機地分布在經(jīng)過零的一條直線上下。返回3/26/202470應用統(tǒng)計方法第四章殘差序列的獨立性分析：分析殘差序列是否存在后期值與前期值相關的現(xiàn)象。D.W檢驗返回3/26/202471應用統(tǒng)計方法第四章3/26/202472應用統(tǒng)計方法第四章樣本奇異值的診斷：樣本奇異值是樣本數(shù)據(jù)中那些遠離均值的樣本數(shù)據(jù)點。它們會對回歸方程的擬合產(chǎn)生較大偏差影響。一般認為，如果某樣本點對應的標準化殘差的值超出了-3—+3的范圍，就可以判定該樣本數(shù)據(jù)為奇異值。返回3/26/202473應用統(tǒng)計方法第四章異方差診斷：線性回歸模型要求殘差序列服從等方差的正態(tài)分布一般通過繪制殘差序列與解釋變量的散點圖或計算殘差與解釋變量間的相關系數(shù)。如果殘差序列和解釋變量的平方根成正比例變化，可以對解釋變量作開方處理；如果殘差序列與解釋變量成比例變化，可以對解釋變量取對數(shù)；如果殘差序列與解釋變量的平方成比例的變化，可以對解釋變量求倒數(shù)。還可以用WLS（加權最小二乘）法消除異方差。返回3/26/202474應用統(tǒng)計方法第四章七、預測和控制所謂預測就是給定解釋變量x樣本外的某一特征值x0=(1，x01,x02,…,x0p),對因變量的值y0以及E(y0)進行估計。1、y0的點預測：2、y0的（1-α）的預測區(qū)間：3/26/202475應用統(tǒng)計方法第四章第二節(jié)可化為多元線性回歸的問題在自然科學中，y關于x的數(shù)量關系多數(shù)都不是簡單的線性關系，而是各種各樣的非線性關系，于是我們常會遇到非線性回歸模型，在非線性回歸模型中，一種類型是可以通過變量變換化為線性模型，然后按線性模型加以解決；另一種類型的非線性模型是用任何變量變換辦法都不能或不方便直接化為線性模型求得參數(shù)的估計值。3/26/202476應用統(tǒng)計方法第四章多項式函數(shù)Y=β0+β

1x+β

2