教育应用统计方法第四章-回归分析

第四章回归分析多元回归方法：在众多的相关的变量中，根据问题的要求，考察其中一个或几个变量与其余变量的依赖关系。多元回归问题：如果只考察某一个变量（常称为响应变量，因变量，指标）与其余多个变量（自变量或因素）的相互依赖关系。多因变量的多元回归问题（多对多回归）3/26/20241应用统计方法第四章例如：若某公司管理人员要预测来年该公司的销售额y时，研究认为影响销售额的因素不只是广告宣传费x1,还有个人可支配收入x2,价格x3,研究与发展费用x4,各种投资x5,销售费用x6.3/26/20242应用统计方法第四章多元线性回归回归变量的选择与逐步回归。可化为多元线性回归的问题3/26/20243应用统计方法第四章第一节多元线性回归3/26/20244应用统计方法第四章3/26/20245应用统计方法第四章一、多元线性回归模型的基本假定解释变量x1,x2,…,xm是确定性变量，不是随机变量，而且解释变量之间互不相关随机误差项具有零均值和同方差

随机误差项在不同样本点之间是相互独立的，不存在序列相关

3/26/20246应用统计方法第四章随机误差项与解释变量之间不相关随机误差项服从零均值，同方差的正态分布

3/26/20247应用统计方法第四章二、建立回归方程设令即3/26/20248应用统计方法第四章3/26/20249应用统计方法第四章3/26/202410应用统计方法第四章3/26/202411应用统计方法第四章3/26/202412应用统计方法第四章3/26/202413应用统计方法第四章3/26/202414应用统计方法第四章3/26/202415应用统计方法第四章例2中，方差分析表为：y3/26/202416应用统计方法第四章3/26/202417应用统计方法第四章3/26/202418应用统计方法第四章3/26/202419应用统计方法第四章3/26/202420应用统计方法第四章3/26/202421应用统计方法第四章3/26/202422应用统计方法第四章3/26/202423应用统计方法第四章3/26/202424应用统计方法第四章3/26/202425应用统计方法第四章3/26/202426应用统计方法第四章3/26/202427应用统计方法第四章3/26/202428应用统计方法第四章3/26/202429应用统计方法第四章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应用统计方法第四章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应用统计方法第四章

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应用统计方法第四章3/26/202433应用统计方法第四章3/26/202434应用统计方法第四章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应用统计方法第四章

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拟合的很好，x1,x2对y的影响显著3/26/202436应用统计方法第四章

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应用统计方法第四章

StepwiseSelection:Step3Variablex2Entered:R-Square=0.9823andC(p)=3.0182AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel32667.79035889.26345166.83<.000Error947.972735.33030CorrectedTotal122715.763083/26/202438应用统计方法第四章

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应用统计方法第四章

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应用统计方法第四章三.回归变量的选择与逐步回归（1）enter:强迫进入法（2）stepwise:逐步选择法（3）remove:强迫消除法（4）backward:向后剔除法（5）forward:向前引入法3/26/202441应用统计方法第四章3/26/202442应用统计方法第四章3/26/202443应用统计方法第四章3/26/202444应用统计方法第四章3/26/202445应用统计方法第四章3/26/202446应用统计方法第四章3/26/202447应用统计方法第四章3/26/202448应用统计方法第四章3/26/202449应用统计方法第四章3/26/202450应用统计方法第四章3/26/202451应用统计方法第四章3/26/202452应用统计方法第四章3/26/202453应用统计方法第四章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应用统计方法第四章

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应用统计方法第四章

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应用统计方法第四章3/26/202457应用统计方法第四章3/26/202458应用统计方法第四章3/26/202459应用统计方法第四章3/26/202460应用统计方法第四章3/26/202461应用统计方法第四章3/26/202462应用统计方法第四章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应用统计方法第四章

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应用统计方法第四章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应用统计方法第四章

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应用统计方法第四章

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应用统计方法第四章回归方程的残差分析残差序列的正态性分析残差序列的随机性分析残差序列的独立性分析奇异值诊断异方差诊断

返回3/26/202468应用统计方法第四章残差序列的正态性分析：通过绘制标准化残差序列的带正态曲线的直方图或累计概率图来分析，确定残差是否接近正态返回3/26/202469应用统计方法第四章残差序列的随机性分析：可以绘制残差序列和对应的预测值序列的散点图。如果残差序列是随机的，那么残差序列应与预测值序列无关，残差序列点将随机地分布在经过零的一条直线上下。返回3/26/202470应用统计方法第四章残差序列的独立性分析：分析残差序列是否存在后期值与前期值相关的现象。D.W检验返回3/26/202471应用统计方法第四章3/26/202472应用统计方法第四章样本奇异值的诊断：样本奇异值是样本数据中那些远离均值的样本数据点。它们会对回归方程的拟合产生较大偏差影响。一般认为，如果某样本点对应的标准化残差的值超出了-3—+3的范围，就可以判定该样本数据为奇异值。返回3/26/202473应用统计方法第四章异方差诊断：线性回归模型要求残差序列服从等方差的正态分布一般通过绘制残差序列与解释变量的散点图或计算残差与解释变量间的相关系数。如果残差序列和解释变量的平方根成正比例变化，可以对解释变量作开方处理；如果残差序列与解释变量成比例变化，可以对解释变量取对数；如果残差序列与解释变量的平方成比例的变化，可以对解释变量求倒数。还可以用WLS（加权最小二乘）法消除异方差。返回3/26/202474应用统计方法第四章七、预测和控制所谓预测就是给定解释变量x样本外的某一特征值x0=(1，x01,x02,…,x0p),对因变量的值y0以及E(y0)进行估计。1、y0的点预测：2、y0的（1-α）的预测区间：3/26/202475应用统计方法第四章第二节可化为多元线性回归的问题在自然科学中，y关于x的数量关系多数都不是简单的线性关系，而是各种各样的非线性关系，于是我们常会遇到非线性回归模型，在非线性回归模型中，一种类型是可以通过变量变换化为线性模型，然后按线性模型加以解决；另一种类型的非线性模型是用任何变量变换办法都不能或不方便直接化为线性模型求得参数的估计值。3/26/202476应用统计方法第四章多项式函数Y=β0+β

1x+β

2