《多重线性回归》PPT课件

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1、第1讲 多重线性回归 主要内容1 多重线性回归模型简介2 回归系数的估计3 方程的假设检验4 决定系数与剩余标准差5 偏回归系数的假设检验6 指标的量化7 回归与t检验、方差分析的关系8 标准偏回归系数与自变量的贡献表表3.1 某地某地13岁男童身高，体重，肺活量的实测数据岁男童身高，体重，肺活量的实测数据(部分部分)编号身高(cm)x1体重(kg)x2肺活量(L)y1135.132.01.753163.646.22.755156.237.12.757167.841.52.759145.033.02.5011165.549.53.0013153.341.02.7515160.547.22.25

2、17147.640.52.0019155.144.72.7521143.031.51.7523160.840.42.7525158.237.52.0027144.534.72.2529156.532.01.75问题：o 身高、体重与肺活量有无线性关系？身高、体重与肺活量有无线性关系？o 用身高和体重预测肺活量有多高的精度？用身高和体重预测肺活量有多高的精度？o 单独用身高、或体重是否也能达到同样效果？单独用身高、或体重是否也能达到同样效果？o 身高的贡献大，还是体重的贡献大？身高的贡献大，还是体重的贡献大？1.多重线性回归模型简介o 多重回归n multiple regressionn mul

3、tiple linear regressiono 因变量n dependent variablen response variable(响应变量)o 自变量n independent variablen explanatory variable(解释变量)回归模型o 因变量y,自变量为x1,x2,xmo a为截距(intercept)，又称常数项(constant),表示各自变量均为0时y的估计值o bi 称为偏回归系数(partial regression coefficient)，简称为回归系数o 称为 y 的估计值或预测值(predicted value)mmxbxbxbay 2211y

4、 例 题：o 根据某地29名13岁男童的身高x1(cm)，体重x2(kg)和肺活量y(L)建立的回归方程为：2105406.0005017.05657.0 xxy yo 当x1=150，x2=32时，=1.9168，表示对所有身高为150cm，体重为32kg的13岁男童，估计平均肺活量为1.9168(L)。2.回归系数的估计o 最小二乘法(least square,LS)o 基本思想n 残差平方和(sum of squares for residuals)最小 nimminiiixbxbxbbyyyQ122211012 表3.2 例3.1资料的估计值与残差 编号编号ye编号编号ye11.751

5、.8420-0.092022.001.77960.220432.752.7527-0.002742.501.98030.519752.752.22360.526462.002.1381-0.138172.752.51960.230481.501.8612-0.361292.501.94580.5542102.252.19040.0596113.002.94060.0594121.251.6037-0.3537132.752.41990.3301141.751.9268-0.1768152.252.7912-0.5412161.751.9318-0.1818172.002.3643-0.3643

6、182.252.5653-0.3153192.752.62890.1211202.002.2668-0.2668211.751.8546-0.1046222.252.01650.2335232.752.42510.3249242.502.31330.1867252.002.2552-0.2552261.752.1330-0.3830272.252.03510.2149282.502.34530.1547291.751.9494-0.1994y y 估计值与残差有下列性质：0)(11 niiniiieyy niiniiieyy1212)(3.Y的总变异分解o未引进回归时的总变异：(sum of

7、squares about the mean of Y)o引进回归以后的变异(剩余):(sum of squares about regression)o回归的贡献，回归平方和：(sum of squares due to regression)2)(YY 2)(YY 2)(YY回归方程的方差分析表 QUmmn1变异来源变异来源SS自由度自由度MSF总总lyyn-1回归回归UmU/m剩余剩余Qn-m-1Q/(n-m-1)例3.1资料回归方程的方差分析 变异来源SS自由度自由度MSFP总5.6336206928回归3.0757339421.5378669715.6319 F =0.0000Pro

8、b F =0.0000 Residual|2.55788675 26 .098380259 Residual|2.55788675 26 .098380259 R-squared =0.5460R-squared =0.5460-+-+-Adj R-squared=0.5110Adj R-squared=0.5110 Total|5.63362069 28 .201200739 Root MSE =.31366 Total|5.63362069 28 .201200739 Root MSE =.31366-y|Coef.Std.Err.t P|t|95%Conf.Interval y|Coef

9、.Std.Err.t P|t|95%Conf.Interval-+-+-x1|.0050165 .0105754 0.47 0.639 -.0167216 .0267547 x1|.0050165 .0105754 0.47 0.639 -.0167216 .0267547 x2|.0540611 .0159838 3.38 0.002 .021206 .0869162 x2|.0540611 .0159838 3.38 0.002 .021206 .0869162 _cons|-.5656643 1.240127 -0.46 0.652 -3.114782 1.983454 _cons|-.

10、5656643 1.240127 -0.46 0.652 -3.114782 1.983454-6 标准偏回归系数与自变量的贡献,iiiiiiiyyylsbbbbls在多重回归方程中，偏回归系数间是不能直接比较大小的，因为，偏回归系数是有单位的。为判断自变量对因变量的贡献大小，必须将它们标准化，变成没有单位的标准偏回归系数。lii是变量是变量xi的离均差平方和。的离均差平方和。STATA的输出结果.reg y x1 x2,beta Source|SS df MS Number of obs=29Source|SS df MS Number of obs=29-+-F(2,26)=15.63-+

11、-F(2,26)=15.63 Model|3.07573394 2 1.53786697 Prob F =0.0000 Model|3.07573394 2 1.53786697 Prob F =0.0000 Residual|2.55788675 26 .098380259 R-squared =0.5460 Residual|2.55788675 26 .098380259 R-squared =0.5460-+-Adj R-squared=0.5110-+-Adj R-squared=0.5110 Total|5.63362069 28 .201200739 Root MSE =.313

12、66 Total|5.63362069 28 .201200739 Root MSE =.31366-y|Coef.Std.Err.t P|t|Beta y|Coef.Std.Err.t P|t|Beta-+-+-x1|.0050165 .0105754 0.47 0.639 x1|.0050165 .0105754 0.47 0.639 .0935215.0935215 x2|.0540611 .0159838 3.38 0.002 x2|.0540611 .0159838 3.38 0.002 .6668242.6668242 _cons|-.5656643 1.240127 -0.46

13、0.652 ._cons|-.5656643 1.240127 -0.46 0.652 .-一元回归分析的结果.reg y x1-y|Coef.Std.Err.t P|t|95%Conf.Interval-+-x1|.0315609 .0083471 3.78 0.001 .0144341 .0486878 _cons|-2.608541 1.275414 -2.05 0.051 -5.225474 .008393-.reg y x2-y|Coef.Std.Err.t P|t|95%Conf.Interval-+-x2|.0596878 .0105587 5.65 0.000 .0380232

14、 .0813524 _cons|-.0091673 .3961987 -0.02 0.982 -.8221 .8037653-为什么单变量分析时都有统计学意义，而同时放入方程则一个有统计学意义，另一个无统计学意义？自变量的作用X1 YX2事实上，由于自变量间的相关，自变量对因变量的作用不是独立的。一个自变量对因变量的影响除该变量的直接作用外，还要该变量通过其他自变量对y的间接作用。自变量作用的分解 自变量中间变量直接贡献间接贡献与y的相关riy身高x1x2b1=0.09352b2 r12=0.66682 0.7421=0.49480.5884体重x2x1b2=0.66682b1 r12=0.0

15、9352 0.7421=0.06940.7362标准偏回归系数反映的是自变量对y的直接作用；间接作用是该自变量与其它自变量共同作用的结果。3.8 指标的量化 o 性别 如果是男性如果是女性 10 xxbbY10 例 t 检验与回归的关系正常人组II期矽肺组64.26 74.9742.84 88.0652.48 93.4748.19 95.1080.22100.6769.61101.1418.19113.5250.90正常人与矽肺患者血清粘蛋白含量(mg/100mg)资料重新整理 y group 1.64.26 0 2.42.84 0 3.52.48 0 4.48.19 0 5.80.22 0

16、6.69.61 0 7.18.19 0 8.50.9 0 9.74.97 1 10.88.06 1 11.93.47 1 12.95.1 1 13.100.67 1 14.101.14 1 15.113.52 1 t 检验结果.ttest y,by(group)Two-sample t test with equal variances-Group|Obs Mean Std.Err.Std.Dev.95%Conf.Interval-+-0|8 53.33625 6.662102 18.84327 37.58288 69.08962 1|7 95.27571 4.535631 12.00015

17、84.17742 106.374-+-combined|15 72.908 6.871658 26.61382 58.16976 87.64624-+-diff|-41.93946 8.307497 -59.88672 -23.99221-Degrees of freedom:13 Ho:mean(0)-mean(1)=diff=0 Ha:diff 0 t=-5.0484 t=-5.0484 t=-5.0484 P|t|=0.0002 P t=0.9999与方差分析结果等价.anova y group Number of obs=15 R-squared =0.6622 Root MSE =1

18、6.0516 Adj R-squared=0.6362 Source|Partial SS df MS F Prob F-+-Model|6566.62918 1 6566.62918 25.49 0.0002|group|6566.62918 1 6566.62918 25.49 0.0002|Residual|3349.50389 13 257.654145 -+-Total|9916.13307 14 708.29522 与回归分析结果的比较.reg y group Source|SS df MS Number of obs=15-+-F(1,13)=25.49 Model|6566.6

19、2918 1 6566.62918 Prob F =0.0002 Residual|3349.50389 13 257.654145 R-squared =0.6622-+-Adj R-squared=0.6362 Total|9916.13307 14 708.29522 Root MSE =16.052-y|Coef.Std.Err.t P|t|95%Conf.Interval-+-group|41.93946 8.307497 5.05 0.000 23.99221 59.88672 _cons|53.33625 5.675101 9.40 0.000 41.07594 65.59656

20、-53.3362541.93946ygroup 回归系数与各组均数的关系53.3362541.939460 53.336251:53.3362541.9394695.27571ygroupgroupygroupy：指标的量化 o 血型(A，B，AB，O)x1=0,x2=0,x3=0 表示O型 x1=1,x2=0,x3=0 表示A型 x1=0,x2=1,x3=0 表示B型 x1=0,x2=0,x3=1 表示AB型哑变量(dummy)又称指示变量(indicator variables)3322110 xbxbxbbY 方差分析与回归分析正常人组I期矽肺组II期矽肺组64.2665.46 74.9

21、742.8460.63 88.0652.4869.73 93.4748.1974.97 95.1080.2280.44100.6769.6197.58101.1418.1995.20113.5250.9096.39血清粘蛋白含量(mg/100mg)各组均数.tab group,sum(y)|Summary of y group|Mean Std.Dev.Freq.-+-0|53.336251 18.84327 8 1|80.050001 14.766198 8 2|95.275713 12.000153 7-+-Total|75.392174 23.069605 23指标的量化 o 组别(0，

22、1，2)x1=0,x2=0 表示0组(正常人)x1=1,x2=0 表示1组(矽肺I期)x1=0,x2=1 表示2组(矽肺II期)哑变量(dummy)又称指示变量(indicator variables)01122Ybb xb x 资料整理正常人组I期矽肺组II期矽肺组64.26065.461 74.97242.84060.631 88.06252.48069.731 93.47248.19074.971 95.10280.22080.441100.67269.61097.581101.14218.19095.201113.52250.90096.391血清粘蛋白含量(mg/100mg)方差分析

23、的结果.anova y g Number of obs=23 R-squared =0.5836 Root MSE =15.6138 Adj R-squared=0.5419 Source|Partial SS df MS F Prob F -+-Model|6832.7588 2 3416.3794 14.01 0.0002|group|6832.7588 2 3416.3794 14.01 0.0002|Residual|4875.78815 20 243.789407 -+-Total|11708.5469 22 532.206679 回归分析的结果.reg y g2 g3 Source

24、|SS df MS Number of obs=23-+-F(2,20)=14.01 Model|6832.7588 2 3416.3794 Prob F =0.0002 Residual|4875.78815 20 243.789407 R-squared =0.5836-+-Adj R-squared=0.5419 Total|11708.5469 22 532.206679 Root MSE =15.614-y|Coef.Std.Err.t P|t|95%Conf.Interval-+-g2|26.71375 7.806878 3.42 0.003 10.42889 42.99861 g

25、3|41.93946 8.080887 5.19 0.000 25.08303 58.7959 _cons|53.33625 5.520297 9.66 0.000 41.82111 64.85139-系数与均数23232353.33625 53.3362526.71375 53.3362541 939460,0,II1,0,0,1,ggYggYggY (正常人)(正常人)(矽肺I期)(矽肺I期)。(矽肺 期)。(矽肺 期)2353.3362526.7137541.93946Ygg 男婴男婴女婴女婴身高身高体重体重体表面积体表面积身高身高体重体重体表面积体表面积543.002446543.00

26、2117502.251928532.252200512.502094512.501906563.502506513.001850523.002121513.001632769.503845777.503934809.0043807710.004180749.504314779.504246809.004078749.003358768.004134737.5038099613.5058309112.0053589714.0060139113.0056109916.0064109415.0060749211.0052839212.0052909415.0061019112.505291协方差分析

27、与回归分析heightweightygenderheightweightygender543.0024461543.0021170502.2519281532.2522000512.5020941512.5019060563.5025061513.0018500523.0021211513.0016320769.5038451777.5039340809.00438017710.0041800749.5043141779.5042460809.0040781749.0033580768.0041341737.50380909613.50583019112.00535809714.0060131

28、9113.00561009916.00641019415.00607409211.00528319212.00529009415.00610119112.5052910资料整理协方差分析.anova y height weight gender,cate(gender)Number of obs=30 R-squared =0.9845 Root MSE =203.667 Adj R-squared=0.9827 Source|Partial SS df MS F Prob F -+-Model|68508456.5 3 22836152.2 550.53 0.0000|height|9259

29、56.904 1 925956.904 22.32 0.0001 weight|374288.752 1 374288.752 9.02 0.0058 gender|144515.841 1 144515.841 3.48 0.0733|Residual|1078488.66 26 41480.3332 -+-Total|69586945.2 29 2399549.83 .reg y w h g Source|SS df MS Number of obs=30-+-F(3,26)=550.53 Model|68508456.5 3 22836152.2 Prob F =0.0000 Resid

30、ual|1078488.66 26 41480.3332 R-squared =0.9845-+-Adj R-squared=0.9827 Total|69586945.2 29 2399549.83 Root MSE =203.67-y|Coef.Std.Err.t P|t|95%Conf.Interval-+-weight|131.7348 43.85493 3.00 0.006 41.58975 221.8799 height|53.97971 11.425 4.72 0.000 30.49528 77.46414 gender|139.0977 74.52177 1.87 0.073

31、-14.08405 292.2793 _cons|-1226.631 493.6082 -2.49 0.020 -2241.257 -212.0045-与回归分析比较o线性回归分析与 t 检验等价o线性回归分析与方差分析等价o线性回归分析与协方差分析等价o回归分析适用于：计量资料(计量、分类、等级)方程左边 方程右边衡量回归方程的标准 o 复相关系数R o 校正复相关系数Radj o 剩余标准差总误差总回归SSSSSSSSR 12 总误差MSMSRpnnRadj 1111122pxxxys21 模拟数据X1X2X3X4YX1X2X3X4Y137261911.5166191410.2151140

32、3419.82410322619.8218291713.72211393825.31912153321.610717209.72711132722.3188342214.83210211519.12911282120.7178181611.71811163219.62610352319.41610153420.3146141810.6187231411.12813213425.52311292920.7199132918.72513414028.91210193819.3329121518.3238251715.63611371821.52811333224.7319251417.721918

33、1915.32913143828.33514243429.81810113521.6例3.2资料的一切可能回归(24-1=15个)2adjRpxxxys21 参数参数个数个数方程中变量方程中变量R2 Cp AIC2X10.365290.3441319.787412834.0097.45623X20.915120.912292.64619354.7433.07465X30.051890.0202929.557574247.00110.29764X40.586000.5722012.906691839.0083.782623X1,X20.920780.915322.55491331.2232.86

34、640X1,X30.375960.3329220.125702788.0098.91384X1,X40.993390.992930.213283.82-46.59486X2,X30.916010.910212.70887352.7434.73893X2,X40.922130.916762.51133325.1232.31589X3,X40.609070.5821112.607801737.0083.948024X1,X2,X30.921230.912792.63099331.1734.68250X1,X2,X40.993810.993140.993140.206890.206893.933.93-46.69119-46.69119X1,X3,X40.993600.992920.213694.85-45.65645X2,X3,X40.923480.915282.55590321.0333.755905X1,X2,X3,X40.994010.994010.993130.207425.00-45.77377