生物統(tǒng)計學第3版杜榮騫 課后習題答案 多元回歸及復相關分析

1、第十一章 多元回歸及復相關分析11.1 嗜酸乳桿菌（Lactobacillus acidophilus Lakcid） 是存在于腸道中的一種重要益生菌，為研究腸道中的條件對該菌生存的影響，設計了在體外不同的膽汁鹽濃度和不同時間該菌的存活數(shù)（活菌數(shù)/mL），結果如下表59：時間/h膽汁鹽/（g ·kg-1）123417.20×1081.04×1091.76×1092.04×1096.40×1068.40×1062.62×1031.74×10321.64×1091.92×1099.60&#

2、215;1087.40×1081.22×1079.20×1062.09×1031.89×10331.30×1091.42×1093.46×1086.00×1082.26×1062.04×1061.86×1031.82×10349.80×1087.80×1081.02×1083.82×1081.30×1061.26×1061.32×1031.22×103以該菌的存活數(shù)為因變量，膽汁鹽濃度和

3、時間為自變量，求二元回歸方程并檢驗偏回歸系數(shù)的顯著性。答：程序和結果如下：options linesize=76 nodate;data mulreg; infile e:dataer11-1e.dat; input num time bile ;run;proc reg; model num=time bile;run; The SAS System The REG Procedure Model: MODEL1 Dependent Variable: num Analysis of Variance Sum of Mean Source DF Squares Square F Value

4、Pr > F Model 2 9.070013E18 4.535006E18 27.66 <.0001 Error 29 4.754238E18 1.639392E17 Corrected Total 31 1.382425E19 Root MSE 404894110 R-Square 0.6561 Dependent Mean 524158580 Adj R-Sq 0.6324 Coeff Var 77.24649 Parameter Estimates Parameter StandardVariable DF Estimate Error t Value Pr > |t

5、|Intercept 1 2020493645 237390215 8.51 <.0001time 1 -144947822 64019380 -2.26 0.0312bile 1 -453586204 64019380 -7.09 <.0001由以上結果得出回歸方程：其中：X1為時間，X2為膽汁鹽濃度。從偏回歸系數(shù)的t檢驗結果可以得知，時間在0.05水平上顯著，而膽汁鹽濃度的顯著性概率P <0.000 1。 11.2 10名浙江女大學士的身體體積、身高和體重的測量結果列在下表中77，以身高和體重為自變量，身體體積為因變量，計算二元回歸方程，并檢驗偏回歸系數(shù)的顯著性。（注：對

6、于二元回歸來說，只有10組觀測值數(shù)量有些少，作為練習，姑且不去考慮樣本的大小。）身體體積/m3身高/cm體重/kg0.055 29165.055.00.043 24151.845.00.051 74159.053.50.054 58164.055.00.049 62158.550.50.046 07155.047.00.053 87158.356.00.052 45161.553.50.047 49157.548.00.060 96169.062.0答：程序不再給出，結果如下： The SAS System The REG Procedure Model: MODEL1 Dependent V

7、ariable: v Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 0.00023670 0.00011835 1553.36 <.0001 Error 7 5.333339E-7 7.619056E-8 Corrected Total 9 0.00023724 Root MSE 0.00027603 R-Square 0.9978 Dependent Mean 0.05153 Adj R-Sq 0.9971 Coeff Var 0.53565 Parameter E

8、stimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 -0.03651 0.00484 -7.54 0.0001 h 1 0.00031062 0.00004217 7.37 0.0002 w 1 0.00072984 0.00004228 17.26 <.0001由參數(shù)估計列可以得到回歸方程：其中X1為身高，X2為體重，身高和體重的偏回歸系數(shù)都極顯著。11.3 社鼠頭骨若干特征的度量值與年齡存在相關性，下表列出了40只社鼠的鑒定年齡（a）和頭骨8個特征的度量值(mm)

9、78：序號鑒定年齡YX1X2X3X4X5X6X7X81334.6033.6231.2616.105.448.746.126.742334.5033.4431.6815.924.829.005.826.483437.3636.3634.2817.465.489.966.086.724436.9435.8034.1017.145.289.805.466.625538.0037.7235.7417.465.149.925.846.686538.3037.4435.6417.085.1410.265.726.907539.7239.1836.7217.845.6010.505.766.628127.3

10、426.4223.5013.464.707.594.505.129436.7836.3634.5216.485.369.445.966.7810437.1236.1234.2416.445.149.525.906.3811334.7833.5631.4015.465.148.425.685.8812231.3830.8628.5614.545.087.825.786.0013436.5035.7233.4816.425.068.905.446.4014233.8032.9230.7016.885.088.245.666.0015232.2831.1428.5015.384.887.685.60

11、5.3816437.8837.0634.5416.605.669.925.526.8417232.7431.8229.5815.305.148.006.005.0818130.0028.5626.1813.924.987.125.105.1219233.2232.1029.6215.584.968.005.565.6620437.0836.9033.7817.385.729.606.046.6821335.3234.3232.1815.705.008.886.026.4622232.6631.0828.9215.344.767.805.725.4223232.6431.5029.4614.64

12、5.087.405.745.2024232.6831.5029.1814.944.767.865.825.6825130.9430.2027.7014.365.227.225.704.9226436.8435.9634.0417.025.369.086.166.0027537.5836.8834.4416.725.4610.005.606.3628537.8837.0634.5416.605.669.925.526.8429334.2833.3431.3016.645.189.225.586.4630335.8035.0032.7016.645.8210.005.686.0031334.123

13、3.1031.1415.685.469.325.626.0032334.2233.2631.6016.005.229.125.566.2833437.5436.8034.6216.445.2410.005.746.7034333.9433.3831.3616.845.088.725.706.2435334.0033.0230.5415.565.128.865.966.4236231.5430.4628.0415.204.927.785.465.6837538.1037.6234.8617.445.7210.166.147.1638230.5030.0027.9214.845.007.125.7

14、05.3039232.2630.8228.6215.304.947.825.505.4640437.3836.2034.2216.905.309.445.546.42 注： X1：顱全長。X2：顱基長。X3：基底長。X4：顴寬。X5：眶間寬。X6：齒隙長。X7：上裂齒長。X8：門齒孔長。計算多元回歸方程，復相關系數(shù)，并用逐步回歸方法選出包含3個自變量的回歸方程。答：（1）計算多元回歸方程的程序和結果：options linesize=76 nodate;data mulreg; infile 'e:dataer11-3e.dat' input y x1-x8 ;run;proc

15、 reg; model y=x1-x8;run; The SAS System The REG Procedure Model: MODEL1 Dependent Variable: y Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 8 53.17231 6.64654 64.33 <.0001 Error 31 3.20269 0.10331 Corrected Total 39 56.37500 Root MSE 0.32142 R-Square 0.9432 Dep

16、endent Mean 3.12500 Adj R-Sq 0.9285 Coeff Var 10.28553 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 -6.14927 1.68879 -3.64 0.0010 x1 1 -0.22296 0.20853 -1.07 0.2932 x2 1 0.56813 0.25038 2.27 0.0304 x3 1 0.01771 0.19207 0.09 0.9271 x4 1 -0.12007 0.

17、12562 -0.96 0.3466 x5 1 -0.39754 0.31415 -1.27 0.2151 x6 1 0.20935 0.19346 1.08 0.2875 x7 1 -0.34198 0.23671 -1.44 0.1586 x8 1 0.21464 0.20076 1.07 0.2932從參數(shù)估計列可以得到回歸方程：復相關系數(shù)：（2）逐步回歸分析：options linesize=76 nodate;data stepreg; infile 'e:dataer11-3e.dat' input y x1-x8;run;proc reg; model y=x1-

18、x8/selection=stepwise slentry=0.05 slstay=0.05;run; The SAS System The REG Procedure Model: MODEL1 Dependent Variable: y Stepwise Selection: Step 1 Variable x2 Entered: R-Square = 0.9188 and C(p) = 8.2905 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 51.79923 51

19、.79923 430.17 <.0001 Error 38 4.57577 0.12041 Corrected Total 39 56.37500 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -10.24579 0.64700 30.19713 250.78 <.0001 x2 0.39483 0.01904 51.79923 430.17 <.0001 Bounds on condition number: 1, 1- Stepwise Selection

20、: Step 2 Variable x7 Entered: R-Square = 0.9294 and C(p) = 4.5012 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 52.39734 26.19867 243.70 <.0001 Error 37 3.97766 0.10750 Corrected Total 39 56.37500 Parameter Standard Variable Estimate Error Type II SS F Value

21、Pr > F Intercept -8.33902 1.01352 7.27767 67.70 <.0001 x2 0.41889 0.02068 44.11123 410.32 <.0001 x7 -0.47751 0.20245 0.59811 5.56 0.0237 Bounds on condition number: 1.3218, 5.2873- Stepwise Selection: Step 3 Variable x8 Entered: R-Square = 0.9369 and C(p) = 2.4570 Analysis of Variance Sum o

22、f Mean Source DF Squares Square F Value Pr > F Model 3 52.81516 17.60505 178.04 <.0001 Error 36 3.55984 0.09888 Corrected Total 39 56.37500 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -8.42672 0.97297 7.41726 75.01 <.0001 x2 0.35766 0.03579 9.87513 99.8

23、7 <.0001 x7 -0.45988 0.19435 0.55367 5.60 0.0235 x8 0.33639 0.16365 0.41782 4.23 0.0471 Bounds on condition number: 4.3043, 28.581- All variables left in the model are significant at the 0.0500 level. No other variable met the 0.0500 significance level for entry into the model. Summary of Stepwis

24、e Selection Variable Variable Number Partial Model Step Entered Removed Vars In R-Square R-Square C(p) F Value Pr > F 1 x2 1 0.9188 0.9188 8.2905 430.17 <.0001 2 x7 2 0.0106 0.9294 4.5012 5.56 0.0237 3 x8 3 0.0074 0.9369 2.4570 4.23 0.0471引入方程中的三個變量沒有剔除，最終保留在方程中的三個變量，在0.05水平上全都是顯著的。方程如下：11.4 下

25、表給出了高山姬鼠頭骨8個特征的測量值和鑒定年齡79，用逐步回歸方法從8個特征中選出與鑒定年齡關系最密切的變量，并對結果做回歸的方差分析。序號鑒定年齡/a頭 骨 特 征 /mmX1X2X3X4X5X6X7X81530.6430.0028.3414.324.308.784.525.662328.7828.5626.7814.004.568.064.345.463328.0027.1225.0413.864.487.564.345.024226.6426.1624.5213.144.687.064.464.865226.0825.5023.7613.284.526.944.364.946429.40

26、28.7027.8614.144.868.244.685.487124.8224.0422.0612.444.526.384.344.748226.5625.7423.7813.024.587.164.185.149227.1826.2624.4413.064.747.344.205.2010226.4625.8224.1213.064.587.064.204.5011429.6228.8227.0413.524.448.284.345.4812530.1029.8828.2414.024.668.824.385.4613531.1830.6229.0614.604.868.864.825.9

27、214327.5426.9225.3014.144.587.544.525.1615328.4027.9426.3013.844.467.844.545.6816328.1227.6425.9613.764.427.964.365.1417227.5027.0025.3613.164.447.684.325.4418429.1828.3626.4614.704.707.864.605.4619530.3429.9228.2415.004.789.264.386.0420532.5032.0230.1415.345.148.964.786.1021531.2830.9629.0215.084.7

28、29.184.626.0022227.3826.8825.1413.384.587.244.425.2023124.4223.8822.1212.404.626.284.204.4624226.8826.2224.4413.344.627.564.165.0025227.5027.0025.3613.164.447.684.325.4426328.3427.6625.7813.824.887.764.525.6027328.5827.7225.7814.584.767.004.085.2428328.4828.0426.2813.784.767.804.345.6829328.8028.082

29、6.3014.004.827.264.605.92 注：X1：顱全長。X2：顱基長。X3：基底長。X4：顴寬。 X5：眶間距。X6：齒隙長。X7：上裂齒長。X8：門齒孔長。答：結果如下： The SAS System The REG Procedure Model: MODEL1 Dependent Variable: y Stepwise Selection: Step 1 Variable x1 Entered: R-Square = 0.9111 and C(p) = 11.3797 Analysis of Variance Sum of Mean Source DF Squares S

30、quare F Value Pr > F Model 1 39.96265 39.96265 276.71 <.0001 Error 27 3.89942 0.14442 Corrected Total 28 43.86207 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -14.87681 1.08114 27.34609 189.35 <.0001 x1 0.63413 0.03812 39.96265 276.71 <.0001 Bounds on

31、 condition number: 1, 1- Stepwise Selection: Step 2 Variable x6 Entered: R-Square = 0.9259 and C(p) = 7.3289 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 40.61122 20.30561 162.40 <.0001 Error 26 3.25085 0.12503 Corrected Total 28 43.86207 Parameter Standard

32、Variable Estimate Error Type II SS F Value Pr > F Intercept -13.31331 1.21786 14.94162 119.50 <.0001 x1 0.44066 0.09205 2.86530 22.92 <.0001 x6 0.50325 0.22096 0.64857 5.19 0.0312 Bounds on condition number: 6.7351, 26.941 Stepwise Selection: Step 3 Variable x8 Entered: R-Square = 0.9375 an

33、d C(p) = 4.5706 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 41.12125 13.70708 125.03 <.0001 Error 25 2.74082 0.10963 Corrected Total 28 43.86207 Parameter Standard Variable Estimate Error Type II SS F Value Pr > F Intercept -13.50669 1.14392 15.28437 139

34、.41 <.0001 x1 0.56648 0.10408 3.24772 29.62 <.0001 x6 0.51347 0.20696 0.67482 6.16 0.0202 x8 -0.64309 0.29816 0.51003 4.65 0.0408 Bounds on condition number: 9.8194, 62.516- All variables left in the model are significant at the 0.0500 level. No other variable met the 0.0500 significance level

35、 for entry into the model. Summary of Stepwise Selection Variable Variable Number Partial Model Step Entered Removed Vars In R-Square R-Square C(p) F Value Pr > F 1 x1 1 0.9111 0.9111 11.3797 276.71 <.0001 2 x6 2 0.0148 0.9259 7.3289 5.19 0.0312 3 x8 3 0.0116 0.9375 4.5706 4.65 0.0408在0.05水平上篩選出三個變量，它們分別是：X1，X6和