生物统计学第3版杜荣骞 课后习题答案 多元回归及复相关分析

1、第十一章 多元回归及复相关分析11.1 嗜酸乳杆菌（Lactobacillus acidophilus Lakcid） 是存在于肠道中的一种重要益生菌，为研究肠道中的条件对该菌生存的影响，设计了在体外不同的胆汁盐浓度和不同时间该菌的存活数（活菌数/mL），结果如下表59：时间/h胆汁盐/（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以该菌的存活数为因变量，胆汁盐浓度和

3、时间为自变量，求二元回归方程并检验偏回归系数的显著性。答：程序和结果如下：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由以上结果得出回归方程：其中：X1为时间，X2为胆汁盐浓度。从偏回归系数的t检验结果可以得知，时间在0.05水平上显著，而胆汁盐浓度的显著性概率P <0.000 1。 11.2 10名浙江女大学士的身体体积、身高和体重的测量结果列在下表中77，以身高和体重为自变量，身体体积为因变量，计算二元回归方程，并检验偏回归系数的显著性。（注：对

6、于二元回归来说，只有10组观测值数量有些少，作为练习，姑且不去考虑样本的大小。）身体体积/m3身高/cm体重/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：程序不再给出，结果如下： 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由参数估计列可以得到回归方程：其中X1为身高，X2为体重，身高和体重的偏回归系数都极显著。11.3 社鼠头骨若干特征的度量值与年龄存在相关性，下表列出了40只社鼠的鉴定年龄（a）和头骨8个特征的度量值(mm)

9、78：序号鉴定年龄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：颅全长。X2：颅基长。X3：基底长。X4：颧宽。X5：眶间宽。X6：齿隙长。X7：上裂齿长。X8：门齿孔长。计算多元回归方程，复相关系数，并用逐步回归方法选出包含3个自变量的回归方程。答：（1）计算多元回归方程的程序和结果：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从参数估计列可以得到回归方程：复相关系数：（2）逐步回归分析：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引入方程中的三个变量没有剔除，最终保留在方程中的三个变量，在0.05水平上全都是显著的。方程如下：11.4 下

25、表给出了高山姬鼠头骨8个特征的测量值和鉴定年龄79，用逐步回归方法从8个特征中选出与鉴定年龄关系最密切的变量，并对结果做回归的方差分析。序号鉴定年龄/a头 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：颅全长。X2：颅基长。X3：基底长。X4：颧宽。 X5：眶间距。X6：齿隙长。X7：上裂齿长。X8：门齿孔长。答：结果如下： 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水平上筛选出三个变量，它们分别是：X1，X6和