生物统计学答案 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 SystemThe REG ProcedureModel: MODEL1Dependent Variable: numAnalysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FM

4、odel 2 9.070013E18 4.535006E18 27.66 <.0001Error 29 4.754238E18 1.639392E17Corrected Total 31 1.382425E19Root MSE 404894110 R-Square 0.6561Dependent Mean 524158580 Adj R-Sq 0.6324Coeff Var 77.24649Parameter EstimatesParameter StandardVariable DF Estimate Error t Value Pr > |t|Intercept 1 20204

5、93645 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：对于二元回归来说，只有10组观测值数量有

6、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 SystemThe REG ProcedureModel: MODEL1Dependent Variable: vAnalysis of V

7、arianceSum of MeanSource DF Squares Square F Value Pr > FModel 2 0.00023670 0.00011835 1553.36 <.0001Error 7 5.333339E-7 7.619056E-8Corrected Total 9 0.00023724Root MSE 0.00027603 R-Square 0.9978Dependent Mean 0.05153 Adj R-Sq 0.9971Coeff Var 0.53565Parameter EstimatesParameter StandardVariabl

8、e DF Estimate Error t Value Pr > |t|Intercept 1 -0.03651 0.00484 -7.54 0.0001h 1 0.00031062 0.00004217 7.37 0.0002w 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)78：序号鉴定年龄YX1X2X3X4X5X6X7X81334.6033.62

9、31.2616.105.448.746.126.742334.5033.4431.6815.924.829.005.826.483437.3636.3634.2817.465.489.966.086.724436.9435.8034.1017.145.289.805.466.625538.0037.7235.7417.465.149.925.846.686538.3037.4435.6417.085.1410.265.726.907539.7239.1836.7217.845.6010.505.766.628127.3426.4223.5013.464.707.594.505.129436.7

10、836.3634.5216.485.369.445.966.7810437.1236.1234.2416.445.149.525.906.3811334.7833.5631.4015.465.148.425.685.8812231.3830.8628.5614.545.087.825.786.0013436.5035.7233.4816.425.068.905.446.4014233.8032.9230.7016.885.088.245.666.0015232.2831.1428.5015.384.887.685.605.3816437.8837.0634.5416.605.669.925.5

11、26.8417232.7431.8229.5815.305.148.006.005.0818130.0028.5626.1813.924.987.125.105.1219233.2232.1029.6215.584.968.005.565.6620437.0836.9033.7817.385.729.606.046.6821335.3234.3232.1815.705.008.886.026.4622232.6631.0828.9215.344.767.805.725.4223232.6431.5029.4614.645.087.405.745.2024232.6831.5029.1814.9

12、44.767.865.825.6825130.9430.2027.7014.365.227.225.704.9226436.8435.9634.0417.025.369.086.166.0027537.5836.8834.4416.725.4610.005.606.3628537.8837.0634.5416.605.669.925.526.8429334.2833.3431.3016.645.189.225.586.4630335.8035.0032.7016.645.8210.005.686.0031334.1233.1031.1415.685.469.325.626.0032334.22

13、33.2631.6016.005.229.125.566.2833437.5436.8034.6216.445.2410.005.746.7034333.9433.3831.3616.845.088.725.706.2435334.0033.0230.5415.565.128.865.966.4236231.5430.4628.0415.204.927.785.465.6837538.1037.6234.8617.445.7210.166.147.1638230.5030.0027.9214.845.007.125.705.3039232.2630.8228.6215.304.947.825.

14、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 reg;model y=x1-x8;run;The SAS SystemThe

15、REG ProcedureModel: MODEL1Dependent Variable: yAnalysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FModel 8 53.17231 6.64654 64.33 <.0001Error 31 3.20269 0.10331Corrected Total 39 56.37500Root MSE 0.32142 R-Square 0.9432Dependent Mean 3.12500 Adj R-Sq 0.9285Coeff Var 10.28553P

16、arameter EstimatesParameter StandardVariable DF Estimate Error t Value Pr > |t|Intercept 1 -6.14927 1.68879 -3.64 0.0010x1 1 -0.22296 0.20853 -1.07 0.2932x2 1 0.56813 0.25038 2.27 0.0304x3 1 0.01771 0.19207 0.09 0.9271x4 1 -0.12007 0.12562 -0.96 0.3466x5 1 -0.39754 0.31415 -1.27 0.2151x6 1 0.2093

17、5 0.19346 1.08 0.2875x7 1 -0.34198 0.23671 -1.44 0.1586x8 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-x8/selection=stepwiseslentry=0.05 slstay=0.05;run;The SAS SystemThe RE

18、G ProcedureModel: MODEL1Dependent Variable: yStepwise Selection: Step 1Variable x2 Entered: R-Square = 0.9188 and C(p) = 8.2905Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FModel 1 51.79923 51.79923 430.17 <.0001Error 38 4.57577 0.12041Corrected Total 39 56.37500Paramet

19、er StandardVariable Estimate Error Type II SS F Value Pr > FIntercept -10.24579 0.64700 30.19713 250.78 <.0001x2 0.39483 0.01904 51.79923 430.17 <.0001Bounds on condition number: 1, 1-Stepwise Selection: Step 2Variable x7 Entered: R-Square = 0.9294 and C(p) = 4.5012Analysis of VarianceSum o

20、f MeanSource DF Squares Square F Value Pr > FModel 2 52.39734 26.19867 243.70 <.0001Error 37 3.97766 0.10750Corrected Total 39 56.37500Parameter StandardVariable Estimate Error Type II SS F Value Pr > FIntercept -8.33902 1.01352 7.27767 67.70 <.0001x2 0.41889 0.02068 44.11123 410.32 <

21、.0001x7 -0.47751 0.20245 0.59811 5.56 0.0237Bounds on condition number: 1.3218, 5.2873-Stepwise Selection: Step 3Variable x8 Entered: R-Square = 0.9369 and C(p) = 2.4570Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FModel 3 52.81516 17.60505 178.04 <.0001Error 36 3.55984

22、 0.09888Corrected Total 39 56.37500Parameter StandardVariable Estimate Error Type II SS F Value Pr > FIntercept -8.42672 0.97297 7.41726 75.01 <.0001x2 0.35766 0.03579 9.87513 99.87 <.0001x7 -0.45988 0.19435 0.55367 5.60 0.0235x8 0.33639 0.16365 0.41782 4.23 0.0471Bounds on condition number

23、: 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 themodel.Summary of Stepwise SelectionVariable Variable Number Partial ModelStep Entered Removed Vars In R-Square R-Square C(p) F Value Pr > F1 x

24、2 1 0.9188 0.9188 8.2905 430.17 <.00012 x7 2 0.0106 0.9294 4.5012 5.56 0.02373 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

25、.308.784.525.662328.7828.5626.7814.004.568.064.345.463328.0027.1225.0413.864.487.564.345.024226.6426.1624.5213.144.687.064.464.865226.0825.5023.7613.284.526.944.364.946429.4028.7027.8614.144.868.244.685.487124.8224.0422.0612.444.526.384.344.748226.5625.7423.7813.024.587.164.185.149227.1826.2624.4413

26、.064.747.344.205.2010226.4625.8224.1213.064.587.064.204.5011429.6228.8227.0413.524.448.284.345.4812530.1029.8828.2414.024.668.824.385.4613531.1830.6229.0614.604.868.864.825.9214327.5426.9225.3014.144.587.544.525.1615328.4027.9426.3013.844.467.844.545.6816328.1227.6425.9613.764.427.964.365.1417227.50

27、27.0025.3613.164.447.684.325.4418429.1828.3626.4614.704.707.864.605.4619530.3429.9228.2415.004.789.264.386.0420532.5032.0230.1415.345.148.964.786.1021531.2830.9629.0215.084.729.184.626.0022227.3826.8825.1413.384.587.244.425.2023124.4223.8822.1212.404.626.284.204.4624226.8826.2224.4413.344.627.564.16

28、5.0025227.5027.0025.3613.164.447.684.325.4426328.3427.6625.7813.824.887.764.525.6027328.5827.7225.7814.584.767.004.085.2428328.4828.0426.2813.784.767.804.345.6829328.8028.0826.3014.004.827.264.605.92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 SystemThe REG ProcedureMode

29、l: MODEL1Dependent Variable: yStepwise Selection: Step 1Variable x1 Entered: R-Square = 0.9111 and C(p) = 11.3797Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FModel 1 39.96265 39.96265 276.71 <.0001Error 27 3.89942 0.14442Corrected Total 28 43.86207Parameter StandardVar

30、iable Estimate Error Type II SS F Value Pr > FIntercept -14.87681 1.08114 27.34609 189.35 <.0001x1 0.63413 0.03812 39.96265 276.71 <.0001Bounds on condition number: 1, 1-Stepwise Selection: Step 2Variable x6 Entered: R-Square = 0.9259 and C(p) = 7.3289Analysis of VarianceSum of MeanSource D

31、F Squares Square F Value Pr > FModel 2 40.61122 20.30561 162.40 <.0001Error 26 3.25085 0.12503Corrected Total 28 43.86207Parameter StandardVariable Estimate Error Type II SS F Value Pr > FIntercept -13.31331 1.21786 14.94162 119.50 <.0001x1 0.44066 0.09205 2.86530 22.92 <.0001x6 0.503

32、25 0.22096 0.64857 5.19 0.0312Bounds on condition number: 6.7351, 26.941Stepwise Selection: Step 3Variable x8 Entered: R-Square = 0.9375 and C(p) = 4.5706Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FModel 3 41.12125 13.70708 125.03 <.0001Error 25 2.74082 0.10963Correct

33、ed Total 28 43.86207Parameter StandardVariable Estimate Error Type II SS F Value Pr > FIntercept -13.50669 1.14392 15.28437 139.41 <.0001x1 0.56648 0.10408 3.24772 29.62 <.0001x6 0.51347 0.20696 0.67482 6.16 0.0202x8 -0.64309 0.29816 0.51003 4.65 0.0408Bounds on condition number: 9.8194, 62

34、.516-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 themodel.Summary of Stepwise SelectionVariable Variable Number Partial ModelStep Entered Removed Vars In R-Square R-Square C(p) F Value Pr > F1 x1 1 0.9111 0

35、.9111 11.3797 276.71 <.00012 x6 2 0.0148 0.9259 7.3289 5.19 0.03123 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X8。回归方程为：方差分析表：变差来源平方和自由度均方FP回 归41.121 25313.707 08125.03<0.000 1误 差2.740 82250.109 63总 和43.862 072811.5 土壤根际微生物的生物量氮与季节变化有如下关联80：月份生物量氮/（10-4mg ·100g-1)56.

36、5767.4478.72810.68911.55109.15115.87124.42生物量氮与月份之间存在怎样的回归关系？求出回归方程。答：先绘出散点图，然后求回归方程。从散点图上可见，生物量氮与月份呈抛物线关系，应当用一元二次方程拟合。程序与结果如下：options linesize=76 nodate;data stepreg;infile 'e:dataer11-5e.dat'input x1 y;x2=x1*2;run;proc reg;model y=x1 x2; run;The SAS SystemThe REG ProcedureModel: MODEL1Depe

37、ndent Variable: yAnalysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FModel 2 35.61535 17.80767 15.61 0.0071Error 5 5.70225 1.14045Corrected Total 7 41.31760Root MSE 1.06792 R-Square 0.8620Dependent Mean 8.05000 Adj R-Sq 0.8068Coeff Var 13.26607Parameter EstimatesParameter Standa

38、rdVariable DF Estimate Error t Value Pr > |t|Intercept 1 -19.57060 5.70767 -3.43 0.0187x1 1 7.29381 1.41032 5.17 0.0035x2 1 -0.44357 0.08239 -5.38 0.0030回归方程为：一次项和二次项的回归系数都是极显著的。11.6 两种农药“呋喃丹”和“铁灭克”，在不同 pH条件下对土壤磷酸酶活性（mg/g）的影响如下表所示81：缓冲液pH 呋喃丹(Y1)铁灭克(Y2)7.90.190.108.31.370.798.71.311.099.11.651.21

39、9.31.491.299.61.120.8710.01.070.7810.50.310.2211.00.120.10分别绘出呋喃丹和铁灭克对pH的散点图，计算出回归方程并求出磷酸酶活性达到最大值时的pH值，以及在该pH时磷酸酶的活性值。答：计算程序与上题一样，不再给出，只给出结果。（1）呋喃丹：The SAS SystemThe REG ProcedureModel: MODEL1Dependent Variable: y1Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FModel 2 2.258

40、59 1.12929 12.37 0.0074Error 6 0.54770 0.09128Corrected Total 8 2.80629Root MSE 0.30213 R-Square 0.8048Dependent Mean 0.95889 Adj R-Sq 0.7398Coeff Var 31.50847Parameter EstimatesParameter StandardVariable DF Estimate Error t Value Pr > |t|Intercept 1 -41.71019 9.89391 -4.22 0.0056x1 1 9.34395 2.1

41、0852 4.43 0.0044x2 1 -0.50595 0.11147 -4.54 0.0039回归方程为：一次项和二次项的回归系数都是极显著的。最大值的计算：1.011 9 X9.343 95 X9.234 06 Y1.431 13故当pH9.234 06时磷酸酶活性有最大值，其最大值为1.431 13。（2）铁灭克：The SAS SystemThe REG ProcedureModel: MODEL2Dependent Variable: y2Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr >

42、 FModel 2 1.46564 0.73282 15.38 0.0044Error 6 0.28596 0.04766Corrected Total 8 1.75160Root MSE 0.21831 R-Square 0.8367Dependent Mean 0.71667 Adj R-Sq 0.7823Coeff Var 30.46194Parameter EstimatesParameter StandardVariable DF Estimate Error t Value Pr > |t|Intercept 1 -35.50332 7.14903 -4.97 0.0025x

43、1 1 7.88157 1.52355 5.17 0.0021x2 1 -0.42419 0.08054 -5.27 0.0019回归方程为：一次项和二次项的回归系数都是极显著的。最大值的计算：0.848 38 X7.881 57 X9.290 14 Y1.107 13故当pH9.290 14时磷酸酶活性有最大值，其最大值为1.107 13。11.7 “武运粳7号”考种相关数据见下表82：序号产量/（kg ·hm-2）千粒重/g每穗总粒数/粒亩有效穗/（104·hm-2）株高/cm19 787.525.9125.7372.30102.529 390.025.8131.336

44、3.75105.639 607.526.3122.5370.8099.349 547.525.9128.3377.7098.959 237.026.5127.8358.65103.568 947.525.8137.5340.05100.378 277.525.7118.2372.9098.888 475.526.2113.6373.9597.698 415.025.9118.9373.0597.3108 040.025.4118.5356.7095.3118 167.526.1121.3333.6095.6127 845.025.3124.7345.7595.1137 927.525.8121

45、.6343.5094.7147 327.525.6112.5343.2094.5157 305.025.9103.8362.4093.6167 125.025.4123.1319.2092.5177 140.026.1113.8308.5589.6186 945.026.4111.5306.4590.5以产量为因变量，计算多元回归方程，通过逐步回归筛选出对产量影响的重要因素。答：（1）多元回归方程见下表：The SAS SystemThe REG ProcedureModel: MODEL1Dependent Variable: yAnalysis of VarianceSum of Mean

46、Source DF Squares Square F Value Pr > FModel 4 14038414 3509603 50.79 <.0001Error 13 898239 69095Corrected Total 17 14936652Root MSE 262.85981 R-Square 0.9399Dependent Mean 8305.97222 Adj R-Sq 0.9214Coeff Var 3.16471Parameter EstimatesParameter StandardVariable DF Estimate Error t Value Pr >

47、; |t|Intercept 1 -31245 5522.29553 -5.66 <.0001x1 1 839.98368 215.86358 3.89 0.0019x2 1 65.89770 14.60597 4.51 0.0006x3 1 23.22349 5.13090 4.53 0.0006x4 1 17.38756 37.83632 0.46 0.6534从参数估计列可以得出回归方程：（2）用逐步回归方法筛选最优回归方程： 首先以sle0.25和sls0.25显著水平进行筛选，结果见下表：The SAS SystemThe REG ProcedureModel: MODEL1D

48、ependent Variable: yStepwise Selection: Step 1Variable x4 Entered: R-Square = 0.8102 and C(p) = 27.0270Analysis of VarianceSum of MeanSource DF Squares Square F Value Pr > FModel 1 12101880 12101880 68.31 <.0001Error 16 2834772 177173Corrected Total 17 14936652Parameter StandardVariable Estimate Error Type II SS F Value Pr > FIntercep