1、第十一章 多元回歸及復(fù)相關(guān)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(shè)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(jié)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(jié)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由以上結(jié)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(jié)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(xué)士的身體體積、身高和體重的測量結(jié)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：對于二元回歸來說，只有10組觀測值數(shù)量有

6、些少，作為練習(xí)，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(jié)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(guān)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)78：序號鑒定年齡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：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(fù)相關(guān)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(jié)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從參數(shù)估計列可以得到回歸方程：復(fù)相關(guān)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-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(guān)系最密切的變量，并對結(jié)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

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：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(jié)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水平上篩選出三個變量，它們分別是：X1，X6和X8。回歸方程為：方差分析表：變差來源平方和自由度均方FP回 歸41.121 25313.707 08125.03<0.000 1誤 差2.740 82250.109 63總 和43.862 072811.5 土壤根際微生物的生物量氮與季節(jié)變化有如下關(guān)聯(lián)80：月份生物量氮/（10-4mg ·100g-1)56.

36、5767.4478.72810.68911.55109.15115.87124.42生物量氮與月份之間存在怎樣的回歸關(guān)系？求出回歸方程。答：先繪出散點圖，然后求回歸方程。從散點圖上可見，生物量氮與月份呈拋物線關(guān)系，應(yīng)當(dāng)用一元二次方程擬合。程序與結(jié)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回歸方程為：一次項和二次項的回歸系數(shù)都是極顯著的。11.6 兩種農(nóng)藥“呋喃丹”和“鐵滅克”，在不同 pH條件下對土壤磷酸酶活性（mg/g）的影響如下表所示81：緩沖液pH 呋喃丹(Y1)鐵滅克(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分別繪出呋喃丹和鐵滅克對pH的散點圖，計算出回歸方程并求出磷酸酶活性達到最大值時的pH值，以及在該pH時磷酸酶的活性值。答：計算程序與上題一樣，不再給出，只給出結(jié)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回歸方程為：一次項和二次項的回歸系數(shù)都是極顯著的。最大值的計算：1.011 9 X9.343 95 X9.234 06 Y1.431 13故當(dāng)pH9.234 06時磷酸酶活性有最大值，其最大值為1.431 13。（2）鐵滅克：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回歸方程為：一次項和二次項的回歸系數(shù)都是極顯著的。最大值的計算：0.848 38 X7.881 57 X9.290 14 Y1.107 13故當(dāng)pH9.290 14時磷酸酶活性有最大值，其最大值為1.107 13。11.7 “武運粳7號”考種相關(guān)數(shù)據(jù)見下表82：序號產(chǎn)量/（kg ·hm-2）千粒重/g每穗總粒數(shù)/粒畝有效穗/（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以產(chǎn)量為因變量，計算多元回歸方程，通過逐步回歸篩選出對產(chǎn)量影響的重要因素。答：（1）多元回歸方程見下表：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從參數(shù)估計列可以得出回歸方程：（2）用逐步回歸方法篩選最優(yōu)回歸方程： 首先以sle0.25和sls0.25顯著水平進行篩選，結(jié)果見下表：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