用最小二乘法求解線性模型及對模型的分析

1、用最小二乘法求解線性模型及對模型的分析作者:鄧春亮1、研究30名兒童體重為因變量與身高為自變量的關(guān)系，兒童體重與身高的記錄如下表：編號體重Y (kg)身高X(cm) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 22.60 21.50 19.10 21.80 21.50 20.10 18.80 22.00 21.30 24.00 23.30 22.50 22.90 19.50 22.90 22.30 22.70 23.50 21.50 25.50 25.00 26.10 27.9

2、0 26.80 27.20 24.40 24.40 23.00 26.30 28.80 119.80 121.70 121.40 124.40 120.00 117.00 118.00 118.80 124.20 124.80 124.70 123.10 125.30 124.20 127.40 128.20 126.10 128.60 129.40 126.90 126.50 128.20 131.40 130.80 133.90 130.40 131.30 130.20 136.00 138.00試用計算機(jī)完成下面統(tǒng)計分析：（1） 應(yīng)用最小二乘法求經(jīng)驗回歸方程；（2） 以擬合值為橫坐標(biāo)，殘

3、差為縱坐標(biāo)，作殘差圖，分析Gauss-Markou假設(shè)對本例的適用性；（3） 考慮因變量的變換,再對新變量和重復(fù)（1）和（2）的統(tǒng)計分析;（4） 將Box-Cox變換應(yīng)用到本例，計算變換參數(shù)的值，并做討論。說明：第一題的數(shù)據(jù)和結(jié)果文件見附件1，下面第二題的數(shù)據(jù)文件和結(jié)果文件見附件2，必要時可參看。解：（1）在SPSS窗口中錄入數(shù)據(jù)，首先進(jìn)行異常值檢測，探查對回歸估計有異常大影響的數(shù)據(jù)。先利用SPSS畫出體重與身高的散點(diǎn)圖圖1從圖1可以看出沒有明顯不一致的點(diǎn)。也可以通過SPSS軟件計算COOK統(tǒng)計量，看下表表1編號殘差學(xué)生化殘差centerCOOK統(tǒng)計量 1 2 3 4 5 6 7 8 9 10

4、 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1.88378 .03312 -2.24835 -.73361 .70477 .49003 -1.20506 1.67887 -1.15460 1.30835 .64786 .48000 .01081 -2.95460 -.81887 -1.73494 -.50526 -.69298 -3.00905 1.97867 1.63670 2.06506 2.60078 1.73783 .91306 -.50413 -.85971 -1.82512 -.81662 .89321.

5、 1.27241 .02204 -1.49944 -.48247 .47518 .34183 -.82971 1.14538 -.75974 .85964 .42576 .31710 .00710 -1.94417 -.53756 -1.14067 -.33146 -.45611 -1.98610 1.29824 1.07368 1.35771 1.73550 1.15517 .62283 -.33434 -.57329 -1.20912 -.57199 .64671. .05491 .02770 .03138 .00489 .05161 .11182 .08920 .07294 .00594

6、 .00310 .00351 .01355 .00143 .00594 .00139 .00434 .00008 .00643 .01183 .00038 .00003 .00434 .03249 .02522 .07268 .02088 .03121 .01887 .11878 .17316. .07835 .00002 .07778 .00463 .01048 .00992 .04807 .07800 .01180 .01397 .00347 .00247 .00000 .07726 .00520 .02547 .00190 .00431 .09329 .02940 .01989 .036

7、08 .10611 .04150 .02300 .00320 .01134 .04026 .02935 .05442.從上面數(shù)據(jù)看殘差值和中心化的杠桿率center的值沒有異常大的，數(shù)據(jù),這里(= center+1/n)， COOK統(tǒng)計量值也沒有異常大的數(shù)據(jù)，一般來說，殘差值和杠桿率越大，COOK統(tǒng)計量就越大，殘差值和杠桿率越小，COOK統(tǒng)計量就越小?？梢娺@些數(shù)據(jù)是比較一致的。接下來對這些數(shù)據(jù)求解經(jīng)驗回歸方程。然后利用最小二乘法，在SPSS中Analyze菜單下依次選擇Regress:2-Stage Least Square,選擇因變量和自變量執(zhí)行可輸出結(jié)果如下表：表2MODEL: MOD_3

8、.Equation number: 1Dependent variable. 體重YListwise Deletion of Missing DataMultiple R .80301R Square .64483Adjusted R Square .63215Standard Error 1.55047 Analysis of Variance: DF Sum of Squares Mean SquareRegression 1 122.20765 122.20765Residuals 28 67.31102 2.40396F = 50.83587 Signif F = .0000- Var

9、iables in the Equation -Variable B SE B Beta T Sig T身高X .395087 .055412 .803014 7.130 .0000(Constant) -26.615154 7.007449 -3.798 .0007Correlation Matrix of Parameter Estimates 身高X身高X 1.0000000這里可以看出所求經(jīng)驗回歸方程的常數(shù)項(Constant) 為-26.615154，身高X的系數(shù)為0.395087。故經(jīng)驗回歸方程為：=-26.615154+0.395087（2）通過SPSS,可得擬合值與殘差如下表表

10、3：擬合值與殘差表體重 Y 身高X擬合值 殘差 22.60 21.50 19.10 21.80 21.50 20.10 18.80 22.00 21.30 24.00 23.30 22.50 22.90 19.50 22.90 22.30 22.70 23.50 21.50 25.50 25.00 26.10 27.90 26.80 27.20 24.40 24.40 23.00 26.30 28.80119.80 121.70 121.40 124.40 120.00 117.00 118.00 118.80 124.20 124.80 124.70 123.10 125.30 124.20

11、 127.40 128.20 126.10 128.60 129.40 126.90 126.50 128.20 131.40 130.80 133.90 130.40 131.30 130.20 136.00 138.00 20.71622 21.46688 21.34835 22.53361 20.79523 19.60997 20.00506 20.32113 22.45460 22.69165 22.65214 22.02000 22.88919 22.45460 23.71887 24.03494 23.20526 24.19298 24.50905 23.52133 23.3633

12、0 24.03494 25.29922 25.06217 26.28694 24.90413 25.25971 24.82512 27.11662 27.90679 1.88378 .03312 -2.24835 -.73361 .70477 .49003 -1.20506 1.67887 -1.15460 1.30835 .64786 .48000 .01081 -2.95460 -.81887 -1.73494 -.50526 -.69298 -3.00905 1.97867 1.63670 2.06506 2.60078 1.73783 .91306 -.50413 -.85971 -1

13、.82512 -.81662 .89321以擬合值為橫坐標(biāo)，殘差為縱標(biāo)，得殘差圖圖2從圖中可以看出，殘差圖沒有明顯的不一致的征兆，則可以認(rèn)為Gauss-Markou假設(shè)對本例基本上是合理的。（3）作變換,這時用同樣的方法可求得經(jīng)驗回歸方程為：=-0.314471+0.040641其預(yù)測值與殘差如下表U擬合值殘差4.754.644.374.674.644.484.344.694.624.904.834.744.794.424.794.724.764.854.645.055.005.115.285.185.224.944.944.805.135.37.4.554264.631484.619294.

14、741214.562394.440474.481114.513624.733084.757474.753404.688384.777794.733084.863134.895654.810304.911904.944414.842814.826564.895655.025705.001315.127304.985055.021634.976935.212645.29392.4.554264.631484.619294.741214.562394.440474.481114.513624.733084.757474.753404.688384.777794.733084.863134.89565

15、4.810304.911904.944414.842814.826564.895655.025705.001315.127304.985055.021634.976935.212645.29392.以擬合值為橫坐標(biāo)，殘差為縱坐標(biāo)，作殘差圖得圖3從圖3看，此時的殘差圖也沒有明顯的不一致的趨勢，認(rèn)為Gauss-Markou假設(shè)對本例基本上是合理的。（4）將因變量進(jìn)行Box-Cox變換，變換后原來的因變量變?yōu)橛嬎悴煌祵?yīng)的殘差平方和，這里分別取值為-1.5，-1，-0.5，0，0.5，1，1.5，計算分別計算，然后計算對應(yīng)的殘差平方和，這里n=30,計算得到如表所示，這里表示。表5編號UZ1Z2Z

16、3Z4Z5Z6Z71234567891011121314151617181920212223242526272829304.754.644.374.674.644.484.344.694.624.904.834.744.794.424.794.724.764.854.645.055.005.115.285.185.224.944.944.805.135.37.1707.191705.941702.581706.301705.941704.111702.091706.531705.701708.571707.911707.081707.501703.221707.501706.861707.30

17、1708.101705.941709.851709.441710.301711.541710.811711.081708.931708.931707.611710.451712.08.513.21511.99508.85512.34511.99510.25508.40512.56511.76514.59513.92513.10513.52509.43513.52512.89513.31514.12511.99515.91515.49516.39517.72516.93517.23514.96514.96513.62516.55518.32.176.17174.98172.05175.31174

18、.98173.33171.64175.53174.76177.56176.88176.06176.48172.57176.48175.85176.27177.07174.98178.92178.48179.43180.86180.00180.32177.93177.93176.58179.59181.52.72.2571.0968.3571.4271.0969.5367.9871.6370.8873.6472.9672.1572.5668.8372.5671.9472.3573.1671.0975.0574.5975.5977.1376.2076.5474.0374.0372.6675.767

19、7.87.36.1435.0132.4535.3235.0133.5432.1235.5334.8137.5436.8436.0436.4432.8936.4435.8436.2437.0435.0138.9938.5139.5641.2340.2140.5837.9337.9336.5439.7542.04.21.6020.5018.1020.8020.5019.1017.8021.0020.3023.0022.3021.5021.9018.5021.9021.3021.7022.5020.5024.5024.0025.1026.9025.8026.2023.4023.4022.0025.3

20、027.80.14.7413.6711.4213.9613.6712.3411.1514.1513.4816.1415.4414.6415.0411.7915.0414.4514.8415.6413.6717.6917.1718.3320.2719.0819.5116.5516.5515.1418.5421.27.通過SPSS軟件運(yùn)行得到的方差分析表，可知道相應(yīng)的殘差平方和，具體數(shù)據(jù)如下表所示：表6-1.5-1-0.500.511.5RSS73.14370.51468.63867.49267.05167.31168.277通過表6的簡單比較可以看出當(dāng)時，殘差平方和達(dá)到最小，因此我們可以近似地認(rèn)

21、為0.5就是變換參數(shù)的最優(yōu)選擇.2、研究兒童的體重與身高和胸圍之間的關(guān)系是具有一定現(xiàn)實意義的，因為這種關(guān)系使我們能夠用簡單的方法從和的值去估計一個兒童的體重，下表是一組觀測數(shù)據(jù):表編號體重身高胸圍 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3022.6021.5019.1021.8021.5020.1018.8022.0021.3024.0023.3022.5022.9019.5022.9022.3022.7023.5021.5025.5025.0026.1027.9026.802

22、7.2024.4024.4023.0026.3028.80119.80121.70121.40124.40120.00117.00118.00118.80124.20124.80124.70123.10125.30124.20127.40128.20126.10128.60129.40126.90126.50128.20131.40130.80133.90130.40131.30130.20136.00138.0060.5055.5056.5060.5057.7057.0057.1061.7058.4060.8060.0060.0065.2053.7059.5060.1057.4060.405

23、2.0061.5063.9063.0063.1061.5065.8062.6059.5062.5060.0063.70試用計算機(jī)完成下面的統(tǒng)計分析：（1） 先假設(shè)Y與和有如下線性關(guān)系：，做最小二乘分析，并做相應(yīng)的殘差圖。試計算Box-Cox變換參數(shù)的值.（2） 對（1）中計算出的變換參數(shù)值，做相應(yīng)的Box-Cox變換，并對變換后的因變量做對和的最小二乘回歸，并做殘差圖。解：（1）先計算中心化杠桿率center和COOK統(tǒng)計量的值表2-1編號擬合值殘差學(xué)生化殘差centerCOOK統(tǒng)計量12345678910111213141516171819202122232425262728293021.5

24、170320.2753020.5475322.8902020.5637519.4149619.7496521.6526622.0707623.1181422.7988622.3212424.8592320.3703823.4239623.8798422.2761524.1077821.3076323.9982724.7471324.9290125.9204325.1624727.6435225.4410224.5881725.3451426.1720728.107691.082971.22470-1.44753-1.09020.93625.68504-.94965.34734-.77076.8

25、8186.50114.17876-1.95923-.87038-.52396-1.57984.42385-.60778.192371.50173.252871.170991.979571.63753-.44352-1.04102-.18817-2.34514.12793.69231.999761.12888-1.31902-.97103.85394.64630-.88474.33076-.68711.78544.44527.15985-1.85996-.83580-.46553-1.40437.38119-.54071.210881.33663.231521.054401.797031.471

26、37-.42282-.93760-.17089-2.10925.12316.67800.07509.07238.05156.00889.05330.11302.09125.12873.01058.00882.00418.01640.12356.14264.00413.00510.02725.00666.33436.00754.06029.02950.04463.02553.13059.02995.04540.02738.14686.17443.04052.05021.05380.01386.02305.02387.03713.00705.00723.00905.00258.00045.2145

27、9.04973.00281.02627.00312.00406.00862.02538.00185.02484.09102.04514.01168.01980.00083.09585.00111.04018從表中2-1的計算結(jié)果可以看出，第19個觀測的杠桿率最高為0.33436.。因此，第19個樣本點(diǎn)最有可能對模型擬合造成較大的影響。然后求解經(jīng)驗回歸方程，從運(yùn)行結(jié)果的方差分析表2-2（ANOVA(b)）可以看出F統(tǒng)計量的P-值（Sig.）為0.000，這表明模型在總體中是顯著的。表2-2 ANOVA(b)表2-3從回歸系數(shù)計算分析表2-3（Coefficients(a)），可知道回歸方程的常數(shù)

28、項為-36.133，自變量身高和胸圍相對應(yīng)的未標(biāo)準(zhǔn)化的回歸系數(shù)（Unstandardized Coefficients）分別為0.299、0.362，因而回歸方程為且從表中可知3個系數(shù)的t統(tǒng)計量的P值均為0.000，這表明模型在總體中是顯著的。以擬合值為橫坐標(biāo)，殘差為縱坐標(biāo)，作殘差圖：圖2-1 殘差圖從圖2-1可以看出，殘差圖從左至右逐漸散開呈漏斗狀，這是誤差方差不相等的征兆。考慮將因變量進(jìn)行Box-Cox變換，跟第一題的（4）問同樣。這里同樣分別取值為-1.5，-1，-0.5，0，0.5，1，1.5，計算分別計算，然后計算對應(yīng)的殘差平方和，這里n=30,計算得到如表1-5所示，然后計算對應(yīng)自變量和的殘差平方和。得方差分析表如下從上面的方差分析表中可以得到對應(yīng)的殘差平方和表2-4-1.5-1-0.500.511.5RSS37.90036.24835.20634.74734.85735.53436.787從這個表中可的簡單比較可以看出當(dāng)時，殘差平方和達(dá)到最小，而對應(yīng)的殘差平方和次之為34.857，且從的方差分析表可知它們對應(yīng)的P值都為0.000，都具有顯著性。現(xiàn)在再看和時,對應(yīng)因變量和對應(yīng)的回歸系數(shù)分析表。從上面兩個表可知，因變量為，