關于黑櫻桃樹的回歸分析問題(修改后)new

1、關于黑櫻桃樹的回歸分析問題在R的基本數據包dataset中，trees是由Girth, Height, Volume這三個變量的31個觀測組成的數據框，分別代表31棵被砍伐的黑櫻桃樹的離地面4英尺6處的直徑、高度、體積。下面是R統(tǒng)計軟件給出的信息：1. names(trees)1 Girth Height Volume2. summary(trees) Girth Height Volume Min. : 8.30 Min. :63 Min. :10.20 1st Qu.:11.05 1st Qu.:72 1st Qu.:19.40 Median :12.90 Median :76 Media

2、n :24.20 Mean :13.25 Mean :76 Mean :30.17 3rd Qu.:15.25 3rd Qu.:80 3rd Qu.:37.30 3. boxplot(trees) 4. cor(trees) Girth Height VolumeGirth 1.0000000 0.5192801 0.9671194Height 0.5192801 1.0000000 0.5982497Volume 0.9671194 0.5982497 1.0000000(1) summary(m1|t|) (Intercept) 62.0313 4.3833 14.152 1.49e-14

3、 *Girth 1.0544 0.3222 3.272 0.00276 * -Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 1 Residual standard error: 5.538 on 29 degrees of freedomMultiple R-squared: 0.2697, Adjusted R-squared: 0.2445 F-statistic: 10.71 on 1 and 29 DF, p-value: 0.0027585.(2) summary(m2|t|) (Intercept) -36.9435 3.3651 -10

4、.98 7.62e-12 *Girth 5.0659 0.2474 20.48 2e-16 *-Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 1 Residual standard error: 4.252 on 29 degrees of freedomMultiple R-squared: 0.9353, Adjusted R-squared: 0.9331 F-statistic: 419.4 on 1 and 29 DF, p-value: summary(m3|t|) (Intercept) -87.1236 29.2731 -2.976

5、0.005835 * Height 1.5433 0.3839 4.021 0.000378 *-Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 1 Residual standard error: 13.4 on 29 degrees of freedomMultiple R-squared: 0.3579, Adjusted R-squared: 0.3358 F-statistic: 16.16 on 1 and 29 DF, p-value: 0.0003784 由回歸分析的結果可得：雖然 VolumeHeight, HeightGirth之間

6、有顯著性關系，但是相關系數都不大，因此沒有進一步分析的必要。而VolumeGirth 相關系數r20.9所以可以繼續(xù)作為研究對象進行深入的分析。6.對于VolumeGirth：由5中可得，y=a1+a2*x,則a1的估計值為-36.9435，a2的估計值為5.0659，得到回歸方程為y=-36.9435+5.0659*x(y:Volume,x:Girth)plot(VolumeGirth,trees)abline(lm(VolumeGirth,trees) anova(m2)Analysis of Variance TableResponse: Volume Df Sum Sq Mean Sq

7、 F value Pr(F) Girth 1 7581.8 7581.8 419.36 r=residuals(m2)plot(r) plot(cooks.distance(m2)由殘差圖分析可得：第31個點有可能為強影響點，用程序將第31個點的殘差標出： x=matrix(c(8.3,10.3,8.6,10.3,8.8,10.2,10.5,16.4,10.7,18.8,10.8,19.7,11,15.6,11,18.2,11.1,22.6,11.2,19.9,11.3,24.2,11.4,21,11.4,21.4,11.7,21.3,12,19.1,12.9,22.2,12.9,33.8,1

8、3.3,27.4,13.7,25.7,13.8,24.9,14,34.5,14.2,31.7,14.5,36.3,16,38.3,16.3,42.6,17.3,55.4,17.5,55.7,17.9,58.3,18,51.5,18,51,20.6,77),nrow=31,ncol=2,byrow=T,dimnames=list(1:31,c(Girth,Volume) output=data.frame(x) m22=lm(VolumeGirth,output) summary(m22) r=residuals(m22) plot(r) text(31,r31,labels=31)用簡單的程序

9、將第31個點去掉： i=1:31 output2=data.frame(xi!=31,) m222=lm(VolumeGirth,output2) summary(m222)Call:lm(formula = Volume Girth, data = output2)Residuals: Min 1Q Median 3Q Max -7.5036 -2.3834 -0.0489 2.3951 6.3726 Coefficients: Estimate Std. Error t value Pr(|t|) (Intercept) -33.3104 3.2784 -10.16 6.76e-11 *G

10、irth 4.7619 0.2464 19.33 2e-16 *-Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 1 Residual standard error: 3.813 on 28 degrees of freedomMultiple R-squared: 0.9303, Adjusted R-squared: 0.9278 F-statistic: 373.6 on 1 and 28 DF, p-value: new=data.frame(Girth=22.0) lm.pred=predict(m2,new,interval=predict

11、ion,level=0.95) lm.pred fit lwr upr1 74.50538 64.62254 84.38822 lm.pred=predict(m2,new,interval=confidence,level=0.95) lm.pred fit lwr upr1 74.50538 69.81016 79.20069.添加變量： fm1=lm(VolumeGirth+Height,data=trees) summary(fm1)Call:lm(formula = Volume Girth + Height, data = trees)Residuals: Min 1Q Media

12、n 3Q Max -6.4065 -2.6493 -0.2876 2.2003 8.4847 Coefficients: Estimate Std. Error t value Pr(|t|) (Intercept) -57.9877 8.6382 -6.713 2.75e-07 *Girth 4.7082 0.2643 17.816 2e-16 *Height 0.3393 0.1302 2.607 0.0145 * -Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 1 Residual standard error: 3.882 on 28 deg

13、rees of freedomMultiple R-squared: 0.948, Adjusted R-squared: 0.9442 F-statistic: 255 on 2 and 28 DF, p-value: anova(fm1)Analysis of Variance TableResponse: Volume Df Sum Sq Mean Sq F value Pr(F) Girth 1 7581.8 7581.8 503.1503 hist(resid(fm1),freq=F,main=殘差的直方圖) fm1.new=update(fm1,.+Girth:Height) su

14、mmary(fm1.new)Call:lm(formula = Volume Girth + Height + Girth:Height, data = trees)Residuals: Min 1Q Median 3Q Max -6.5821 -1.0673 0.3026 1.5641 4.6649 Coefficients: Estimate Std. Error t value Pr(|t|) (Intercept) 69.39632 23.83575 2.911 0.00713 * Girth -5.85585 1.92134 -3.048 0.00511 * Height -1.29

15、708 0.30984 -4.186 0.00027 *Girth:Height 0.13465 0.02438 5.524 7.48e-06 *-Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 1 Residual standard error: 2.709 on 27 degrees of freedomMultiple R-squared: 0.9756, Adjusted R-squared: 0.9728 F-statistic: 359.3 on 3 and 27 DF, p-value: 2.2e-16 由新的fm1可以看出：Girth 與 Height 均對Volume有影響，且相關系數比僅用Girth 對Volume進行回歸更大，所以y=b0+b1*x1+b2*x2,b0的估計