R语言--逐步回归分析 R语言--逐步回归分析

【R语言--逐步回归分析】逐步回归分析是以AIC信息统计量为准则，通过选择最小的AIC信息统计量，来达到删除或增加变量的目的。R语言中用于逐步回归分析的函数 step()，drop1()，add1()。
1.载入数据首先对数据进行多元线性回归分析

tdata<-data.frame( x1=c( 7, 1,11,11, 7,11, 3, 1, 2,21, 1,11,10), x2=c(26,29,56,31,52,55,71,31,54,47,40,66,68), x3=c( 6,15, 8, 8, 6, 9,17,22,18, 4,23, 9, 8), x4=c(60,52,20,47,33,22, 6,44,22,26,34,12,12), Y =c(78.5,74.3,104.3,87.6,95.9,109.2,102.7,72.5, 93.1,115.9,83.8,113.3,109.4) ) tlm<-lm(Y~x1+x2+x3+x4,data=https://www.it610.com/article/tdata) summary(tlm)

多元线性回归结果分析

Call: lm(formula = Y ~ x1 + x2 + x3 + x4, data = https://www.it610.com/article/tdata)Residuals: Min1QMedian3QMax -3.1750 -1.67090.25081.37833.9254 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)62.405470.07100.8910.3991 x11.55110.74482.0830.0708 . x20.51020.72380.7050.5009 x30.10190.75470.1350.8959 x4-0.14410.7091-0.2030.8441 --- Signif. codes:0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 2.446 on 8 degrees of freedom Multiple R-squared:0.9824,Adjusted R-squared:0.9736 F-statistic: 111.5 on 4 and 8 DF,p-value: 4.756e-07

通过观察，回归方程的系数都没有通过显著性检验
2.逐步回归分析###

tstep<-step(tlm) summary(tstep)

Start:AIC=26.94 Y ~ x1 + x2 + x3 + x4Df Sum of SqRSSAIC - x310.1091 47.973 24.974 - x410.2470 48.111 25.011 - x212.9725 50.836 25.728 47.864 26.944 - x1125.9509 73.815 30.576Step:AIC=24.97 Y ~ x1 + x2 + x4Df Sum of SqRSSAIC 47.97 24.974 - x419.9357.90 25.420 - x2126.7974.76 28.742 - x11820.91 868.88 60.629

结果分析：当用x1 x2 x3 x4作为回归方程的系数时，AIC的值为26.94
去掉x3 回归方程的AIC值为24.974；
去掉x4 回归方程的AIC值为25.011；
……
由于去x3可以使得AIC达到最小值，因此R会自动去掉x3;
去掉x3之后 AIC的值都增加逐步回归分析终止得到当前最优的回归方程

Call: lm(formula = Y ~ x1 + x2 + x4, data = https://www.it610.com/article/tdata)Residuals: Min1QMedian3QMax -3.0919 -1.80160.25621.28183.8982 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)71.648314.14245.066 0.000675 *** x11.45190.117012.410 5.78e-07 *** x20.41610.18562.242 0.051687 . x4-0.23650.1733-1.365 0.205395 --- Signif. codes:0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 2.309 on 9 degrees of freedom Multiple R-squared:0.9823,Adjusted R-squared:0.9764 F-statistic: 166.8 on 3 and 9 DF,p-value: 3.323e-08

回归系数的显著性水平有所提高但是x2 x4的显著性水平仍然不理想
3.逐步回归分析的优化

drop1(tstep)

结果分析：

Single term deletionsModel: Y ~ x1 + x2 + x4 Df Sum of SqRSSAIC 47.97 24.974 x11820.91 868.88 60.629 x2126.7974.76 28.742 x419.9357.90 25.420

如果去掉x4 AIC的值从24.974增加到25.420 是三个变量中增加最小的
4.进一步进行多元回归分析

tlm<-lm(Y~x1+x2,data=https://www.it610.com/article/tdata) summary(tlm)

结果分析：

Call: lm(formula = Y ~ x1 + x2, data = https://www.it610.com/article/tdata)Residuals: Min1Q Median3QMax -2.893 -1.574 -1.3021.3634.048 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 52.577352.2861723.00 5.46e-10 *** x11.468310.1213012.11 2.69e-07 *** x20.662250.0458514.44 5.03e-08 *** --- Signif. codes:0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 2.406 on 10 degrees of freedom Multiple R-squared:0.9787,Adjusted R-squared:0.9744 F-statistic: 229.5 on 2 and 10 DF,p-value: 4.407e-09

所有的检验均为显著。
因此所得回归方程为y=52.57735+ 1.46831x1+ 0.66225x2.