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=tdata)
summary(tlm)

多元线性回归结果分析

Call:
lm(formula = Y ~ x1 + x2 + x3 + x4, data = tdata)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.1750 -1.6709  0.2508  1.3783  3.9254 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)  62.4054    70.0710   0.891   0.3991  
x1            1.5511     0.7448   2.083   0.0708 .
x2            0.5102     0.7238   0.705   0.5009  
x3            0.1019     0.7547   0.135   0.8959  
x4           -0.1441     0.7091  -0.203   0.8441  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual 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 + x4

       Df Sum of Sq    RSS    AIC
- x3    1    0.1091 47.973 24.974
- x4    1    0.2470 48.111 25.011
- x2    1    2.9725 50.836 25.728
<none>              47.864 26.944
- x1    1   25.9509 73.815 30.576

Step:  AIC=24.97
Y ~ x1 + x2 + x4

       Df Sum of Sq    RSS    AIC
<none>               47.97 24.974
- x4    1      9.93  57.90 25.420
- x2    1     26.79  74.76 28.742
- x1    1    820.91 868.88 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 = tdata)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.0919 -1.8016  0.2562  1.2818  3.8982 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  71.6483    14.1424   5.066 0.000675 ***
x1            1.4519     0.1170  12.410 5.78e-07 ***
x2            0.4161     0.1856   2.242 0.051687 .  
x4           -0.2365     0.1733  -1.365 0.205395    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual 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 deletions

Model:
Y ~ x1 + x2 + x4
       Df Sum of Sq    RSS    AIC
<none>               47.97 24.974
x1      1    820.91 868.88 60.629
x2      1     26.79  74.76 28.742
x4      1      9.93  57.90 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=tdata)
summary(tlm)

结果分析：

Call:
lm(formula = Y ~ x1 + x2, data = tdata)

Residuals:
   Min     1Q Median     3Q    Max 
-2.893 -1.574 -1.302  1.363  4.048 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 52.57735    2.28617   23.00 5.46e-10 ***
x1           1.46831    0.12130   12.11 2.69e-07 ***
x2           0.66225    0.04585   14.44 5.03e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual 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.